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src/HOL/Unix/Unix.thy

author | wenzelm |

Tue, 30 Jan 2001 23:53:46 +0100 | |

changeset 11004 | af8008e4de96 |

parent 10979 | 3da4543034e7 |

child 11072 | 8f47967ecc80 |

permissions | -rw-r--r-- |

tuned;

(* Title: HOL/Unix/Unix.thy ID: $Id$ Author: Markus Wenzel, TU Muenchen License: GPL (GNU GENERAL PUBLIC LICENSE) *) header {* Unix file-systems \label{sec:unix-file-system} *} theory Unix = Nested_Environment + List_Prefix: text {* We give a simple mathematical model of the basic structures underlying the Unix file-system, together with a few fundamental operations that could be imagined to be performed internally by the Unix kernel. This forms the basis for the set of Unix system-calls to be introduced later (see \secref{sec:unix-trans}), which are the actual interface offered to processes running in user-space. \medskip Basically, any Unix file is either a \emph{plain file} or a \emph{directory}, consisting of some \emph{content} plus \emph{attributes}. The content of a plain file is plain text. The content of a directory is a mapping from names to further files.\footnote{In fact, this is the only way that names get associated with files. In Unix files do \emph{not} have a name in itself. Even more, any number of names may be associated with the very same file due to \emph{hard links} (although this is excluded from our model).} Attributes include information to control various ways to access the file (read, write etc.). Our model will be quite liberal in omitting excessive detail that is easily seen to be ``irrelevant'' for the aspects of Unix file-systems to be discussed here. First of all, we ignore character and block special files, pipes, sockets, hard links, symbolic links, and mount points. *} subsection {* Names *} text {* User ids and file name components shall be represented by natural numbers (without loss of generality). We do not bother about encoding of actual names (e.g.\ strings), nor a mapping between user names and user ids as would be present in a reality. *} types uid = nat name = nat path = "name list" subsection {* Attributes *} text {* Unix file attributes mainly consist of \emph{owner} information and a number of \emph{permission} bits which control access for ``user'', ``group'', and ``others'' (see the Unix man pages @{text "chmod(2)"} and @{text "stat(2)"} for more details). \medskip Our model of file permissions only considers the ``others'' part. The ``user'' field may be omitted without loss of overall generality, since the owner is usually able to change it anyway by performing @{text chmod}.\footnote{The inclined Unix expert may try to figure out some exotic arrangements of a real-world Unix file-system such that the owner of a file is unable to apply the @{text chmod} system call.} We omit ``group'' permissions as a genuine simplification as we just do not intend to discuss a model of multiple groups and group membership, but pretend that everyone is member of a single global group.\footnote{A general HOL model of user group structures and related issues is given in \cite{Naraschewski:2001}.} *} datatype perm = Readable | Writable | Executable -- "(ignored)" types perms = "perm set" record att = owner :: uid others :: perms text {* For plain files @{term Readable} and @{term Writable} specify read and write access to the actual content, i.e.\ the string of text stored here. For directories @{term Readable} determines if the set of entry names may be accessed, and @{term Writable} controls the ability to create or delete any entries (both plain files or sub-directories). As another simplification, we ignore the @{term Executable} permission altogether. In reality it would indicate executable plain files (also known as ``binaries''), or control actual lookup of directory entries (recall that mere directory browsing is controlled via @{term Readable}). Note that the latter means that in order to perform any file-system operation whatsoever, all directories encountered on the path would have to grant @{term Executable}. We ignore this detail and pretend that all directories give @{term Executable} permission to anybody. *} subsection {* Files *} text {* In order to model the general tree structure of a Unix file-system we use the arbitrarily branching datatype @{typ "('a, 'b, 'c) env"} from the standard library of Isabelle/HOL \cite{Bauer-et-al:2001:HOL-Library}. This type provides constructors @{term Val} and @{term Env} as follows: \medskip {\def\isastyleminor{\isastyle} \begin{tabular}{l} @{term [source] "Val :: 'a \<Rightarrow> ('a, 'b, 'c) env"} \\ @{term [source] "Env :: 'b \<Rightarrow> ('c \<Rightarrow> ('a, 'b, 'c) env option) \<Rightarrow> ('a, 'b, 'c) env"} \\ \end{tabular}} \medskip Here the parameter @{typ 'a} refers to plain values occurring at leaf positions, parameter @{typ 'b} to information kept with inner branch nodes, and parameter @{typ 'c} to the branching type of the tree structure. For our purpose we use the type instance with @{typ "att \<times> string"} (representing plain files), @{typ att} (for attributes of directory nodes), and @{typ name} (for the index type of directory nodes). *} types file = "(att \<times> string, att, name) env" text {* \medskip The HOL library also provides @{term lookup} and @{term update} operations for general tree structures with the subsequent primitive recursive characterizations. \medskip {\def\isastyleminor{\isastyle} \begin{tabular}{l} @{term [source] "lookup :: ('a, 'b, 'c) env \<Rightarrow> 'c list \<Rightarrow> ('a, 'b, 'c) env option"} \\ @{term [source] "update :: 'c list \<Rightarrow> ('a, 'b, 'c) env option \<Rightarrow> ('a, 'b, 'c) env \<Rightarrow> ('a, 'b, 'c) env"} \\ \end{tabular}} @{thm [display, indent = 2, eta_contract = false] lookup_eq [no_vars]} @{thm [display, indent = 2, eta_contract = false] update_eq [no_vars]} Several further properties of these operations are proven in \cite{Bauer-et-al:2001:HOL-Library}. These will be routinely used later on without further notice. \bigskip Apparently, the elements of type @{typ file} contain an @{typ att} component in either case. We now define a few auxiliary operations to manipulate this field uniformly, following the conventions for record types in Isabelle/HOL \cite{Nipkow-et-al:2000:HOL}. *} constdefs attributes :: "file \<Rightarrow> att" "attributes file \<equiv> (case file of Val (att, text) \<Rightarrow> att | Env att dir \<Rightarrow> att)" attributes_update :: "att \<Rightarrow> file \<Rightarrow> file" "attributes_update att file \<equiv> (case file of Val (att', text) \<Rightarrow> Val (att, text) | Env att' dir \<Rightarrow> Env att dir)" lemma [simp]: "attributes (Val (att, text)) = att" by (simp add: attributes_def) lemma [simp]: "attributes (Env att dir) = att" by (simp add: attributes_def) lemma [simp]: "attributes (file \<lparr>attributes := att\<rparr>) = att" by (cases file) (simp_all add: attributes_def attributes_update_def split_tupled_all) lemma [simp]: "(Val (att, text)) \<lparr>attributes := att'\<rparr> = Val (att', text)" by (simp add: attributes_update_def) lemma [simp]: "(Env att dir) \<lparr>attributes := att'\<rparr> = Env att' dir" by (simp add: attributes_update_def) subsection {* Initial file-systems *} text {* Given a set of \emph{known users} a file-system shall be initialized by providing an empty home directory for each user, with read-only access for everyone else. (Note that we may directly use the user id as home directory name, since both types have been identified.) Certainly, the very root directory is owned by the super user (who has user id 0). *} constdefs init :: "uid set \<Rightarrow> file" "init users \<equiv> Env \<lparr>owner = 0, others = {Readable}\<rparr> (\<lambda>u. if u \<in> users then Some (Env \<lparr>owner = u, others = {Readable}\<rparr> empty) else None)" subsection {* Accessing file-systems *} text {* The main internal file-system operation is access of a file by a user, requesting a certain set of permissions. The resulting @{typ "file option"} indicates if a file had been present at the corresponding @{typ path} and if access was granted according to the permissions recorded within the file-system. Note that by the rules of Unix file-system security (e.g.\ \cite{Tanenbaum:1992}) both the super-user and owner may always access a file unconditionally (in our simplified model). *} constdefs access :: "file \<Rightarrow> path \<Rightarrow> uid \<Rightarrow> perms \<Rightarrow> file option" "access root path uid perms \<equiv> (case lookup root path of None \<Rightarrow> None | Some file \<Rightarrow> if uid = 0 \<or> uid = owner (attributes file) \<or> perms \<subseteq> others (attributes file) then Some file else None)" text {* \medskip Successful access to a certain file is the main prerequisite for system-calls to be applicable (cf.\ \secref{sec:unix-trans}). Any modification of the file-system is then performed using the basic @{term update} operation. \medskip We see that @{term access} is just a wrapper for the basic @{term lookup} function, with additional checking of attributes. Subsequently we establish a few auxiliary facts that stem from the primitive @{term lookup} used within @{term access}. *} lemma access_empty_lookup: "access root path uid {} = lookup root path" by (simp add: access_def split: option.splits) lemma access_some_lookup: "access root path uid perms = Some file \<Longrightarrow> lookup root path = Some file" by (simp add: access_def split: option.splits if_splits) lemma access_update_other: "path' \<parallel> path \<Longrightarrow> access (update path' opt root) path uid perms = access root path uid perms" proof - assume "path' \<parallel> path" then obtain y z xs ys zs where "y \<noteq> z" and "path' = xs @ y # ys" and "path = xs @ z # zs" by (blast dest: parallel_decomp) hence "lookup (update path' opt root) path = lookup root path" by (blast intro: lookup_update_other) thus ?thesis by (simp only: access_def) qed section {* File-system transitions \label{sec:unix-trans} *} subsection {* Unix system calls \label{sec:unix-syscall} *} text {* According to established operating system design (cf.\ \cite{Tanenbaum:1992}) user space processes may only initiate system operations by a fixed set of \emph{system-calls}. This enables the kernel to enforce certain security policies in the first place.\footnote{Incidently, this is the very same principle employed by any ``LCF-style'' theorem proving system according to Milner's principle of ``correctness by construction'', such as Isabelle/HOL itself.} \medskip In our model of Unix we give a fixed datatype @{text operation} for the syntax of system-calls, together with an inductive definition of file-system state transitions of the form @{text "root \<midarrow>x\<rightarrow> root'"} for the operational semantics. *} datatype operation = Read uid string path | Write uid string path | Chmod uid perms path | Creat uid perms path | Unlink uid path | Mkdir uid perms path | Rmdir uid path | Readdir uid "name set" path text {* The @{typ uid} field of an operation corresponds to the \emph{effective user id} of the underlying process, although our model never mentions processes explicitly. The other parameters are provided as arguments by the caller; the @{term path} one is common to all kinds of system-calls. *} consts uid_of :: "operation \<Rightarrow> uid" primrec "uid_of (Read uid text path) = uid" "uid_of (Write uid text path) = uid" "uid_of (Chmod uid perms path) = uid" "uid_of (Creat uid perms path) = uid" "uid_of (Unlink uid path) = uid" "uid_of (Mkdir uid path perms) = uid" "uid_of (Rmdir uid path) = uid" "uid_of (Readdir uid names path) = uid" consts path_of :: "operation \<Rightarrow> path" primrec "path_of (Read uid text path) = path" "path_of (Write uid text path) = path" "path_of (Chmod uid perms path) = path" "path_of (Creat uid perms path) = path" "path_of (Unlink uid path) = path" "path_of (Mkdir uid perms path) = path" "path_of (Rmdir uid path) = path" "path_of (Readdir uid names path) = path" text {* \medskip Note that we have omitted explicit @{text Open} and @{text Close} operations, pretending that @{term Read} and @{term Write} would already take care of this behind the scenes. Thus we have basically treated actual sequences of real system-calls @{text open}--@{text read}/@{text write}--@{text close} as atomic. In principle, this could make big a difference in a model with explicit concurrent processes. On the other hand, even on a real Unix system the exact scheduling of concurrent @{text open} and @{text close} operations does \emph{not} directly affect the success of corresponding @{text read} or @{text write}. Unix allows several processes to have files opened at the same time, even for writing! Certainly, the result from reading the contents later may be hard to predict, but the system-calls involved here will succeed in any case. \bigskip The operational semantics of system calls is now specified via transitions of the file-system configuration. This is expressed as an inductive relation (although there is no actual recursion involved here). *} consts transition :: "(file \<times> operation \<times> file) set" syntax "_transition" :: "file \<Rightarrow> operation \<Rightarrow> file \<Rightarrow> bool" ("_ \<midarrow>_\<rightarrow> _" [90, 1000, 90] 100) translations "root \<midarrow>x\<rightarrow> root'" \<rightleftharpoons> "(root, x, root') \<in> transition" inductive transition intros read: "access root path uid {Readable} = Some (Val (att, text)) \<Longrightarrow> root \<midarrow>(Read uid text path)\<rightarrow> root" write: "access root path uid {Writable} = Some (Val (att, text')) \<Longrightarrow> root \<midarrow>(Write uid text path)\<rightarrow> update path (Some (Val (att, text))) root" chmod: "access root path uid {} = Some file \<Longrightarrow> uid = 0 \<or> uid = owner (attributes file) \<Longrightarrow> root \<midarrow>(Chmod uid perms path)\<rightarrow> update path (Some (file \<lparr>attributes := attributes file \<lparr>others := perms\<rparr>\<rparr>)) root" creat: "path = parent_path @ [name] \<Longrightarrow> access root parent_path uid {Writable} = Some (Env att parent) \<Longrightarrow> access root path uid {} = None \<Longrightarrow> root \<midarrow>(Creat uid perms path)\<rightarrow> update path (Some (Val (\<lparr>owner = uid, others = perms\<rparr>, []))) root" unlink: "path = parent_path @ [name] \<Longrightarrow> access root parent_path uid {Writable} = Some (Env att parent) \<Longrightarrow> access root path uid {} = Some (Val plain) \<Longrightarrow> root \<midarrow>(Unlink uid path)\<rightarrow> update path None root" mkdir: "path = parent_path @ [name] \<Longrightarrow> access root parent_path uid {Writable} = Some (Env att parent) \<Longrightarrow> access root path uid {} = None \<Longrightarrow> root \<midarrow>(Mkdir uid perms path)\<rightarrow> update path (Some (Env \<lparr>owner = uid, others = perms\<rparr> empty)) root" rmdir: "path = parent_path @ [name] \<Longrightarrow> access root parent_path uid {Writable} = Some (Env att parent) \<Longrightarrow> access root path uid {} = Some (Env att' empty) \<Longrightarrow> root \<midarrow>(Rmdir uid path)\<rightarrow> update path None root" readdir: "access root path uid {Readable} = Some (Env att dir) \<Longrightarrow> names = dom dir \<Longrightarrow> root \<midarrow>(Readdir uid names path)\<rightarrow> root" text {* \medskip Certainly, the above specification is central to the whole formal development. Any of the results to be established later on are only meaningful to the outside world if this transition system provides an adequate model of real Unix systems. This kind of ``reality-check'' of a formal model is the well-known problem of \emph{validation}. If in doubt, one may consider to compare our definition with the informal specifications given the corresponding Unix man pages, or even peek at an actual implementation such as \cite{Torvalds-et-al:Linux}. Another common way to gain confidence into the formal model is to run simple simulations (see \secref{sec:unix-examples}), and check the results with that of experiments performed on a real Unix system. *} subsection {* Basic properties of single transitions \label{sec:unix-single-trans} *} text {* The transition system @{text "root \<midarrow>x\<rightarrow> root'"} defined above determines a unique result @{term root'} from given @{term root} and @{term x} (this is holds rather trivially, since there is even only one clause for each operation). This uniqueness statement will simplify our subsequent development to some extent, since we only have to reason about a partial function rather than a general relation. *} theorem transition_uniq: "root \<midarrow>x\<rightarrow> root' \<Longrightarrow> root \<midarrow>x\<rightarrow> root'' \<Longrightarrow> root' = root''" proof - assume root: "root \<midarrow>x\<rightarrow> root'" assume "root \<midarrow>x\<rightarrow> root''" thus "root' = root''" proof cases case read with root show ?thesis by cases auto next case write with root show ?thesis by cases auto next case chmod with root show ?thesis by cases auto next case creat with root show ?thesis by cases auto next case unlink with root show ?thesis by cases auto next case mkdir with root show ?thesis by cases auto next case rmdir with root show ?thesis by cases auto next case readdir with root show ?thesis by cases auto qed qed text {* \medskip Apparently, file-system transitions are \emph{type-safe} in the sense that the result of transforming an actual directory yields again a directory. *} theorem transition_type_safe: "root \<midarrow>x\<rightarrow> root' \<Longrightarrow> \<exists>att dir. root = Env att dir \<Longrightarrow> \<exists>att dir. root' = Env att dir" proof - assume tr: "root \<midarrow>x\<rightarrow> root'" assume inv: "\<exists>att dir. root = Env att dir" show ?thesis proof (cases "path_of x") case Nil with tr inv show ?thesis by cases (auto simp add: access_def split: if_splits) next case Cons from tr obtain opt where "root' = root \<or> root' = update (path_of x) opt root" by cases auto with inv Cons show ?thesis by (auto simp add: update_eq split: list.splits) qed qed text {* The previous result may be seen as the most basic invariant on the file-system state that is enforced by any proper kernel implementation. So user processes --- being bound to the system-call interface --- may never mess up a file-system such that the root becomes a plain file instead of a directory, which would be a strange situation indeed. *} subsection {* Iterated transitions *} text {* Iterated system transitions via finite sequences of system operations are modeled inductively as follows. In a sense, this relation describes the cumulative effect of the sequence of system-calls issued by a number of running processes over a finite amount of time. *} consts transitions :: "(file \<times> operation list \<times> file) set" syntax "_transitions" :: "file \<Rightarrow> operation list \<Rightarrow> file \<Rightarrow> bool" ("_ =_\<Rightarrow> _" [90, 1000, 90] 100) translations "root =xs\<Rightarrow> root'" \<rightleftharpoons> "(root, xs, root') \<in> transitions" inductive transitions intros nil: "root =[]\<Rightarrow> root" cons: "root \<midarrow>x\<rightarrow> root' \<Longrightarrow> root' =xs\<Rightarrow> root'' \<Longrightarrow> root =(x # xs)\<Rightarrow> root''" text {* \medskip We establish a few basic facts relating iterated transitions with single ones, according to the recursive structure of lists. *} lemma transitions_nil_eq: "root =[]\<Rightarrow> root' = (root = root')" proof assume "root =[]\<Rightarrow> root'" thus "root = root'" by cases simp_all next assume "root = root'" thus "root =[]\<Rightarrow> root'" by (simp only: transitions.nil) qed lemma transitions_cons_eq: "root =(x # xs)\<Rightarrow> root'' = (\<exists>root'. root \<midarrow>x\<rightarrow> root' \<and> root' =xs\<Rightarrow> root'')" proof assume "root =(x # xs)\<Rightarrow> root''" thus "\<exists>root'. root \<midarrow>x\<rightarrow> root' \<and> root' =xs\<Rightarrow> root''" by cases auto next assume "\<exists>root'. root \<midarrow>x\<rightarrow> root' \<and> root' =xs\<Rightarrow> root''" thus "root =(x # xs)\<Rightarrow> root''" by (blast intro: transitions.cons) qed text {* The next two rules show how to ``destruct'' known transition sequences. Note that the second one actually relies on the uniqueness property of the basic transition system (see \secref{sec:unix-single-trans}). *} lemma transitions_nilD: "root =[]\<Rightarrow> root' \<Longrightarrow> root' = root" by (simp add: transitions_nil_eq) lemma transitions_consD: "root =(x # xs)\<Rightarrow> root'' \<Longrightarrow> root \<midarrow>x\<rightarrow> root' \<Longrightarrow> root' =xs\<Rightarrow> root''" proof - assume "root =(x # xs)\<Rightarrow> root''" then obtain r' where r': "root \<midarrow>x\<rightarrow> r'" and root'': "r' =xs\<Rightarrow> root''" by cases simp_all assume "root \<midarrow>x\<rightarrow> root'" with r' have "r' = root'" by (rule transition_uniq) with root'' show "root' =xs\<Rightarrow> root''" by simp qed text {* \medskip The following fact shows how an invariant @{term Q} of single transitions with property @{term P} may be transferred to iterated transitions. The proof is rather obvious by rule induction over the definition of @{term "root =xs\<Rightarrow> root'"}. *} lemma transitions_invariant: "(\<And>r x r'. r \<midarrow>x\<rightarrow> r' \<Longrightarrow> Q r \<Longrightarrow> P x \<Longrightarrow> Q r') \<Longrightarrow> root =xs\<Rightarrow> root' \<Longrightarrow> Q root \<Longrightarrow> \<forall>x \<in> set xs. P x \<Longrightarrow> Q root'" proof - assume r: "\<And>r x r'. r \<midarrow>x\<rightarrow> r' \<Longrightarrow> Q r \<Longrightarrow> P x \<Longrightarrow> Q r'" assume "root =xs\<Rightarrow> root'" thus "Q root \<Longrightarrow> (\<forall>x \<in> set xs. P x) \<Longrightarrow> Q root'" (is "PROP ?P root xs root'") proof (induct ?P root xs root') fix root assume "Q root" thus "Q root" . next fix root root' root'' and x xs assume root': "root \<midarrow>x\<rightarrow> root'" assume hyp: "PROP ?P root' xs root''" assume Q: "Q root" assume P: "\<forall>x \<in> set (x # xs). P x" hence "P x" by simp with root' Q have Q': "Q root'" by (rule r) from P have "\<forall>x \<in> set xs. P x" by simp with Q' show "Q root''" by (rule hyp) qed qed text {* As an example of applying the previous result, we transfer the basic type-safety property (see \secref{sec:unix-single-trans}) from single transitions to iterated ones, which is a rather obvious result indeed. *} theorem transitions_type_safe: "root =xs\<Rightarrow> root' \<Longrightarrow> \<exists>att dir. root = Env att dir \<Longrightarrow> \<exists>att dir. root' = Env att dir" proof - case antecedent with transition_type_safe show ?thesis proof (rule transitions_invariant) show "\<forall>x \<in> set xs. True" by blast qed qed section {* Executable sequences *} text {* An inductively defined relation such as the one of @{text "root \<midarrow>x\<rightarrow> root'"} (see \secref{sec:unix-syscall}) has two main aspects. First of all, the resulting system admits a certain set of transition rules (introductions) as given in the specification. Furthermore, there is an explicit least-fixed-point construction involved, which results in induction (and case-analysis) rules to eliminate known transitions in an exhaustive manner. \medskip Subsequently, we explore our transition system in an experimental style, mainly using the introduction rules with basic algebraic properties of the underlying structures. The technique closely resembles that of Prolog combined with functional evaluation in a very simple manner. Just as the ``closed-world assumption'' is left implicit in Prolog, we do not refer to induction over the whole transition system here. So this is still purely positive reasoning about possible executions; exhaustive reasoning will be employed only later on (see \secref{sec:unix-bogosity}), when we shall demonstrate that certain behavior is \emph{not} possible. *} subsection {* Possible transitions *} text {* Rather obviously, a list of system operations can be executed within a certain state if there is a result state reached by an iterated transition. *} constdefs can_exec :: "file \<Rightarrow> operation list \<Rightarrow> bool" "can_exec root xs \<equiv> \<exists>root'. root =xs\<Rightarrow> root'" lemma can_exec_nil: "can_exec root []" by (unfold can_exec_def) (blast intro: transitions.intros) lemma can_exec_cons: "root \<midarrow>x\<rightarrow> root' \<Longrightarrow> can_exec root' xs \<Longrightarrow> can_exec root (x # xs)" by (unfold can_exec_def) (blast intro: transitions.intros) text {* \medskip In case that we already know that a certain sequence can be executed we may destruct it backwardly into individual transitions. *} lemma can_exec_snocD: "\<And>root. can_exec root (xs @ [y]) \<Longrightarrow> \<exists>root' root''. root =xs\<Rightarrow> root' \<and> root' \<midarrow>y\<rightarrow> root''" (is "PROP ?P xs" is "\<And>root. ?A root xs \<Longrightarrow> ?C root xs") proof (induct xs) fix root { assume "?A root []" thus "?C root []" by (simp add: can_exec_def transitions_nil_eq transitions_cons_eq) next fix x xs assume hyp: "PROP ?P xs" assume asm: "?A root (x # xs)" show "?C root (x # xs)" proof - from asm obtain r root'' where x: "root \<midarrow>x\<rightarrow> r" and xs_y: "r =(xs @ [y])\<Rightarrow> root''" by (auto simp add: can_exec_def transitions_nil_eq transitions_cons_eq) from xs_y hyp obtain root' r' where xs: "r =xs\<Rightarrow> root'" and y: "root' \<midarrow>y\<rightarrow> r'" by (auto simp add: can_exec_def) from x xs have "root =(x # xs)\<Rightarrow> root'" by (rule transitions.cons) with y show ?thesis by blast qed } qed subsection {* Example executions \label{sec:unix-examples} *} text {* We are now ready to perform a few experiments within our formal model of Unix system-calls. The common technique is to alternate introduction rules of the transition system (see \secref{sec:unix-trans}), and steps to solve any emerging side conditions using algebraic properties of the underlying file-system structures (see \secref{sec:unix-file-system}). *} lemmas eval = access_def init_def theorem "u \<in> users \<Longrightarrow> can_exec (init users) [Mkdir u perms [u, name]]" apply (rule can_exec_cons) -- {* back-chain @{term can_exec} (of @{term [source] Cons}) *} apply (rule mkdir) -- {* back-chain @{term Mkdir} *} apply (force simp add: eval)+ -- {* solve preconditions of @{term Mkdir} *} apply (simp add: eval) -- {* peek at resulting dir (optional) *} txt {* @{subgoals [display]} *} apply (rule can_exec_nil) -- {* back-chain @{term can_exec} (of @{term [source] Nil}) *} done text {* By inspecting the result shown just before the final step above, we may gain confidence that our specification of Unix system-calls actually makes sense. Further common errors are usually exhibited when preconditions of transition rules fail unexpectedly. \medskip Here are a few further experiments, using the same techniques as before. *} theorem "u \<in> users \<Longrightarrow> can_exec (init users) [Creat u perms [u, name], Unlink u [u, name]]" apply (rule can_exec_cons) apply (rule creat) apply (force simp add: eval)+ apply (simp add: eval) apply (rule can_exec_cons) apply (rule unlink) apply (force simp add: eval)+ apply (simp add: eval) txt {* peek at result: @{subgoals [display]} *} apply (rule can_exec_nil) done theorem "u \<in> users \<Longrightarrow> Writable \<in> perms1 \<Longrightarrow> Readable \<in> perms2 \<Longrightarrow> name3 \<noteq> name4 \<Longrightarrow> can_exec (init users) [Mkdir u perms1 [u, name1], Mkdir u' perms2 [u, name1, name2], Creat u' perms3 [u, name1, name2, name3], Creat u' perms3 [u, name1, name2, name4], Readdir u {name3, name4} [u, name1, name2]]" apply (rule can_exec_cons, rule transition.intros, (force simp add: eval)+, (simp add: eval)?)+ txt {* peek at result: @{subgoals [display]} *} apply (rule can_exec_nil) done theorem "u \<in> users \<Longrightarrow> Writable \<in> perms1 \<Longrightarrow> Readable \<in> perms3 \<Longrightarrow> can_exec (init users) [Mkdir u perms1 [u, name1], Mkdir u' perms2 [u, name1, name2], Creat u' perms3 [u, name1, name2, name3], Write u' ''foo'' [u, name1, name2, name3], Read u ''foo'' [u, name1, name2, name3]]" apply (rule can_exec_cons, rule transition.intros, (force simp add: eval)+, (simp add: eval)?)+ txt {* peek at result: @{subgoals [display]} *} apply (rule can_exec_nil) done section {* Odd effects --- treated formally \label{sec:unix-bogosity} *} text {* We are now ready to give a completely formal treatment of the slightly odd situation discussed in the introduction (see \secref{sec:unix-intro}): the file-system can easily reach a state where a user is unable to remove his very own directory, because it is still populated by items placed there by another user in an uncouth manner. *} subsection {* The general procedure \label{sec:unix-inv-procedure} *} text {* The following theorem expresses the general procedure we are following to achieve the main result. *} theorem general_procedure: "(\<And>r r'. Q r \<Longrightarrow> r \<midarrow>y\<rightarrow> r' \<Longrightarrow> False) \<Longrightarrow> (\<And>root. init users =bs\<Rightarrow> root \<Longrightarrow> Q root) \<Longrightarrow> (\<And>r x r'. r \<midarrow>x\<rightarrow> r' \<Longrightarrow> Q r \<Longrightarrow> P x \<Longrightarrow> Q r') \<Longrightarrow> init users =bs\<Rightarrow> root \<Longrightarrow> \<not> (\<exists>xs. (\<forall>x \<in> set xs. P x) \<and> can_exec root (xs @ [y]))" proof - assume cannot_y: "\<And>r r'. Q r \<Longrightarrow> r \<midarrow>y\<rightarrow> r' \<Longrightarrow> False" assume init_inv: "\<And>root. init users =bs\<Rightarrow> root \<Longrightarrow> Q root" assume preserve_inv: "\<And>r x r'. r \<midarrow>x\<rightarrow> r' \<Longrightarrow> Q r \<Longrightarrow> P x \<Longrightarrow> Q r'" assume init_result: "init users =bs\<Rightarrow> root" { fix xs assume Ps: "\<forall>x \<in> set xs. P x" assume can_exec: "can_exec root (xs @ [y])" then obtain root' root'' where xs: "root =xs\<Rightarrow> root'" and y: "root' \<midarrow>y\<rightarrow> root''" by (blast dest: can_exec_snocD) from init_result have "Q root" by (rule init_inv) from preserve_inv xs this Ps have "Q root'" by (rule transitions_invariant) from this y have False by (rule cannot_y) } thus ?thesis by blast qed text {* Here @{prop "P x"} refers to the restriction on file-system operations that are admitted after having reached the critical configuration; according to the problem specification this will become @{prop "uid_of x = user1"} later on. Furthermore, @{term y} refers to the operations we claim to be impossible to perform afterwards, we will take @{term Rmdir} later. Moreover @{term Q} is a suitable (auxiliary) invariant over the file-system; choosing @{term Q} adequately is very important to make the proof work (see \secref{sec:unix-inv-lemmas}). *} subsection {* The particular setup *} text {* We introduce a few global declarations and axioms to describe our particular setup considered here. Thus we avoid excessive use of local parameters in the subsequent development. *} consts users :: "uid set" user1 :: uid user2 :: uid name1 :: name name2 :: name name3 :: name perms1 :: perms perms2 :: perms axioms user1_known: "user1 \<in> users" user1_not_root: "user1 \<noteq> 0" users_neq: "user1 \<noteq> user2" perms1_writable: "Writable \<in> perms1" perms2_not_writable: "Writable \<notin> perms2" lemmas "setup" = user1_known user1_not_root users_neq perms1_writable perms2_not_writable text {* \medskip The @{term bogus} operations are the ones that lead into the uncouth situation; @{term bogus_path} is the key position within the file-system where things go awry. *} constdefs bogus :: "operation list" "bogus \<equiv> [Mkdir user1 perms1 [user1, name1], Mkdir user2 perms2 [user1, name1, name2], Creat user2 perms2 [user1, name1, name2, name3]]" bogus_path :: path "bogus_path \<equiv> [user1, name1, name2]" subsection {* Invariance lemmas \label{sec:unix-inv-lemmas} *} text {* The following invariant over the root file-system describes the bogus situation in an abstract manner: located at a certain @{term path} within the file-system is a non-empty directory that is neither owned and nor writable by @{term user1}. *} constdefs invariant :: "file \<Rightarrow> path \<Rightarrow> bool" "invariant root path \<equiv> (\<exists>att dir. access root path user1 {} = Some (Env att dir) \<and> dir \<noteq> empty \<and> user1 \<noteq> owner att \<and> access root path user1 {Writable} = None)" text {* Following the general procedure (see \secref{sec:unix-inv-procedure}) we will now establish the three key lemmas required to yield the final result. \begin{enumerate} \item The invariant is sufficiently strong to entail the pathological case that @{term user1} is unable to remove the (owned) directory at @{term "[user1, name1]"}. \item The invariant does hold after having executed the @{term bogus} list of operations (starting with an initial file-system configuration). \item The invariant is preserved by any file-system operation performed by @{term user1} alone, without any help by other users. \end{enumerate} As the invariant appears both as assumptions and conclusions in the course of proof, its formulation is rather critical for the whole development to work out properly. In particular, the third step is very sensitive to the invariant being either too strong or too weak. Moreover the invariant has to be sufficiently abstract, lest the proof become cluttered by confusing detail. \medskip The first two lemmas are technically straight forward --- we just have to inspect rather special cases. *} lemma cannot_rmdir: "invariant root bogus_path \<Longrightarrow> root \<midarrow>(Rmdir user1 [user1, name1])\<rightarrow> root' \<Longrightarrow> False" proof - assume "invariant root bogus_path" then obtain file where "access root bogus_path user1 {} = Some file" by (unfold invariant_def) blast moreover assume "root \<midarrow>(Rmdir user1 [user1, name1])\<rightarrow> root'" then obtain att where "access root [user1, name1] user1 {} = Some (Env att empty)" by cases auto hence "access root ([user1, name1] @ [name2]) user1 {} = empty name2" by (simp only: access_empty_lookup lookup_append_some) simp ultimately show False by (simp add: bogus_path_def) qed lemma init_invariant: "init users =bogus\<Rightarrow> root \<Longrightarrow> invariant root bogus_path" proof - note eval = "setup" access_def init_def case antecedent thus ?thesis apply (unfold bogus_def bogus_path_def) apply (drule transitions_consD, rule transition.intros, (force simp add: eval)+, (simp add: eval)?)+ -- "evaluate all operations" apply (drule transitions_nilD) -- "reach final result" apply (simp add: invariant_def eval) -- "check the invariant" done qed text {* \medskip At last we are left with the main effort to show that the ``bogosity'' invariant is preserved by any file-system operation performed by @{term user1} alone. Note that this holds for any @{term path} given, the particular @{term bogus_path} is not required here. *} lemma preserve_invariant: "root \<midarrow>x\<rightarrow> root' \<Longrightarrow> invariant root path \<Longrightarrow> uid_of x = user1 \<Longrightarrow> invariant root' path" proof - assume tr: "root \<midarrow>x\<rightarrow> root'" assume inv: "invariant root path" assume uid: "uid_of x = user1" from inv obtain att dir where inv1: "access root path user1 {} = Some (Env att dir)" and inv2: "dir \<noteq> empty" and inv3: "user1 \<noteq> owner att" and inv4: "access root path user1 {Writable} = None" by (auto simp add: invariant_def) from inv1 have lookup: "lookup root path = Some (Env att dir)" by (simp only: access_empty_lookup) from inv1 inv3 inv4 and user1_not_root have not_writable: "Writable \<notin> others att" by (auto simp add: access_def split: option.splits if_splits) show ?thesis proof cases assume "root' = root" with inv show "invariant root' path" by (simp only:) next assume changed: "root' \<noteq> root" with tr obtain opt where root': "root' = update (path_of x) opt root" by cases auto show ?thesis proof (rule prefix_cases) assume "path_of x \<parallel> path" with inv root' have "\<And>perms. access root' path user1 perms = access root path user1 perms" by (simp only: access_update_other) with inv show "invariant root' path" by (simp only: invariant_def) next assume "path_of x \<le> path" then obtain ys where path: "path = path_of x @ ys" .. show ?thesis proof (cases ys) assume "ys = []" with tr uid inv2 inv3 lookup changed path and user1_not_root have False by cases (auto simp add: access_empty_lookup dest: access_some_lookup) thus ?thesis .. next fix z zs assume ys: "ys = z # zs" have "lookup root' path = lookup root path" proof - from inv2 lookup path ys have look: "lookup root (path_of x @ z # zs) = Some (Env att dir)" by (simp only:) then obtain att' dir' file' where look': "lookup root (path_of x) = Some (Env att' dir')" and dir': "dir' z = Some file'" and file': "lookup file' zs = Some (Env att dir)" by (blast dest: lookup_some_upper) from tr uid changed look' dir' obtain att'' where look'': "lookup root' (path_of x) = Some (Env att'' dir')" by cases (auto simp add: access_empty_lookup lookup_update_some dest: access_some_lookup) with dir' file' have "lookup root' (path_of x @ z # zs) = Some (Env att dir)" by (simp add: lookup_append_some) with look path ys show ?thesis by simp qed with inv show "invariant root' path" by (simp only: invariant_def access_def) qed next assume "path < path_of x" then obtain y ys where path: "path_of x = path @ y # ys" .. obtain dir' where lookup': "lookup root' path = Some (Env att dir')" and inv2': "dir' \<noteq> empty" proof (cases ys) assume "ys = []" with path have parent: "path_of x = path @ [y]" by simp with tr uid inv4 changed obtain file where "root' = update (path_of x) (Some file) root" by cases auto with lookup parent have "lookup root' path = Some (Env att (dir(y\<mapsto>file)))" by (simp only: update_append_some update_cons_nil_env) moreover have "dir(y\<mapsto>file) \<noteq> empty" by simp ultimately show ?thesis .. next fix z zs assume ys: "ys = z # zs" with lookup root' path have "lookup root' path = Some (update (y # ys) opt (Env att dir))" by (simp only: update_append_some) also obtain file' where "update (y # ys) opt (Env att dir) = Env att (dir(y\<mapsto>file'))" proof - have "dir y \<noteq> None" proof - have "dir y = lookup (Env att dir) [y]" by (simp split: option.splits) also from lookup have "\<dots> = lookup root (path @ [y])" by (simp only: lookup_append_some) also have "\<dots> \<noteq> None" proof - from ys obtain us u where rev_ys: "ys = us @ [u]" by (cases ys rule: rev_cases) auto with tr path have "lookup root ((path @ [y]) @ (us @ [u])) \<noteq> None \<or> lookup root ((path @ [y]) @ us) \<noteq> None" by cases (auto dest: access_some_lookup) thus ?thesis by (blast dest!: lookup_some_append) qed finally show ?thesis . qed with ys show ?thesis by (insert that, auto simp add: update_cons_cons_env) qed also have "dir(y\<mapsto>file') \<noteq> empty" by simp ultimately show ?thesis .. qed from lookup' have inv1': "access root' path user1 {} = Some (Env att dir')" by (simp only: access_empty_lookup) from inv3 lookup' and not_writable user1_not_root have "access root' path user1 {Writable} = None" by (simp add: access_def) with inv1' inv2' inv3 show ?thesis by (unfold invariant_def) blast qed qed qed subsection {* Putting it all together \label{sec:unix-main-result} *} text {* The main result is now imminent, just by composing the three invariance lemmas (see \secref{sec:unix-inv-lemmas}) according the the overall procedure (see \secref{sec:unix-inv-procedure}). *} theorem main: "init users =bogus\<Rightarrow> root \<Longrightarrow> \<not> (\<exists>xs. (\<forall>x \<in> set xs. uid_of x = user1) \<and> can_exec root (xs @ [Rmdir user1 [user1, name1]]))" proof - case antecedent with cannot_rmdir init_invariant preserve_invariant show ?thesis by (rule general_procedure) qed end