(* Title: HOL/Unix/Unix.thy Author: Markus Wenzel, TU Muenchen *) section ‹Unix file-systems \label{sec:unix-file-system}› theory Unix imports Nested_Environment "HOL-Library.Sublist" begin 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. ┉ Basically, any Unix file is either a ∗‹plain file› or a ∗‹directory›, consisting of some ∗‹content› plus ∗‹attributes›. The content of a plain file is plain text. The content of a directory is a mapping from names to further files.⁋‹In fact, this is the only way that names get associated with files. In Unix files do ∗‹not› have a name in itself. Even more, any number of names may be associated with the very same file due to ∗‹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. › type_synonym uid = nat type_synonym name = nat type_synonym path = "name list" subsection ‹Attributes› text ‹ Unix file attributes mainly consist of ∗‹owner› information and a number of ∗‹permission› bits which control access for ``user'', ``group'', and ``others'' (see the Unix man pages ‹chmod(2)› and ‹stat(2)› for more details). ┉ 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 ‹chmod›.⁋‹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 ‹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.⁋‹A general HOL model of user group structures and related issues is given in @{cite "Naraschewski:2001"}.› › datatype perm = Readable | Writable | Executable ― ‹(ignored)› type_synonym 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:2002:HOL-Library"}. This type provides constructors @{term Val} and @{term Env} as follows: ┉ {\def\isastyleminor{\isastyle} \begin{tabular}{l} @{term [source] "Val :: 'a ⇒ ('a, 'b, 'c) env"} \\ @{term [source] "Env :: 'b ⇒ ('c ⇒ ('a, 'b, 'c) env option) ⇒ ('a, 'b, 'c) env"} \\ \end{tabular}} ┉ 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 × string"} (representing plain files), @{typ att} (for attributes of directory nodes), and @{typ name} (for the index type of directory nodes). › type_synonym "file" = "(att × string, att, name) env" text ‹ ┉ The HOL library also provides @{term lookup} and @{term update} operations for general tree structures with the subsequent primitive recursive characterizations. ┉ {\def\isastyleminor{\isastyle} \begin{tabular}{l} @{term [source] "lookup :: ('a, 'b, 'c) env ⇒ 'c list ⇒ ('a, 'b, 'c) env option"} \\ @{term [source] "update :: 'c list ⇒ ('a, 'b, 'c) env option ⇒ ('a, 'b, 'c) env ⇒ ('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:2002:HOL-Library"}. These will be routinely used later on without further notice. ━ 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"}. › definition "attributes file = (case file of Val (att, text) ⇒ att | Env att dir ⇒ att)" definition "map_attributes f file = (case file of Val (att, text) ⇒ Val (f att, text) | Env att dir ⇒ Env (f 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 (map_attributes f file) = f (attributes file)" by (cases "file") (simp_all add: attributes_def map_attributes_def split_tupled_all) lemma [simp]: "map_attributes f (Val (att, text)) = Val (f att, text)" by (simp add: map_attributes_def) lemma [simp]: "map_attributes f (Env att dir) = Env (f att) dir" by (simp add: map_attributes_def) subsection ‹Initial file-systems› text ‹ Given a set of ∗‹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). › definition "init users = Env ⦇owner = 0, others = {Readable}⦈ (λu. if u ∈ users then Some (Env ⦇owner = u, others = {Readable}⦈ Map.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). › definition "access root path uid perms = (case lookup root path of None ⇒ None | Some file ⇒ if uid = 0 ∨ uid = owner (attributes file) ∨ perms ⊆ others (attributes file) then Some file else None)" text ‹ ┉ 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. ┉ 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 ⟹ lookup root path = Some file" by (simp add: access_def split: option.splits if_splits) lemma access_update_other: assumes parallel: "path' ∥ path" shows "access (update path' opt root) path uid perms = access root path uid perms" proof - from parallel obtain y z xs ys zs where "y ≠ z" and "path' = xs @ y # ys" and "path = xs @ z # zs" by (blast dest: parallel_decomp) then have "lookup (update path' opt root) path = lookup root path" by (blast intro: lookup_update_other) then show ?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 ∗‹system-calls›. This enables the kernel to enforce certain security policies in the first place.⁋‹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.› ┉ In our model of Unix we give a fixed datatype ‹operation› for the syntax of system-calls, together with an inductive definition of file-system state transitions of the form ‹root ─x→ 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 ∗‹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. › primrec uid_of :: "operation ⇒ uid" where "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" primrec path_of :: "operation ⇒ path" where "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 ‹ ┉ Note that we have omitted explicit ‹Open› and ‹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 ‹open›--‹read›/‹write›--‹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 ‹open› and ‹close› operations does ∗‹not› directly affect the success of corresponding ‹read› or ‹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. ━ 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). › inductive transition :: "file ⇒ operation ⇒ file ⇒ bool" ("_ ─_→ _" [90, 1000, 90] 100) where read: "root ─(Read uid text path)→ root" if "access root path uid {Readable} = Some (Val (att, text))" | "write": "root ─(Write uid text path)→ update path (Some (Val (att, text))) root" if "access root path uid {Writable} = Some (Val (att, text'))" | chmod: "root ─(Chmod uid perms path)→ update path (Some (map_attributes (others_update (λ_. perms)) file)) root" if "access root path uid {} = Some file" and "uid = 0 ∨ uid = owner (attributes file)" | creat: "root ─(Creat uid perms path)→ update path (Some (Val (⦇owner = uid, others = perms⦈, []))) root" if "path = parent_path @ [name]" and "access root parent_path uid {Writable} = Some (Env att parent)" and "access root path uid {} = None" | unlink: "root ─(Unlink uid path)→ update path None root" if "path = parent_path @ [name]" and "access root parent_path uid {Writable} = Some (Env att parent)" and "access root path uid {} = Some (Val plain)" | mkdir: "root ─(Mkdir uid perms path)→ update path (Some (Env ⦇owner = uid, others = perms⦈ Map.empty)) root" if "path = parent_path @ [name]" and "access root parent_path uid {Writable} = Some (Env att parent)" and "access root path uid {} = None" | rmdir: "root ─(Rmdir uid path)→ update path None root" if "path = parent_path @ [name]" and "access root parent_path uid {Writable} = Some (Env att parent)" and "access root path uid {} = Some (Env att' Map.empty)" | readdir: "root ─(Readdir uid names path)→ root" if "access root path uid {Readable} = Some (Env att dir)" and "names = dom dir" text ‹ ┉ 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 ∗‹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 ‹root ─x→ 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: assumes root': "root ─x→ root'" and root'': "root ─x→ root''" shows "root' = root''" using 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 blast+ qed text ‹ ┉ Apparently, file-system transitions are ∗‹type-safe› in the sense that the result of transforming an actual directory yields again a directory. › theorem transition_type_safe: assumes tr: "root ─x→ root'" and inv: "∃att dir. root = Env att dir" shows "∃att dir. root' = Env att dir" 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 ∨ 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 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. › inductive transitions :: "file ⇒ operation list ⇒ file ⇒ bool" ("_ ═_⇒ _" [90, 1000, 90] 100) where nil: "root ═[]⇒ root" | cons: "root ═(x # xs)⇒ root''" if "root ─x→ root'" and "root' ═xs⇒ root''" text ‹ ┉ We establish a few basic facts relating iterated transitions with single ones, according to the recursive structure of lists. › lemma transitions_nil_eq: "root ═[]⇒ root' ⟷ root = root'" proof assume "root ═[]⇒ root'" then show "root = root'" by cases simp_all next assume "root = root'" then show "root ═[]⇒ root'" by (simp only: transitions.nil) qed lemma transitions_cons_eq: "root ═(x # xs)⇒ root'' ⟷ (∃root'. root ─x→ root' ∧ root' ═xs⇒ root'')" proof assume "root ═(x # xs)⇒ root''" then show "∃root'. root ─x→ root' ∧ root' ═xs⇒ root''" by cases auto next assume "∃root'. root ─x→ root' ∧ root' ═xs⇒ root''" then show "root ═(x # xs)⇒ 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 ═[]⇒ root' ⟹ root' = root" by (simp add: transitions_nil_eq) lemma transitions_consD: assumes list: "root ═(x # xs)⇒ root''" and hd: "root ─x→ root'" shows "root' ═xs⇒ root''" proof - from list obtain r' where r': "root ─x→ r'" and root'': "r' ═xs⇒ root''" by cases simp_all from r' hd have "r' = root'" by (rule transition_uniq) with root'' show "root' ═xs⇒ root''" by simp qed text ‹ ┉ 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⇒ root'"}. › lemma transitions_invariant: assumes r: "⋀r x r'. r ─x→ r' ⟹ Q r ⟹ P x ⟹ Q r'" and trans: "root ═xs⇒ root'" shows "Q root ⟹ ∀x ∈ set xs. P x ⟹ Q root'" using trans proof induct case nil show ?case by (rule nil.prems) next case (cons root x root' xs root'') note P = ‹∀x ∈ set (x # xs). P x› then have "P x" by simp with ‹root ─x→ root'› and ‹Q root› have Q': "Q root'" by (rule r) from P have "∀x ∈ set xs. P x" by simp with Q' show "Q root''" by (rule cons.hyps) 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: assumes "root ═xs⇒ root'" and "∃att dir. root = Env att dir" shows "∃att dir. root' = Env att dir" using transition_type_safe and assms proof (rule transitions_invariant) show "∀x ∈ set xs. True" by blast qed section ‹Executable sequences› text ‹ An inductively defined relation such as the one of ‹root ─x→ 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. ┉ 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 ∗‹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. › definition "can_exec root xs ⟷ (∃root'. root ═xs⇒ root')" lemma can_exec_nil: "can_exec root []" unfolding can_exec_def by (blast intro: transitions.intros) lemma can_exec_cons: "root ─x→ root' ⟹ can_exec root' xs ⟹ can_exec root (x # xs)" unfolding can_exec_def by (blast intro: transitions.intros) text ‹ ┉ 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: "can_exec root (xs @ [y]) ⟹ ∃root' root''. root ═xs⇒ root' ∧ root' ─y→ root''" proof (induct xs arbitrary: root) case Nil then show ?case by (simp add: can_exec_def transitions_nil_eq transitions_cons_eq) next case (Cons x xs) from ‹can_exec root ((x # xs) @ [y])› obtain r root'' where x: "root ─x→ r" and xs_y: "r ═(xs @ [y])⇒ root''" by (auto simp add: can_exec_def transitions_nil_eq transitions_cons_eq) from xs_y Cons.hyps obtain root' r' where xs: "r ═xs⇒ root'" and y: "root' ─y→ r'" unfolding can_exec_def by blast from x xs have "root ═(x # xs)⇒ root'" by (rule transitions.cons) with y show ?case by blast 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 ∈ users ⟹ 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. ┉ Here are a few further experiments, using the same techniques as before. › theorem "u ∈ users ⟹ 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 ∈ users ⟹ Writable ∈ perms⇩_{1}⟹ Readable ∈ perms⇩_{2}⟹ name⇩_{3}≠ name⇩_{4}⟹ can_exec (init users) [Mkdir u perms⇩_{1}[u, name⇩_{1}], Mkdir u' perms⇩_{2}[u, name⇩_{1}, name⇩_{2}], Creat u' perms⇩_{3}[u, name⇩_{1}, name⇩_{2}, name⇩_{3}], Creat u' perms⇩_{3}[u, name⇩_{1}, name⇩_{2}, name⇩_{4}], Readdir u {name⇩_{3}, name⇩_{4}} [u, name⇩_{1}, name⇩_{2}]]" 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 ∈ users ⟹ Writable ∈ perms⇩_{1}⟹ Readable ∈ perms⇩_{3}⟹ can_exec (init users) [Mkdir u perms⇩_{1}[u, name⇩_{1}], Mkdir u' perms⇩_{2}[u, name⇩_{1}, name⇩_{2}], Creat u' perms⇩_{3}[u, name⇩_{1}, name⇩_{2}, name⇩_{3}], Write u' ''foo'' [u, name⇩_{1}, name⇩_{2}, name⇩_{3}], Read u ''foo'' [u, name⇩_{1}, name⇩_{2}, name⇩_{3}]]" 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: assumes cannot_y: "⋀r r'. Q r ⟹ r ─y→ r' ⟹ False" and init_inv: "⋀root. init users ═bs⇒ root ⟹ Q root" and preserve_inv: "⋀r x r'. r ─x→ r' ⟹ Q r ⟹ P x ⟹ Q r'" and init_result: "init users ═bs⇒ root" shows "¬ (∃xs. (∀x ∈ set xs. P x) ∧ can_exec root (xs @ [y]))" proof - { fix xs assume Ps: "∀x ∈ set xs. P x" assume can_exec: "can_exec root (xs @ [y])" then obtain root' root'' where xs: "root ═xs⇒ root'" and y: "root' ─y→ 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) } then show ?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 = user⇩_{1}"} 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 situation› text ‹ We introduce a few global declarations and axioms to describe our particular situation considered here. Thus we avoid excessive use of local parameters in the subsequent development. › context fixes users :: "uid set" and user⇩_{1}:: uid and user⇩_{2}:: uid and name⇩_{1}:: name and name⇩_{2}:: name and name⇩_{3}:: name and perms⇩_{1}:: perms and perms⇩_{2}:: perms assumes user⇩_{1}_known: "user⇩_{1}∈ users" and user⇩_{1}_not_root: "user⇩_{1}≠ 0" and users_neq: "user⇩_{1}≠ user⇩_{2}" and perms⇩_{1}_writable: "Writable ∈ perms⇩_{1}" and perms⇩_{2}_not_writable: "Writable ∉ perms⇩_{2}" notes facts = user⇩_{1}_known user⇩_{1}_not_root users_neq perms⇩_{1}_writable perms⇩_{2}_not_writable begin definition "bogus = [Mkdir user⇩_{1}perms⇩_{1}[user⇩_{1}, name⇩_{1}], Mkdir user⇩_{2}perms⇩_{2}[user⇩_{1}, name⇩_{1}, name⇩_{2}], Creat user⇩_{2}perms⇩_{2}[user⇩_{1}, name⇩_{1}, name⇩_{2}, name⇩_{3}]]" definition "bogus_path = [user⇩_{1}, name⇩_{1}, name⇩_{2}]" text ‹ 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. › 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 nor writable by @{term user⇩_{1}}. › definition "invariant root path ⟷ (∃att dir. access root path user⇩_{1}{} = Some (Env att dir) ∧ dir ≠ Map.empty ∧ user⇩_{1}≠ owner att ∧ access root path user⇩_{1}{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. ▸ The invariant is sufficiently strong to entail the pathological case that @{term user⇩_{1}} is unable to remove the (owned) directory at @{term "[user⇩_{1}, name⇩_{1}]"}. ▸ The invariant does hold after having executed the @{term bogus} list of operations (starting with an initial file-system configuration). ▸ The invariant is preserved by any file-system operation performed by @{term user⇩_{1}} alone, without any help by other users. 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. ┉ The first two lemmas are technically straight forward --- we just have to inspect rather special cases. › lemma cannot_rmdir: assumes inv: "invariant root bogus_path" and rmdir: "root ─(Rmdir user⇩_{1}[user⇩_{1}, name⇩_{1}])→ root'" shows False proof - from inv obtain "file" where "access root bogus_path user⇩_{1}{} = Some file" unfolding invariant_def by blast moreover from rmdir obtain att where "access root [user⇩_{1}, name⇩_{1}] user⇩_{1}{} = Some (Env att Map.empty)" by cases auto then have "access root ([user⇩_{1}, name⇩_{1}] @ [name⇩_{2}]) user⇩_{1}{} = Map.empty name⇩_{2}" by (simp only: access_empty_lookup lookup_append_some) simp ultimately show False by (simp add: bogus_path_def) qed lemma assumes init: "init users ═bogus⇒ root" shows init_invariant: "invariant root bogus_path" supply eval = facts access_def init_def using init 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 text ‹ ┉ 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 user⇩_{1}} alone. Note that this holds for any @{term path} given, the particular @{term bogus_path} is not required here. › lemma preserve_invariant: assumes tr: "root ─x→ root'" and inv: "invariant root path" and uid: "uid_of x = user⇩_{1}" shows "invariant root' path" proof - from inv obtain att dir where inv1: "access root path user⇩_{1}{} = Some (Env att dir)" and inv2: "dir ≠ Map.empty" and inv3: "user⇩_{1}≠ owner att" and inv4: "access root path user⇩_{1}{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 user⇩_{1}_not_root have not_writable: "Writable ∉ others att" by (auto simp add: access_def split: option.splits) show ?thesis proof cases assume "root' = root" with inv show "invariant root' path" by (simp only:) next assume changed: "root' ≠ 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 ∥ path" with inv root' have "⋀perms. access root' path user⇩_{1}perms = access root path user⇩_{1}perms" by (simp only: access_update_other) with inv show "invariant root' path" by (simp only: invariant_def) next assume "prefix (path_of x) 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 user⇩_{1}_not_root have False by cases (auto simp add: access_empty_lookup dest: access_some_lookup) then show ?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 "strict_prefix 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' ≠ Map.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↦file)))" by (simp only: update_append_some update_cons_nil_env) moreover have "dir(y↦file) ≠ Map.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↦file'))" proof - have "dir y ≠ None" proof - have "dir y = lookup (Env att dir) [y]" by (simp split: option.splits) also from lookup have "… = lookup root (path @ [y])" by (simp only: lookup_append_some) also have "… ≠ None" proof - from ys obtain us u where rev_ys: "ys = us @ [u]" by (cases ys rule: rev_cases) simp with tr path have "lookup root ((path @ [y]) @ (us @ [u])) ≠ None ∨ lookup root ((path @ [y]) @ us) ≠ None" by cases (auto dest: access_some_lookup) then show ?thesis by (fastforce dest!: lookup_some_append) qed finally show ?thesis . qed with ys show ?thesis using that by auto qed also have "dir(y↦file') ≠ Map.empty" by simp ultimately show ?thesis .. qed from lookup' have inv1': "access root' path user⇩_{1}{} = Some (Env att dir')" by (simp only: access_empty_lookup) from inv3 lookup' and not_writable user⇩_{1}_not_root have "access root' path user⇩_{1}{Writable} = None" by (simp add: access_def) with inv1' inv2' inv3 show ?thesis unfolding invariant_def by 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}). › corollary assumes bogus: "init users ═bogus⇒ root" shows "¬ (∃xs. (∀x ∈ set xs. uid_of x = user⇩_{1}) ∧ can_exec root (xs @ [Rmdir user⇩_{1}[user⇩_{1}, name⇩_{1}]]))" proof - from cannot_rmdir init_invariant preserve_invariant and bogus show ?thesis by (rule general_procedure) qed end end