src/HOL/Unix/Unix.thy
author wenzelm
Tue Nov 14 22:17:04 2006 +0100 (2006-11-14)
changeset 21372 4a0a83378669
parent 21226 a607ae87ee81
child 21404 eb85850d3eb7
permissions -rw-r--r--
converted to 'inductive2';
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(*  Title:      HOL/Unix/Unix.thy
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    ID:         $Id$
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    Author:     Markus Wenzel, TU Muenchen
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*)
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header {* Unix file-systems \label{sec:unix-file-system} *}
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theory Unix
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imports Nested_Environment List_Prefix
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begin
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text {*
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  We give a simple mathematical model of the basic structures
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  underlying the Unix file-system, together with a few fundamental
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  operations that could be imagined to be performed internally by the
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  Unix kernel.  This forms the basis for the set of Unix system-calls
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  to be introduced later (see \secref{sec:unix-trans}), which are the
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  actual interface offered to processes running in user-space.
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  \medskip Basically, any Unix file is either a \emph{plain file} or a
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  \emph{directory}, consisting of some \emph{content} plus
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  \emph{attributes}.  The content of a plain file is plain text.  The
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  content of a directory is a mapping from names to further
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  files.\footnote{In fact, this is the only way that names get
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  associated with files.  In Unix files do \emph{not} have a name in
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  itself.  Even more, any number of names may be associated with the
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  very same file due to \emph{hard links} (although this is excluded
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  from our model).}  Attributes include information to control various
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  ways to access the file (read, write etc.).
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  Our model will be quite liberal in omitting excessive detail that is
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  easily seen to be ``irrelevant'' for the aspects of Unix
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  file-systems to be discussed here.  First of all, we ignore
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  character and block special files, pipes, sockets, hard links,
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  symbolic links, and mount points.
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*}
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subsection {* Names *}
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text {*
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  User ids and file name components shall be represented by natural
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  numbers (without loss of generality).  We do not bother about
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  encoding of actual names (e.g.\ strings), nor a mapping between user
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  names and user ids as would be present in a reality.
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*}
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types
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  uid = nat
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  name = nat
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  path = "name list"
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subsection {* Attributes *}
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text {*
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  Unix file attributes mainly consist of \emph{owner} information and
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  a number of \emph{permission} bits which control access for
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  ``user'', ``group'', and ``others'' (see the Unix man pages @{text
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  "chmod(2)"} and @{text "stat(2)"} for more details).
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  \medskip Our model of file permissions only considers the ``others''
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  part.  The ``user'' field may be omitted without loss of overall
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  generality, since the owner is usually able to change it anyway by
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  performing @{text chmod}.\footnote{The inclined Unix expert may try
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  to figure out some exotic arrangements of a real-world Unix
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  file-system such that the owner of a file is unable to apply the
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  @{text chmod} system call.}  We omit ``group'' permissions as a
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  genuine simplification as we just do not intend to discuss a model
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  of multiple groups and group membership, but pretend that everyone
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  is member of a single global group.\footnote{A general HOL model of
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  user group structures and related issues is given in
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  \cite{Naraschewski:2001}.}
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*}
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datatype perm =
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    Readable
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  | Writable
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  | Executable    -- "(ignored)"
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types perms = "perm set"
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record att =
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  owner :: uid
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  others :: perms
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text {*
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  For plain files @{term Readable} and @{term Writable} specify read
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  and write access to the actual content, i.e.\ the string of text
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  stored here.  For directories @{term Readable} determines if the set
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  of entry names may be accessed, and @{term Writable} controls the
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  ability to create or delete any entries (both plain files or
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  sub-directories).
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  As another simplification, we ignore the @{term Executable}
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  permission altogether.  In reality it would indicate executable
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  plain files (also known as ``binaries''), or control actual lookup
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  of directory entries (recall that mere directory browsing is
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  controlled via @{term Readable}).  Note that the latter means that
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  in order to perform any file-system operation whatsoever, all
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  directories encountered on the path would have to grant @{term
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  Executable}.  We ignore this detail and pretend that all directories
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  give @{term Executable} permission to anybody.
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*}
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subsection {* Files *}
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text {*
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  In order to model the general tree structure of a Unix file-system
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  we use the arbitrarily branching datatype @{typ "('a, 'b, 'c) env"}
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  from the standard library of Isabelle/HOL
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  \cite{Bauer-et-al:2002:HOL-Library}.  This type provides
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  constructors @{term Val} and @{term Env} as follows:
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  \medskip
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  {\def\isastyleminor{\isastyle}
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  \begin{tabular}{l}
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  @{term [source] "Val :: 'a \<Rightarrow> ('a, 'b, 'c) env"} \\
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  @{term [source] "Env :: 'b \<Rightarrow> ('c \<Rightarrow> ('a, 'b, 'c) env option) \<Rightarrow> ('a, 'b, 'c) env"} \\
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  \end{tabular}}
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  \medskip
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  Here the parameter @{typ 'a} refers to plain values occurring at
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  leaf positions, parameter @{typ 'b} to information kept with inner
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  branch nodes, and parameter @{typ 'c} to the branching type of the
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  tree structure.  For our purpose we use the type instance with @{typ
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  "att \<times> string"} (representing plain files), @{typ att} (for
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  attributes of directory nodes), and @{typ name} (for the index type
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  of directory nodes).
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*}
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types
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  "file" = "(att \<times> string, att, name) env"
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text {*
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  \medskip The HOL library also provides @{term lookup} and @{term
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  update} operations for general tree structures with the subsequent
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  primitive recursive characterizations.
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  \medskip
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  {\def\isastyleminor{\isastyle}
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  \begin{tabular}{l}
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  @{term [source] "lookup :: ('a, 'b, 'c) env \<Rightarrow> 'c list \<Rightarrow> ('a, 'b, 'c) env option"} \\
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  @{term [source]
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  "update :: 'c list \<Rightarrow> ('a, 'b, 'c) env option \<Rightarrow> ('a, 'b, 'c) env \<Rightarrow> ('a, 'b, 'c) env"} \\
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  \end{tabular}}
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  @{thm [display, indent = 2, eta_contract = false] lookup_eq [no_vars]}
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  @{thm [display, indent = 2, eta_contract = false] update_eq [no_vars]}
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  Several further properties of these operations are proven in
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  \cite{Bauer-et-al:2002:HOL-Library}.  These will be routinely used
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  later on without further notice.
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  \bigskip Apparently, the elements of type @{typ "file"} contain an
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  @{typ att} component in either case.  We now define a few auxiliary
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  operations to manipulate this field uniformly, following the
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  conventions for record types in Isabelle/HOL
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  \cite{Nipkow-et-al:2000:HOL}.
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*}
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definition
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  "attributes file =
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    (case file of
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      Val (att, text) \<Rightarrow> att
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    | Env att dir \<Rightarrow> att)"
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  "map_attributes f file =
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    (case file of
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      Val (att, text) \<Rightarrow> Val (f att, text)
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    | Env att dir \<Rightarrow> Env (f att) dir)"
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lemma [simp]: "attributes (Val (att, text)) = att"
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  by (simp add: attributes_def)
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lemma [simp]: "attributes (Env att dir) = att"
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  by (simp add: attributes_def)
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lemma [simp]: "attributes (map_attributes f file) = f (attributes file)"
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  by (cases "file") (simp_all add: attributes_def map_attributes_def
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    split_tupled_all)
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lemma [simp]: "map_attributes f (Val (att, text)) = Val (f att, text)"
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  by (simp add: map_attributes_def)
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lemma [simp]: "map_attributes f (Env att dir) = Env (f att) dir"
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  by (simp add: map_attributes_def)
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subsection {* Initial file-systems *}
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text {*
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  Given a set of \emph{known users} a file-system shall be initialized
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  by providing an empty home directory for each user, with read-only
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  access for everyone else.  (Note that we may directly use the user
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  id as home directory name, since both types have been identified.)
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  Certainly, the very root directory is owned by the super user (who
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  has user id 0).
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*}
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definition
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  "init users =
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    Env \<lparr>owner = 0, others = {Readable}\<rparr>
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      (\<lambda>u. if u \<in> users then Some (Env \<lparr>owner = u, others = {Readable}\<rparr> empty)
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        else None)"
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subsection {* Accessing file-systems *}
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text {*
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  The main internal file-system operation is access of a file by a
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  user, requesting a certain set of permissions.  The resulting @{typ
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  "file option"} indicates if a file had been present at the
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  corresponding @{typ path} and if access was granted according to the
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  permissions recorded within the file-system.
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  Note that by the rules of Unix file-system security (e.g.\
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  \cite{Tanenbaum:1992}) both the super-user and owner may always
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  access a file unconditionally (in our simplified model).
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*}
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definition
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  "access root path uid perms =
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    (case lookup root path of
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      None \<Rightarrow> None
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    | Some file \<Rightarrow>
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        if uid = 0
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          \<or> uid = owner (attributes file)
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          \<or> perms \<subseteq> others (attributes file)
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        then Some file
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        else None)"
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text {*
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  \medskip Successful access to a certain file is the main
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  prerequisite for system-calls to be applicable (cf.\
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  \secref{sec:unix-trans}).  Any modification of the file-system is
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  then performed using the basic @{term update} operation.
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  \medskip We see that @{term access} is just a wrapper for the basic
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  @{term lookup} function, with additional checking of
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  attributes. Subsequently we establish a few auxiliary facts that
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  stem from the primitive @{term lookup} used within @{term access}.
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*}
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lemma access_empty_lookup: "access root path uid {} = lookup root path"
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  by (simp add: access_def split: option.splits)
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lemma access_some_lookup:
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  "access root path uid perms = Some file \<Longrightarrow>
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    lookup root path = Some file"
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  by (simp add: access_def split: option.splits if_splits)
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lemma access_update_other:
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  assumes parallel: "path' \<parallel> path"
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  shows "access (update path' opt root) path uid perms = access root path uid perms"
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proof -
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  from parallel obtain y z xs ys zs where
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      "y \<noteq> z" and "path' = xs @ y # ys" and "path = xs @ z # zs"
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    by (blast dest: parallel_decomp)
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  then have "lookup (update path' opt root) path = lookup root path"
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    by (blast intro: lookup_update_other)
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  then show ?thesis by (simp only: access_def)
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qed
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section {* File-system transitions \label{sec:unix-trans} *}
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subsection {* Unix system calls \label{sec:unix-syscall} *}
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text {*
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  According to established operating system design (cf.\
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  \cite{Tanenbaum:1992}) user space processes may only initiate system
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  operations by a fixed set of \emph{system-calls}.  This enables the
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  kernel to enforce certain security policies in the first
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  place.\footnote{Incidently, this is the very same principle employed
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  by any ``LCF-style'' theorem proving system according to Milner's
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  principle of ``correctness by construction'', such as Isabelle/HOL
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  itself.}
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  \medskip In our model of Unix we give a fixed datatype @{text
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  operation} for the syntax of system-calls, together with an
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  inductive definition of file-system state transitions of the form
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  @{text "root \<midarrow>x\<rightarrow> root'"} for the operational semantics.
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*}
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datatype operation =
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    Read uid string path
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  | Write uid string path
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  | Chmod uid perms path
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  | Creat uid perms path
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  | Unlink uid path
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  | Mkdir uid perms path
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  | Rmdir uid path
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  | Readdir uid "name set" path
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text {*
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  The @{typ uid} field of an operation corresponds to the
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  \emph{effective user id} of the underlying process, although our
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  model never mentions processes explicitly.  The other parameters are
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  provided as arguments by the caller; the @{term path} one is common
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  to all kinds of system-calls.
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*}
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consts
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  uid_of :: "operation \<Rightarrow> uid"
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primrec
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  "uid_of (Read uid text path) = uid"
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  "uid_of (Write uid text path) = uid"
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  "uid_of (Chmod uid perms path) = uid"
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  "uid_of (Creat uid perms path) = uid"
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  "uid_of (Unlink uid path) = uid"
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  "uid_of (Mkdir uid path perms) = uid"
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  "uid_of (Rmdir uid path) = uid"
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  "uid_of (Readdir uid names path) = uid"
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consts
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  path_of :: "operation \<Rightarrow> path"
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primrec
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  "path_of (Read uid text path) = path"
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  "path_of (Write uid text path) = path"
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  "path_of (Chmod uid perms path) = path"
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  "path_of (Creat uid perms path) = path"
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  "path_of (Unlink uid path) = path"
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  "path_of (Mkdir uid perms path) = path"
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  "path_of (Rmdir uid path) = path"
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  "path_of (Readdir uid names path) = path"
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text {*
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  \medskip Note that we have omitted explicit @{text Open} and @{text
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  Close} operations, pretending that @{term Read} and @{term Write}
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  would already take care of this behind the scenes.  Thus we have
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  basically treated actual sequences of real system-calls @{text
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  "open"}--@{text read}/@{text write}--@{text close} as atomic.
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  In principle, this could make big a difference in a model with
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  explicit concurrent processes.  On the other hand, even on a real
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  Unix system the exact scheduling of concurrent @{text "open"} and
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  @{text close} operations does \emph{not} directly affect the success
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  of corresponding @{text read} or @{text write}.  Unix allows several
wenzelm@10966
   341
  processes to have files opened at the same time, even for writing!
wenzelm@10966
   342
  Certainly, the result from reading the contents later may be hard to
wenzelm@10966
   343
  predict, but the system-calls involved here will succeed in any
wenzelm@10966
   344
  case.
wenzelm@10966
   345
wenzelm@10966
   346
  \bigskip The operational semantics of system calls is now specified
wenzelm@10966
   347
  via transitions of the file-system configuration.  This is expressed
wenzelm@10966
   348
  as an inductive relation (although there is no actual recursion
wenzelm@10966
   349
  involved here).
wenzelm@10966
   350
*}
wenzelm@10966
   351
wenzelm@21372
   352
inductive2
wenzelm@21372
   353
  transition :: "file \<Rightarrow> operation \<Rightarrow> file \<Rightarrow> bool"
wenzelm@21372
   354
    ("_ \<midarrow>_\<rightarrow> _" [90, 1000, 90] 100)
wenzelm@21372
   355
where
wenzelm@10966
   356
wenzelm@10966
   357
  read:
wenzelm@10966
   358
    "access root path uid {Readable} = Some (Val (att, text)) \<Longrightarrow>
wenzelm@21372
   359
      root \<midarrow>(Read uid text path)\<rightarrow> root" |
wenzelm@10966
   360
  write:
wenzelm@10966
   361
    "access root path uid {Writable} = Some (Val (att, text')) \<Longrightarrow>
wenzelm@21372
   362
      root \<midarrow>(Write uid text path)\<rightarrow> update path (Some (Val (att, text))) root" |
wenzelm@10966
   363
wenzelm@10966
   364
  chmod:
wenzelm@10966
   365
    "access root path uid {} = Some file \<Longrightarrow>
wenzelm@10966
   366
      uid = 0 \<or> uid = owner (attributes file) \<Longrightarrow>
wenzelm@10966
   367
      root \<midarrow>(Chmod uid perms path)\<rightarrow> update path
wenzelm@21372
   368
        (Some (map_attributes (others_update (K_record perms)) file)) root" |
wenzelm@10966
   369
wenzelm@10966
   370
  creat:
wenzelm@10966
   371
    "path = parent_path @ [name] \<Longrightarrow>
wenzelm@10966
   372
      access root parent_path uid {Writable} = Some (Env att parent) \<Longrightarrow>
wenzelm@10966
   373
      access root path uid {} = None \<Longrightarrow>
wenzelm@10966
   374
      root \<midarrow>(Creat uid perms path)\<rightarrow> update path
wenzelm@21372
   375
        (Some (Val (\<lparr>owner = uid, others = perms\<rparr>, []))) root" |
wenzelm@10966
   376
  unlink:
wenzelm@10966
   377
    "path = parent_path @ [name] \<Longrightarrow>
wenzelm@10966
   378
      access root parent_path uid {Writable} = Some (Env att parent) \<Longrightarrow>
wenzelm@10966
   379
      access root path uid {} = Some (Val plain) \<Longrightarrow>
wenzelm@21372
   380
      root \<midarrow>(Unlink uid path)\<rightarrow> update path None root" |
wenzelm@10966
   381
wenzelm@10966
   382
  mkdir:
wenzelm@10966
   383
    "path = parent_path @ [name] \<Longrightarrow>
wenzelm@10966
   384
      access root parent_path uid {Writable} = Some (Env att parent) \<Longrightarrow>
wenzelm@10966
   385
      access root path uid {} = None \<Longrightarrow>
wenzelm@10966
   386
      root \<midarrow>(Mkdir uid perms path)\<rightarrow> update path
wenzelm@21372
   387
        (Some (Env \<lparr>owner = uid, others = perms\<rparr> empty)) root" |
wenzelm@10966
   388
  rmdir:
wenzelm@10966
   389
    "path = parent_path @ [name] \<Longrightarrow>
wenzelm@10966
   390
      access root parent_path uid {Writable} = Some (Env att parent) \<Longrightarrow>
wenzelm@10966
   391
      access root path uid {} = Some (Env att' empty) \<Longrightarrow>
wenzelm@21372
   392
      root \<midarrow>(Rmdir uid path)\<rightarrow> update path None root" |
wenzelm@10966
   393
wenzelm@10966
   394
  readdir:
wenzelm@10966
   395
    "access root path uid {Readable} = Some (Env att dir) \<Longrightarrow>
wenzelm@10966
   396
      names = dom dir \<Longrightarrow>
wenzelm@10966
   397
      root \<midarrow>(Readdir uid names path)\<rightarrow> root"
wenzelm@10966
   398
wenzelm@10966
   399
text {*
wenzelm@10966
   400
  \medskip Certainly, the above specification is central to the whole
wenzelm@10966
   401
  formal development.  Any of the results to be established later on
wenzelm@10966
   402
  are only meaningful to the outside world if this transition system
wenzelm@10966
   403
  provides an adequate model of real Unix systems.  This kind of
wenzelm@10966
   404
  ``reality-check'' of a formal model is the well-known problem of
wenzelm@10966
   405
  \emph{validation}.
wenzelm@10966
   406
wenzelm@10966
   407
  If in doubt, one may consider to compare our definition with the
wenzelm@10966
   408
  informal specifications given the corresponding Unix man pages, or
wenzelm@10966
   409
  even peek at an actual implementation such as
wenzelm@10966
   410
  \cite{Torvalds-et-al:Linux}.  Another common way to gain confidence
wenzelm@10966
   411
  into the formal model is to run simple simulations (see
wenzelm@10966
   412
  \secref{sec:unix-examples}), and check the results with that of
wenzelm@10966
   413
  experiments performed on a real Unix system.
wenzelm@10966
   414
*}
wenzelm@10966
   415
wenzelm@10966
   416
wenzelm@10966
   417
subsection {* Basic properties of single transitions \label{sec:unix-single-trans} *}
wenzelm@10966
   418
wenzelm@10966
   419
text {*
wenzelm@10966
   420
  The transition system @{text "root \<midarrow>x\<rightarrow> root'"} defined above
wenzelm@10966
   421
  determines a unique result @{term root'} from given @{term root} and
wenzelm@10966
   422
  @{term x} (this is holds rather trivially, since there is even only
wenzelm@10966
   423
  one clause for each operation).  This uniqueness statement will
wenzelm@10966
   424
  simplify our subsequent development to some extent, since we only
wenzelm@10966
   425
  have to reason about a partial function rather than a general
wenzelm@10966
   426
  relation.
wenzelm@10966
   427
*}
wenzelm@10966
   428
wenzelm@18372
   429
theorem transition_uniq:
wenzelm@18372
   430
  assumes root': "root \<midarrow>x\<rightarrow> root'"
wenzelm@18372
   431
    and root'': "root \<midarrow>x\<rightarrow> root''"
wenzelm@18372
   432
  shows "root' = root''"
wenzelm@18372
   433
  using root''
wenzelm@18372
   434
proof cases
wenzelm@18372
   435
  case read
wenzelm@18372
   436
  with root' show ?thesis by cases auto
wenzelm@18372
   437
next
wenzelm@18372
   438
  case write
wenzelm@18372
   439
  with root' show ?thesis by cases auto
wenzelm@18372
   440
next
wenzelm@18372
   441
  case chmod
wenzelm@18372
   442
  with root' show ?thesis by cases auto
wenzelm@18372
   443
next
wenzelm@18372
   444
  case creat
wenzelm@18372
   445
  with root' show ?thesis by cases auto
wenzelm@18372
   446
next
wenzelm@18372
   447
  case unlink
wenzelm@18372
   448
  with root' show ?thesis by cases auto
wenzelm@18372
   449
next
wenzelm@18372
   450
  case mkdir
wenzelm@18372
   451
  with root' show ?thesis by cases auto
wenzelm@18372
   452
next
wenzelm@18372
   453
  case rmdir
wenzelm@18372
   454
  with root' show ?thesis by cases auto
wenzelm@18372
   455
next
wenzelm@18372
   456
  case readdir
wenzelm@18372
   457
  with root' show ?thesis by cases fastsimp+
wenzelm@10966
   458
qed
wenzelm@10966
   459
wenzelm@10966
   460
text {*
wenzelm@10966
   461
  \medskip Apparently, file-system transitions are \emph{type-safe} in
wenzelm@10966
   462
  the sense that the result of transforming an actual directory yields
wenzelm@10966
   463
  again a directory.
wenzelm@10966
   464
*}
wenzelm@10966
   465
wenzelm@10966
   466
theorem transition_type_safe:
wenzelm@18372
   467
  assumes tr: "root \<midarrow>x\<rightarrow> root'"
wenzelm@18372
   468
    and inv: "\<exists>att dir. root = Env att dir"
wenzelm@18372
   469
  shows "\<exists>att dir. root' = Env att dir"
wenzelm@18372
   470
proof (cases "path_of x")
wenzelm@18372
   471
  case Nil
wenzelm@18372
   472
  with tr inv show ?thesis
wenzelm@18372
   473
    by cases (auto simp add: access_def split: if_splits)
wenzelm@18372
   474
next
wenzelm@18372
   475
  case Cons
wenzelm@18372
   476
  from tr obtain opt where
wenzelm@18372
   477
      "root' = root \<or> root' = update (path_of x) opt root"
wenzelm@18372
   478
    by cases auto
wenzelm@18372
   479
  with inv Cons show ?thesis
wenzelm@18372
   480
    by (auto simp add: update_eq split: list.splits)
wenzelm@10966
   481
qed
wenzelm@10966
   482
wenzelm@10966
   483
text {*
wenzelm@10966
   484
  The previous result may be seen as the most basic invariant on the
wenzelm@10966
   485
  file-system state that is enforced by any proper kernel
wenzelm@10966
   486
  implementation.  So user processes --- being bound to the
wenzelm@10966
   487
  system-call interface --- may never mess up a file-system such that
wenzelm@10966
   488
  the root becomes a plain file instead of a directory, which would be
wenzelm@10966
   489
  a strange situation indeed.
wenzelm@10966
   490
*}
wenzelm@10966
   491
wenzelm@10966
   492
wenzelm@10966
   493
subsection {* Iterated transitions *}
wenzelm@10966
   494
wenzelm@10966
   495
text {*
wenzelm@10966
   496
  Iterated system transitions via finite sequences of system
wenzelm@10966
   497
  operations are modeled inductively as follows.  In a sense, this
wenzelm@10966
   498
  relation describes the cumulative effect of the sequence of
wenzelm@10966
   499
  system-calls issued by a number of running processes over a finite
wenzelm@10966
   500
  amount of time.
wenzelm@10966
   501
*}
wenzelm@10966
   502
wenzelm@21372
   503
inductive2
wenzelm@21372
   504
  transitions :: "file \<Rightarrow> operation list \<Rightarrow> file \<Rightarrow> bool"
wenzelm@21372
   505
    ("_ =_\<Rightarrow> _" [90, 1000, 90] 100)
wenzelm@21372
   506
where
wenzelm@10966
   507
    nil: "root =[]\<Rightarrow> root"
wenzelm@21372
   508
  | cons: "root \<midarrow>x\<rightarrow> root' \<Longrightarrow> root' =xs\<Rightarrow> root'' \<Longrightarrow> root =(x # xs)\<Rightarrow> root''"
wenzelm@10966
   509
wenzelm@10966
   510
text {*
wenzelm@10966
   511
  \medskip We establish a few basic facts relating iterated
wenzelm@10966
   512
  transitions with single ones, according to the recursive structure
wenzelm@10966
   513
  of lists.
wenzelm@10966
   514
*}
wenzelm@10966
   515
wenzelm@10966
   516
lemma transitions_nil_eq: "root =[]\<Rightarrow> root' = (root = root')"
wenzelm@10966
   517
proof
wenzelm@10966
   518
  assume "root =[]\<Rightarrow> root'"
wenzelm@18372
   519
  then show "root = root'" by cases simp_all
wenzelm@10966
   520
next
wenzelm@10966
   521
  assume "root = root'"
wenzelm@18372
   522
  then show "root =[]\<Rightarrow> root'" by (simp only: transitions.nil)
wenzelm@10966
   523
qed
wenzelm@10966
   524
wenzelm@10966
   525
lemma transitions_cons_eq:
wenzelm@10966
   526
  "root =(x # xs)\<Rightarrow> root'' = (\<exists>root'. root \<midarrow>x\<rightarrow> root' \<and> root' =xs\<Rightarrow> root'')"
wenzelm@10966
   527
proof
wenzelm@10966
   528
  assume "root =(x # xs)\<Rightarrow> root''"
wenzelm@18372
   529
  then show "\<exists>root'. root \<midarrow>x\<rightarrow> root' \<and> root' =xs\<Rightarrow> root''"
wenzelm@10966
   530
    by cases auto
wenzelm@10966
   531
next
wenzelm@10966
   532
  assume "\<exists>root'. root \<midarrow>x\<rightarrow> root' \<and> root' =xs\<Rightarrow> root''"
wenzelm@18372
   533
  then show "root =(x # xs)\<Rightarrow> root''"
wenzelm@10966
   534
    by (blast intro: transitions.cons)
wenzelm@10966
   535
qed
wenzelm@10966
   536
wenzelm@10966
   537
text {*
wenzelm@10966
   538
  The next two rules show how to ``destruct'' known transition
wenzelm@10966
   539
  sequences.  Note that the second one actually relies on the
wenzelm@10966
   540
  uniqueness property of the basic transition system (see
wenzelm@10966
   541
  \secref{sec:unix-single-trans}).
wenzelm@10966
   542
*}
wenzelm@10966
   543
wenzelm@10966
   544
lemma transitions_nilD: "root =[]\<Rightarrow> root' \<Longrightarrow> root' = root"
wenzelm@10966
   545
  by (simp add: transitions_nil_eq)
wenzelm@10966
   546
wenzelm@10966
   547
lemma transitions_consD:
wenzelm@18372
   548
  assumes list: "root =(x # xs)\<Rightarrow> root''"
wenzelm@18372
   549
    and hd: "root \<midarrow>x\<rightarrow> root'"
wenzelm@18372
   550
  shows "root' =xs\<Rightarrow> root''"
wenzelm@10966
   551
proof -
wenzelm@18372
   552
  from list obtain r' where r': "root \<midarrow>x\<rightarrow> r'" and root'': "r' =xs\<Rightarrow> root''"
wenzelm@10966
   553
    by cases simp_all
wenzelm@18372
   554
  from r' hd have "r' = root'" by (rule transition_uniq)
wenzelm@10966
   555
  with root'' show "root' =xs\<Rightarrow> root''" by simp
wenzelm@10966
   556
qed
wenzelm@10966
   557
wenzelm@10966
   558
text {*
wenzelm@10966
   559
  \medskip The following fact shows how an invariant @{term Q} of
wenzelm@10966
   560
  single transitions with property @{term P} may be transferred to
wenzelm@10966
   561
  iterated transitions.  The proof is rather obvious by rule induction
wenzelm@10966
   562
  over the definition of @{term "root =xs\<Rightarrow> root'"}.
wenzelm@10966
   563
*}
wenzelm@10966
   564
wenzelm@10966
   565
lemma transitions_invariant:
wenzelm@18372
   566
  assumes r: "\<And>r x r'. r \<midarrow>x\<rightarrow> r' \<Longrightarrow> Q r \<Longrightarrow>  P x \<Longrightarrow> Q r'"
wenzelm@18372
   567
    and trans: "root =xs\<Rightarrow> root'"
wenzelm@18372
   568
  shows "Q root \<Longrightarrow> \<forall>x \<in> set xs. P x \<Longrightarrow> Q root'"
wenzelm@18372
   569
  using trans
wenzelm@18372
   570
proof induct
wenzelm@18372
   571
  case nil
wenzelm@18372
   572
  show ?case by assumption
wenzelm@18372
   573
next
wenzelm@21372
   574
  case (cons root x root' xs root'')
wenzelm@18372
   575
  note P = `\<forall>x \<in> set (x # xs). P x`
wenzelm@18372
   576
  then have "P x" by simp
wenzelm@18372
   577
  with `root \<midarrow>x\<rightarrow> root'` and `Q root` have Q': "Q root'" by (rule r)
wenzelm@18372
   578
  from P have "\<forall>x \<in> set xs. P x" by simp
wenzelm@18372
   579
  with Q' show "Q root''" by (rule cons.hyps)
wenzelm@10966
   580
qed
wenzelm@10966
   581
wenzelm@10966
   582
text {*
wenzelm@10966
   583
  As an example of applying the previous result, we transfer the basic
wenzelm@10966
   584
  type-safety property (see \secref{sec:unix-single-trans}) from
wenzelm@10966
   585
  single transitions to iterated ones, which is a rather obvious
wenzelm@10966
   586
  result indeed.
wenzelm@10966
   587
*}
wenzelm@10966
   588
wenzelm@10966
   589
theorem transitions_type_safe:
wenzelm@16670
   590
  assumes "root =xs\<Rightarrow> root'"
wenzelm@16670
   591
    and "\<exists>att dir. root = Env att dir"
wenzelm@16670
   592
  shows "\<exists>att dir. root' = Env att dir"
wenzelm@16670
   593
  using transition_type_safe and prems
wenzelm@16670
   594
proof (rule transitions_invariant)
wenzelm@16670
   595
  show "\<forall>x \<in> set xs. True" by blast
wenzelm@10966
   596
qed
wenzelm@10966
   597
wenzelm@10966
   598
wenzelm@10966
   599
section {* Executable sequences *}
wenzelm@10966
   600
wenzelm@10966
   601
text {*
wenzelm@17455
   602
  An inductively defined relation such as the one of @{text "root \<midarrow>x\<rightarrow>
wenzelm@17455
   603
  root'"} (see \secref{sec:unix-syscall}) has two main aspects.  First
wenzelm@17455
   604
  of all, the resulting system admits a certain set of transition
wenzelm@17455
   605
  rules (introductions) as given in the specification.  Furthermore,
wenzelm@17455
   606
  there is an explicit least-fixed-point construction involved, which
wenzelm@17455
   607
  results in induction (and case-analysis) rules to eliminate known
wenzelm@17455
   608
  transitions in an exhaustive manner.
wenzelm@10966
   609
wenzelm@10966
   610
  \medskip Subsequently, we explore our transition system in an
wenzelm@10966
   611
  experimental style, mainly using the introduction rules with basic
wenzelm@10966
   612
  algebraic properties of the underlying structures.  The technique
wenzelm@10966
   613
  closely resembles that of Prolog combined with functional evaluation
wenzelm@10966
   614
  in a very simple manner.
wenzelm@10966
   615
wenzelm@10966
   616
  Just as the ``closed-world assumption'' is left implicit in Prolog,
wenzelm@10966
   617
  we do not refer to induction over the whole transition system here.
wenzelm@10966
   618
  So this is still purely positive reasoning about possible
wenzelm@10966
   619
  executions; exhaustive reasoning will be employed only later on (see
wenzelm@10966
   620
  \secref{sec:unix-bogosity}), when we shall demonstrate that certain
wenzelm@10966
   621
  behavior is \emph{not} possible.
wenzelm@10966
   622
*}
wenzelm@10966
   623
wenzelm@10966
   624
wenzelm@10966
   625
subsection {* Possible transitions *}
wenzelm@10966
   626
wenzelm@10966
   627
text {*
wenzelm@10966
   628
  Rather obviously, a list of system operations can be executed within
wenzelm@10966
   629
  a certain state if there is a result state reached by an iterated
wenzelm@10966
   630
  transition.
wenzelm@10966
   631
*}
wenzelm@10966
   632
wenzelm@19086
   633
definition
wenzelm@20321
   634
  "can_exec root xs = (\<exists>root'. root =xs\<Rightarrow> root')"
wenzelm@10966
   635
wenzelm@10966
   636
lemma can_exec_nil: "can_exec root []"
wenzelm@18730
   637
  unfolding can_exec_def by (blast intro: transitions.intros)
wenzelm@10966
   638
wenzelm@10966
   639
lemma can_exec_cons:
wenzelm@10966
   640
    "root \<midarrow>x\<rightarrow> root' \<Longrightarrow> can_exec root' xs \<Longrightarrow> can_exec root (x # xs)"
wenzelm@18730
   641
  unfolding can_exec_def by (blast intro: transitions.intros)
wenzelm@10966
   642
wenzelm@10966
   643
text {*
wenzelm@10966
   644
  \medskip In case that we already know that a certain sequence can be
wenzelm@10966
   645
  executed we may destruct it backwardly into individual transitions.
wenzelm@10966
   646
*}
wenzelm@10966
   647
wenzelm@18372
   648
lemma can_exec_snocD: "can_exec root (xs @ [y])
wenzelm@10966
   649
    \<Longrightarrow> \<exists>root' root''. root =xs\<Rightarrow> root' \<and> root' \<midarrow>y\<rightarrow> root''"
wenzelm@20503
   650
proof (induct xs arbitrary: root)
wenzelm@18372
   651
  case Nil
wenzelm@18372
   652
  then show ?case
wenzelm@18372
   653
    by (simp add: can_exec_def transitions_nil_eq transitions_cons_eq)
wenzelm@18372
   654
next
wenzelm@18372
   655
  case (Cons x xs)
wenzelm@18372
   656
  from `can_exec root ((x # xs) @ [y])` obtain r root'' where
wenzelm@18372
   657
      x: "root \<midarrow>x\<rightarrow> r" and
wenzelm@18372
   658
      xs_y: "r =(xs @ [y])\<Rightarrow> root''"
wenzelm@18372
   659
    by (auto simp add: can_exec_def transitions_nil_eq transitions_cons_eq)
wenzelm@18372
   660
  from xs_y Cons.hyps obtain root' r' where xs: "r =xs\<Rightarrow> root'" and y: "root' \<midarrow>y\<rightarrow> r'"
wenzelm@18730
   661
    unfolding can_exec_def by blast
wenzelm@18372
   662
  from x xs have "root =(x # xs)\<Rightarrow> root'"
wenzelm@18372
   663
    by (rule transitions.cons)
wenzelm@18372
   664
  with y show ?case by blast
wenzelm@10966
   665
qed
wenzelm@10966
   666
wenzelm@10966
   667
wenzelm@10966
   668
subsection {* Example executions \label{sec:unix-examples} *}
wenzelm@10966
   669
wenzelm@10966
   670
text {*
wenzelm@10966
   671
  We are now ready to perform a few experiments within our formal
wenzelm@10966
   672
  model of Unix system-calls.  The common technique is to alternate
wenzelm@10966
   673
  introduction rules of the transition system (see
wenzelm@10966
   674
  \secref{sec:unix-trans}), and steps to solve any emerging side
wenzelm@10966
   675
  conditions using algebraic properties of the underlying file-system
wenzelm@10966
   676
  structures (see \secref{sec:unix-file-system}).
wenzelm@10966
   677
*}
wenzelm@10966
   678
wenzelm@10966
   679
lemmas eval = access_def init_def
wenzelm@10966
   680
wenzelm@10966
   681
theorem "u \<in> users \<Longrightarrow> can_exec (init users)
wenzelm@10966
   682
    [Mkdir u perms [u, name]]"
wenzelm@10966
   683
  apply (rule can_exec_cons)
wenzelm@10966
   684
    -- {* back-chain @{term can_exec} (of @{term [source] Cons}) *}
wenzelm@10966
   685
  apply (rule mkdir)
wenzelm@10966
   686
    -- {* back-chain @{term Mkdir} *}
wenzelm@10966
   687
  apply (force simp add: eval)+
wenzelm@10966
   688
    -- {* solve preconditions of @{term Mkdir} *}
wenzelm@10966
   689
  apply (simp add: eval)
wenzelm@10966
   690
    -- {* peek at resulting dir (optional) *}
wenzelm@10966
   691
  txt {* @{subgoals [display]} *}
wenzelm@10966
   692
  apply (rule can_exec_nil)
wenzelm@10966
   693
    -- {* back-chain @{term can_exec} (of @{term [source] Nil}) *}
wenzelm@10966
   694
  done
wenzelm@10966
   695
wenzelm@10966
   696
text {*
wenzelm@10966
   697
  By inspecting the result shown just before the final step above, we
wenzelm@10966
   698
  may gain confidence that our specification of Unix system-calls
wenzelm@10966
   699
  actually makes sense.  Further common errors are usually exhibited
wenzelm@10966
   700
  when preconditions of transition rules fail unexpectedly.
wenzelm@10966
   701
wenzelm@10966
   702
  \medskip Here are a few further experiments, using the same
wenzelm@10966
   703
  techniques as before.
wenzelm@10966
   704
*}
wenzelm@10966
   705
wenzelm@10966
   706
theorem "u \<in> users \<Longrightarrow> can_exec (init users)
wenzelm@10966
   707
   [Creat u perms [u, name],
wenzelm@10966
   708
    Unlink u [u, name]]"
wenzelm@10966
   709
  apply (rule can_exec_cons)
wenzelm@10966
   710
  apply (rule creat)
wenzelm@10966
   711
  apply (force simp add: eval)+
wenzelm@10966
   712
  apply (simp add: eval)
wenzelm@10966
   713
  apply (rule can_exec_cons)
wenzelm@10966
   714
  apply (rule unlink)
wenzelm@10966
   715
  apply (force simp add: eval)+
wenzelm@10966
   716
  apply (simp add: eval)
wenzelm@10966
   717
  txt {* peek at result: @{subgoals [display]} *}
wenzelm@10966
   718
  apply (rule can_exec_nil)
wenzelm@10966
   719
  done
wenzelm@10966
   720
wenzelm@17455
   721
theorem "u \<in> users \<Longrightarrow> Writable \<in> perms\<^isub>1 \<Longrightarrow>
wenzelm@17455
   722
  Readable \<in> perms\<^isub>2 \<Longrightarrow> name\<^isub>3 \<noteq> name\<^isub>4 \<Longrightarrow>
wenzelm@10966
   723
  can_exec (init users)
wenzelm@17455
   724
   [Mkdir u perms\<^isub>1 [u, name\<^isub>1],
wenzelm@17455
   725
    Mkdir u' perms\<^isub>2 [u, name\<^isub>1, name\<^isub>2],
wenzelm@17455
   726
    Creat u' perms\<^isub>3 [u, name\<^isub>1, name\<^isub>2, name\<^isub>3],
wenzelm@17455
   727
    Creat u' perms\<^isub>3 [u, name\<^isub>1, name\<^isub>2, name\<^isub>4],
wenzelm@17455
   728
    Readdir u {name\<^isub>3, name\<^isub>4} [u, name\<^isub>1, name\<^isub>2]]"
wenzelm@10966
   729
  apply (rule can_exec_cons, rule transition.intros,
wenzelm@10966
   730
    (force simp add: eval)+, (simp add: eval)?)+
wenzelm@10966
   731
  txt {* peek at result: @{subgoals [display]} *}
wenzelm@10966
   732
  apply (rule can_exec_nil)
wenzelm@10966
   733
  done
wenzelm@10966
   734
wenzelm@17455
   735
theorem "u \<in> users \<Longrightarrow> Writable \<in> perms\<^isub>1 \<Longrightarrow> Readable \<in> perms\<^isub>3 \<Longrightarrow>
wenzelm@10966
   736
  can_exec (init users)
wenzelm@17455
   737
   [Mkdir u perms\<^isub>1 [u, name\<^isub>1],
wenzelm@17455
   738
    Mkdir u' perms\<^isub>2 [u, name\<^isub>1, name\<^isub>2],
wenzelm@17455
   739
    Creat u' perms\<^isub>3 [u, name\<^isub>1, name\<^isub>2, name\<^isub>3],
wenzelm@17455
   740
    Write u' ''foo'' [u, name\<^isub>1, name\<^isub>2, name\<^isub>3],
wenzelm@17455
   741
    Read u ''foo'' [u, name\<^isub>1, name\<^isub>2, name\<^isub>3]]"
wenzelm@10966
   742
  apply (rule can_exec_cons, rule transition.intros,
wenzelm@10966
   743
    (force simp add: eval)+, (simp add: eval)?)+
wenzelm@10966
   744
  txt {* peek at result: @{subgoals [display]} *}
wenzelm@10966
   745
  apply (rule can_exec_nil)
wenzelm@10966
   746
  done
wenzelm@10966
   747
wenzelm@10966
   748
wenzelm@10966
   749
section {* Odd effects --- treated formally \label{sec:unix-bogosity} *}
wenzelm@10966
   750
wenzelm@10966
   751
text {*
wenzelm@10966
   752
  We are now ready to give a completely formal treatment of the
wenzelm@10966
   753
  slightly odd situation discussed in the introduction (see
wenzelm@10966
   754
  \secref{sec:unix-intro}): the file-system can easily reach a state
wenzelm@10966
   755
  where a user is unable to remove his very own directory, because it
wenzelm@10966
   756
  is still populated by items placed there by another user in an
wenzelm@10966
   757
  uncouth manner.
wenzelm@10966
   758
*}
wenzelm@10966
   759
wenzelm@10966
   760
subsection {* The general procedure \label{sec:unix-inv-procedure} *}
wenzelm@10966
   761
wenzelm@10966
   762
text {*
wenzelm@10966
   763
  The following theorem expresses the general procedure we are
wenzelm@10966
   764
  following to achieve the main result.
wenzelm@10966
   765
*}
wenzelm@10966
   766
wenzelm@10966
   767
theorem general_procedure:
wenzelm@18372
   768
  assumes cannot_y: "\<And>r r'. Q r \<Longrightarrow> r \<midarrow>y\<rightarrow> r' \<Longrightarrow> False"
wenzelm@18372
   769
    and init_inv: "\<And>root. init users =bs\<Rightarrow> root \<Longrightarrow> Q root"
wenzelm@18372
   770
    and preserve_inv: "\<And>r x r'. r \<midarrow>x\<rightarrow> r' \<Longrightarrow> Q r \<Longrightarrow> P x \<Longrightarrow> Q r'"
wenzelm@18372
   771
    and init_result: "init users =bs\<Rightarrow> root"
wenzelm@18372
   772
  shows "\<not> (\<exists>xs. (\<forall>x \<in> set xs. P x) \<and> can_exec root (xs @ [y]))"
wenzelm@10966
   773
proof -
wenzelm@10966
   774
  {
wenzelm@10966
   775
    fix xs
wenzelm@10966
   776
    assume Ps: "\<forall>x \<in> set xs. P x"
wenzelm@10966
   777
    assume can_exec: "can_exec root (xs @ [y])"
wenzelm@10966
   778
    then obtain root' root'' where
wenzelm@10966
   779
        xs: "root =xs\<Rightarrow> root'" and y: "root' \<midarrow>y\<rightarrow> root''"
wenzelm@10966
   780
      by (blast dest: can_exec_snocD)
wenzelm@10966
   781
    from init_result have "Q root" by (rule init_inv)
wenzelm@10966
   782
    from preserve_inv xs this Ps have "Q root'"
wenzelm@10966
   783
      by (rule transitions_invariant)
wenzelm@10966
   784
    from this y have False by (rule cannot_y)
wenzelm@10966
   785
  }
wenzelm@18372
   786
  then show ?thesis by blast
wenzelm@10966
   787
qed
wenzelm@10966
   788
wenzelm@10966
   789
text {*
wenzelm@10966
   790
  Here @{prop "P x"} refers to the restriction on file-system
wenzelm@10966
   791
  operations that are admitted after having reached the critical
wenzelm@10966
   792
  configuration; according to the problem specification this will
wenzelm@17455
   793
  become @{prop "uid_of x = user\<^isub>1"} later on.  Furthermore,
wenzelm@17455
   794
  @{term y} refers to the operations we claim to be impossible to
wenzelm@17455
   795
  perform afterwards, we will take @{term Rmdir} later.  Moreover
wenzelm@17455
   796
  @{term Q} is a suitable (auxiliary) invariant over the file-system;
wenzelm@17455
   797
  choosing @{term Q} adequately is very important to make the proof
wenzelm@17455
   798
  work (see \secref{sec:unix-inv-lemmas}).
wenzelm@10966
   799
*}
wenzelm@10966
   800
wenzelm@10966
   801
wenzelm@12079
   802
subsection {* The particular situation *}
wenzelm@10966
   803
wenzelm@10966
   804
text {*
wenzelm@10966
   805
  We introduce a few global declarations and axioms to describe our
wenzelm@12079
   806
  particular situation considered here.  Thus we avoid excessive use
wenzelm@12079
   807
  of local parameters in the subsequent development.
wenzelm@10966
   808
*}
wenzelm@10966
   809
wenzelm@12079
   810
locale situation =
wenzelm@12079
   811
  fixes users :: "uid set"
wenzelm@17455
   812
    and user\<^isub>1 :: uid
wenzelm@17455
   813
    and user\<^isub>2 :: uid
wenzelm@17455
   814
    and name\<^isub>1 :: name
wenzelm@17455
   815
    and name\<^isub>2 :: name
wenzelm@17455
   816
    and name\<^isub>3 :: name
wenzelm@17455
   817
    and perms\<^isub>1 :: perms
wenzelm@17455
   818
    and perms\<^isub>2 :: perms
wenzelm@17455
   819
  assumes user\<^isub>1_known: "user\<^isub>1 \<in> users"
wenzelm@17455
   820
    and user\<^isub>1_not_root: "user\<^isub>1 \<noteq> 0"
wenzelm@17455
   821
    and users_neq: "user\<^isub>1 \<noteq> user\<^isub>2"
wenzelm@17455
   822
    and perms\<^isub>1_writable: "Writable \<in> perms\<^isub>1"
wenzelm@17455
   823
    and perms\<^isub>2_not_writable: "Writable \<notin> perms\<^isub>2"
wenzelm@17455
   824
  notes facts = user\<^isub>1_known user\<^isub>1_not_root users_neq
wenzelm@17455
   825
    perms\<^isub>1_writable perms\<^isub>2_not_writable
wenzelm@21029
   826
begin
wenzelm@10966
   827
wenzelm@21029
   828
definition
wenzelm@19086
   829
  "bogus =
wenzelm@17455
   830
     [Mkdir user\<^isub>1 perms\<^isub>1 [user\<^isub>1, name\<^isub>1],
wenzelm@17455
   831
      Mkdir user\<^isub>2 perms\<^isub>2 [user\<^isub>1, name\<^isub>1, name\<^isub>2],
wenzelm@17455
   832
      Creat user\<^isub>2 perms\<^isub>2 [user\<^isub>1, name\<^isub>1, name\<^isub>2, name\<^isub>3]]"
wenzelm@19086
   833
  "bogus_path = [user\<^isub>1, name\<^isub>1, name\<^isub>2]"
wenzelm@10966
   834
wenzelm@10966
   835
text {*
wenzelm@12079
   836
  The @{term bogus} operations are the ones that lead into the uncouth
wenzelm@12079
   837
  situation; @{term bogus_path} is the key position within the
wenzelm@12079
   838
  file-system where things go awry.
wenzelm@10966
   839
*}
wenzelm@10966
   840
wenzelm@10966
   841
wenzelm@10966
   842
subsection {* Invariance lemmas \label{sec:unix-inv-lemmas} *}
wenzelm@10966
   843
wenzelm@10966
   844
text {*
wenzelm@10966
   845
  The following invariant over the root file-system describes the
wenzelm@10966
   846
  bogus situation in an abstract manner: located at a certain @{term
wenzelm@10966
   847
  path} within the file-system is a non-empty directory that is
wenzelm@17455
   848
  neither owned and nor writable by @{term user\<^isub>1}.
wenzelm@10966
   849
*}
wenzelm@10966
   850
wenzelm@21029
   851
definition
wenzelm@20321
   852
  "invariant root path =
wenzelm@10966
   853
    (\<exists>att dir.
wenzelm@17455
   854
      access root path user\<^isub>1 {} = Some (Env att dir) \<and> dir \<noteq> empty \<and>
wenzelm@17455
   855
      user\<^isub>1 \<noteq> owner att \<and>
wenzelm@17455
   856
      access root path user\<^isub>1 {Writable} = None)"
wenzelm@10966
   857
wenzelm@10966
   858
text {*
wenzelm@10966
   859
  Following the general procedure (see
wenzelm@17455
   860
  \secref{sec:unix-inv-procedure}) we will now establish the three key
wenzelm@17455
   861
  lemmas required to yield the final result.
wenzelm@10966
   862
wenzelm@10966
   863
  \begin{enumerate}
wenzelm@10966
   864
wenzelm@10966
   865
  \item The invariant is sufficiently strong to entail the
wenzelm@17455
   866
  pathological case that @{term user\<^isub>1} is unable to remove the
wenzelm@17455
   867
  (owned) directory at @{term "[user\<^isub>1, name\<^isub>1]"}.
wenzelm@10966
   868
wenzelm@10966
   869
  \item The invariant does hold after having executed the @{term
wenzelm@10966
   870
  bogus} list of operations (starting with an initial file-system
wenzelm@10966
   871
  configuration).
wenzelm@10966
   872
wenzelm@10966
   873
  \item The invariant is preserved by any file-system operation
wenzelm@17455
   874
  performed by @{term user\<^isub>1} alone, without any help by other
wenzelm@17455
   875
  users.
wenzelm@10966
   876
wenzelm@10966
   877
  \end{enumerate}
wenzelm@10966
   878
wenzelm@10966
   879
  As the invariant appears both as assumptions and conclusions in the
wenzelm@10966
   880
  course of proof, its formulation is rather critical for the whole
wenzelm@10966
   881
  development to work out properly.  In particular, the third step is
wenzelm@10966
   882
  very sensitive to the invariant being either too strong or too weak.
wenzelm@10966
   883
  Moreover the invariant has to be sufficiently abstract, lest the
wenzelm@10966
   884
  proof become cluttered by confusing detail.
wenzelm@10966
   885
wenzelm@10966
   886
  \medskip The first two lemmas are technically straight forward ---
wenzelm@10966
   887
  we just have to inspect rather special cases.
wenzelm@10966
   888
*}
wenzelm@10966
   889
wenzelm@21029
   890
lemma cannot_rmdir:
wenzelm@18372
   891
  assumes inv: "invariant root bogus_path"
wenzelm@18372
   892
    and rmdir: "root \<midarrow>(Rmdir user\<^isub>1 [user\<^isub>1, name\<^isub>1])\<rightarrow> root'"
wenzelm@18372
   893
  shows False
wenzelm@10966
   894
proof -
wenzelm@18372
   895
  from inv obtain "file" where "access root bogus_path user\<^isub>1 {} = Some file"
wenzelm@18730
   896
    unfolding invariant_def by blast
wenzelm@10966
   897
  moreover
wenzelm@18372
   898
  from rmdir obtain att where
wenzelm@17455
   899
      "access root [user\<^isub>1, name\<^isub>1] user\<^isub>1 {} = Some (Env att empty)"
wenzelm@10966
   900
    by cases auto
wenzelm@18372
   901
  then have "access root ([user\<^isub>1, name\<^isub>1] @ [name\<^isub>2]) user\<^isub>1 {} = empty name\<^isub>2"
wenzelm@10966
   902
    by (simp only: access_empty_lookup lookup_append_some) simp
wenzelm@10966
   903
  ultimately show False by (simp add: bogus_path_def)
wenzelm@10966
   904
qed
wenzelm@10966
   905
wenzelm@21029
   906
lemma
wenzelm@18372
   907
  assumes init: "init users =bogus\<Rightarrow> root"
wenzelm@18372
   908
  notes eval = facts access_def init_def
wenzelm@18372
   909
  shows init_invariant: "invariant root bogus_path"
wenzelm@18372
   910
  using init
wenzelm@18372
   911
  apply (unfold bogus_def bogus_path_def)
wenzelm@18372
   912
  apply (drule transitions_consD, rule transition.intros,
wenzelm@10966
   913
      (force simp add: eval)+, (simp add: eval)?)+
wenzelm@18372
   914
    -- "evaluate all operations"
wenzelm@18372
   915
  apply (drule transitions_nilD)
wenzelm@18372
   916
    -- "reach final result"
wenzelm@18372
   917
  apply (simp add: invariant_def eval)
wenzelm@18372
   918
    -- "check the invariant"
wenzelm@18372
   919
  done
wenzelm@10966
   920
wenzelm@10966
   921
text {*
wenzelm@10966
   922
  \medskip At last we are left with the main effort to show that the
wenzelm@10966
   923
  ``bogosity'' invariant is preserved by any file-system operation
wenzelm@17455
   924
  performed by @{term user\<^isub>1} alone.  Note that this holds for
wenzelm@17455
   925
  any @{term path} given, the particular @{term bogus_path} is not
wenzelm@10966
   926
  required here.
wenzelm@11004
   927
*}
wenzelm@10966
   928
wenzelm@21029
   929
lemma preserve_invariant:
wenzelm@18372
   930
  assumes tr: "root \<midarrow>x\<rightarrow> root'"
wenzelm@18372
   931
    and inv: "invariant root path"
wenzelm@18372
   932
    and uid: "uid_of x = user\<^isub>1"
wenzelm@18372
   933
  shows "invariant root' path"
wenzelm@10966
   934
proof -
wenzelm@10966
   935
  from inv obtain att dir where
wenzelm@17455
   936
      inv1: "access root path user\<^isub>1 {} = Some (Env att dir)" and
wenzelm@10966
   937
      inv2: "dir \<noteq> empty" and
wenzelm@17455
   938
      inv3: "user\<^isub>1 \<noteq> owner att" and
wenzelm@17455
   939
      inv4: "access root path user\<^isub>1 {Writable} = None"
wenzelm@10966
   940
    by (auto simp add: invariant_def)
wenzelm@10966
   941
wenzelm@10966
   942
  from inv1 have lookup: "lookup root path = Some (Env att dir)"
wenzelm@10966
   943
    by (simp only: access_empty_lookup)
wenzelm@10966
   944
wenzelm@17455
   945
  from inv1 inv3 inv4 and user\<^isub>1_not_root
wenzelm@10966
   946
  have not_writable: "Writable \<notin> others att"
wenzelm@10966
   947
    by (auto simp add: access_def split: option.splits if_splits)
wenzelm@10966
   948
wenzelm@10966
   949
  show ?thesis
wenzelm@10966
   950
  proof cases
wenzelm@10966
   951
    assume "root' = root"
wenzelm@10966
   952
    with inv show "invariant root' path" by (simp only:)
wenzelm@10966
   953
  next
wenzelm@10966
   954
    assume changed: "root' \<noteq> root"
wenzelm@10966
   955
    with tr obtain opt where root': "root' = update (path_of x) opt root"
wenzelm@10966
   956
      by cases auto
wenzelm@10966
   957
    show ?thesis
wenzelm@10966
   958
    proof (rule prefix_cases)
wenzelm@10966
   959
      assume "path_of x \<parallel> path"
wenzelm@10966
   960
      with inv root'
wenzelm@17455
   961
      have "\<And>perms. access root' path user\<^isub>1 perms = access root path user\<^isub>1 perms"
wenzelm@10966
   962
        by (simp only: access_update_other)
wenzelm@10966
   963
      with inv show "invariant root' path"
wenzelm@10966
   964
        by (simp only: invariant_def)
wenzelm@10966
   965
    next
wenzelm@10966
   966
      assume "path_of x \<le> path"
wenzelm@10966
   967
      then obtain ys where path: "path = path_of x @ ys" ..
wenzelm@10966
   968
wenzelm@10966
   969
      show ?thesis
wenzelm@10966
   970
      proof (cases ys)
wenzelm@10966
   971
        assume "ys = []"
wenzelm@17455
   972
        with tr uid inv2 inv3 lookup changed path and user\<^isub>1_not_root
wenzelm@10966
   973
        have False
wenzelm@10966
   974
          by cases (auto simp add: access_empty_lookup dest: access_some_lookup)
wenzelm@18372
   975
        then show ?thesis ..
wenzelm@10966
   976
      next
wenzelm@10966
   977
        fix z zs assume ys: "ys = z # zs"
wenzelm@10966
   978
        have "lookup root' path = lookup root path"
wenzelm@10966
   979
        proof -
wenzelm@10966
   980
          from inv2 lookup path ys
wenzelm@10966
   981
          have look: "lookup root (path_of x @ z # zs) = Some (Env att dir)"
wenzelm@10966
   982
            by (simp only:)
wenzelm@10966
   983
          then obtain att' dir' file' where
wenzelm@10966
   984
              look': "lookup root (path_of x) = Some (Env att' dir')" and
wenzelm@10966
   985
              dir': "dir' z = Some file'" and
wenzelm@10966
   986
              file': "lookup file' zs = Some (Env att dir)"
wenzelm@10966
   987
            by (blast dest: lookup_some_upper)
wenzelm@10966
   988
wenzelm@10966
   989
          from tr uid changed look' dir' obtain att'' where
wenzelm@10966
   990
              look'': "lookup root' (path_of x) = Some (Env att'' dir')"
wenzelm@10966
   991
            by cases (auto simp add: access_empty_lookup lookup_update_some
wenzelm@10966
   992
              dest: access_some_lookup)
wenzelm@10966
   993
          with dir' file' have "lookup root' (path_of x @ z # zs) =
wenzelm@10966
   994
              Some (Env att dir)"
wenzelm@10966
   995
            by (simp add: lookup_append_some)
wenzelm@10966
   996
          with look path ys show ?thesis
wenzelm@10966
   997
            by simp
wenzelm@10966
   998
        qed
wenzelm@10966
   999
        with inv show "invariant root' path"
wenzelm@10966
  1000
          by (simp only: invariant_def access_def)
wenzelm@10966
  1001
      qed
wenzelm@10966
  1002
    next
wenzelm@10966
  1003
      assume "path < path_of x"
wenzelm@10966
  1004
      then obtain y ys where path: "path_of x = path @ y # ys" ..
wenzelm@10966
  1005
wenzelm@10966
  1006
      obtain dir' where
wenzelm@10966
  1007
        lookup': "lookup root' path = Some (Env att dir')" and
wenzelm@10966
  1008
        inv2': "dir' \<noteq> empty"
wenzelm@10966
  1009
      proof (cases ys)
wenzelm@10966
  1010
        assume "ys = []"
wenzelm@10966
  1011
        with path have parent: "path_of x = path @ [y]" by simp
berghofe@17146
  1012
        with tr uid inv4 changed obtain "file" where
wenzelm@10966
  1013
            "root' = update (path_of x) (Some file) root"
wenzelm@10966
  1014
          by cases auto
wenzelm@10979
  1015
        with lookup parent have "lookup root' path = Some (Env att (dir(y\<mapsto>file)))"
wenzelm@10966
  1016
          by (simp only: update_append_some update_cons_nil_env)
wenzelm@10966
  1017
        moreover have "dir(y\<mapsto>file) \<noteq> empty" by simp
wenzelm@10966
  1018
        ultimately show ?thesis ..
wenzelm@10966
  1019
      next
wenzelm@10966
  1020
        fix z zs assume ys: "ys = z # zs"
wenzelm@10966
  1021
        with lookup root' path
wenzelm@10966
  1022
        have "lookup root' path = Some (update (y # ys) opt (Env att dir))"
wenzelm@10966
  1023
          by (simp only: update_append_some)
wenzelm@10966
  1024
        also obtain file' where
wenzelm@10966
  1025
          "update (y # ys) opt (Env att dir) = Env att (dir(y\<mapsto>file'))"
wenzelm@10966
  1026
        proof -
wenzelm@10966
  1027
          have "dir y \<noteq> None"
wenzelm@10966
  1028
          proof -
wenzelm@10966
  1029
            have "dir y = lookup (Env att dir) [y]"
wenzelm@10966
  1030
              by (simp split: option.splits)
wenzelm@10966
  1031
            also from lookup have "\<dots> = lookup root (path @ [y])"
wenzelm@10966
  1032
              by (simp only: lookup_append_some)
wenzelm@10966
  1033
            also have "\<dots> \<noteq> None"
wenzelm@10966
  1034
            proof -
wenzelm@10966
  1035
              from ys obtain us u where rev_ys: "ys = us @ [u]"
berghofe@13601
  1036
                by (cases ys rule: rev_cases) fastsimp+
wenzelm@10966
  1037
              with tr path
wenzelm@10966
  1038
              have "lookup root ((path @ [y]) @ (us @ [u])) \<noteq> None \<or>
wenzelm@10966
  1039
                  lookup root ((path @ [y]) @ us) \<noteq> None"
wenzelm@10966
  1040
                by cases (auto dest: access_some_lookup)
paulson@18447
  1041
              then show ?thesis 
nipkow@18576
  1042
                by (simp, blast dest!: lookup_some_append)
wenzelm@10966
  1043
            qed
wenzelm@10966
  1044
            finally show ?thesis .
wenzelm@10966
  1045
          qed
wenzelm@10966
  1046
          with ys show ?thesis
wenzelm@10966
  1047
            by (insert that, auto simp add: update_cons_cons_env)
wenzelm@10966
  1048
        qed
wenzelm@10966
  1049
        also have "dir(y\<mapsto>file') \<noteq> empty" by simp
wenzelm@10966
  1050
        ultimately show ?thesis ..
wenzelm@10966
  1051
      qed
wenzelm@10966
  1052
wenzelm@17455
  1053
      from lookup' have inv1': "access root' path user\<^isub>1 {} = Some (Env att dir')"
wenzelm@10966
  1054
        by (simp only: access_empty_lookup)
wenzelm@10966
  1055
wenzelm@17455
  1056
      from inv3 lookup' and not_writable user\<^isub>1_not_root
wenzelm@17455
  1057
      have "access root' path user\<^isub>1 {Writable} = None"
wenzelm@10966
  1058
        by (simp add: access_def)
wenzelm@18730
  1059
      with inv1' inv2' inv3 show ?thesis unfolding invariant_def by blast
wenzelm@10966
  1060
    qed
wenzelm@10966
  1061
  qed
wenzelm@10966
  1062
qed
wenzelm@10966
  1063
wenzelm@10966
  1064
wenzelm@10966
  1065
subsection {* Putting it all together \label{sec:unix-main-result} *}
wenzelm@10966
  1066
wenzelm@10966
  1067
text {*
wenzelm@10966
  1068
  The main result is now imminent, just by composing the three
wenzelm@10966
  1069
  invariance lemmas (see \secref{sec:unix-inv-lemmas}) according the the
wenzelm@10966
  1070
  overall procedure (see \secref{sec:unix-inv-procedure}).
wenzelm@10966
  1071
*}
wenzelm@10966
  1072
wenzelm@21029
  1073
corollary
wenzelm@13380
  1074
  assumes bogus: "init users =bogus\<Rightarrow> root"
wenzelm@17455
  1075
  shows "\<not> (\<exists>xs. (\<forall>x \<in> set xs. uid_of x = user\<^isub>1) \<and>
wenzelm@17455
  1076
    can_exec root (xs @ [Rmdir user\<^isub>1 [user\<^isub>1, name\<^isub>1]]))"
wenzelm@10966
  1077
proof -
wenzelm@13380
  1078
  from cannot_rmdir init_invariant preserve_invariant
wenzelm@13380
  1079
    and bogus show ?thesis by (rule general_procedure)
wenzelm@10966
  1080
qed
wenzelm@10966
  1081
wenzelm@10966
  1082
end
wenzelm@21029
  1083
wenzelm@21029
  1084
end