Difference between pages "Tasking Event-B Overview" and "The Proving Perspective (Rodin User Manual)"

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=== Tasking Event-B ===
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{{Navigation|Previous= [[The_Proof_Obligation_Explorer_(Rodin_User_Manual)|The Proof Obligation Explorer]]|Next= [[The_Mathematical_Language_(Rodin_User_Manual)|The Mathematical Language]]|Up= [[index_(Rodin_User_Manual)|User_Manual_index]]}}
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Tasking Event-B can be viewed as an extension of the existing Event-B language. We use the existing approaches of refinement and decomposition to structure a project that is suitable for a Tasking Development. During the modelling phase parameters are introduced to facilitate decomposition. As a result of the decomposition process, parameters become part of the interface that enables event synchronization. We make use of this interface and add information (see [[#Implementing Events]]) to facilitate code generation. The tasking extension consists of the constructs in the following table.
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{{TOCright}}
  
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<center>
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The Proving Perspective is made of a number of windows: the proof tree, the goal, the selected hypotheses, the proof control, the proof information, and the searched hypotheses. In subsequent sections, we study these windows, but before that let us see how one can load a proof.
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{| border="1"
 
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!Construct
 
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!Options
 
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|-
 
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|Machine Type
 
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|DeclaredTask, [http://wiki.event-b.org/index.php/Tasking_Event-B_Overview#Tasking_Machines AutoTask], [http://wiki.event-b.org/index.php/Tasking_Event-B_Overview#Shared_Machines SharedMachine], [http://wiki.event-b.org/index.php/Tasking_Event-B_Overview#The_Environ_Machine Environ]
 
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|-
 
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|[http://wiki.event-b.org/index.php/Tasking_Event-B_Overview#Control_Constructs Control]
 
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|Sequence, Loop, Branch, EventSynch, Output
 
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|-
 
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|[http://wiki.event-b.org/index.php/Tasking_Event-B_Overview#Tasking_Machines Task Type]
 
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|Periodic(n), Triggered, Repeating, OneShot
 
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|-
 
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|Priority
 
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| -
 
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|-
 
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|[http://wiki.event-b.org/index.php/Tasking_Event-B_Overview#Implementing_Events Event Type]
 
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|Branch, Loop, ProcedureDef, ProcedureSynch, Actuating, Sensing
 
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|-
 
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|Parameter Type
 
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|ActualIn, ActualOut, FormalIn, FormalOut
 
−
|-
 
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|Addressed Variable
 
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|Address, Base
 
−
|}
 
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</center>
 
  
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=== Tasking Developments ===
+
== Loading a Proof ==
−
A Tasking Development is modelling component that is generated programmatically, at the direction of the user. The Tasking Development consists of a number of machines (and perhaps associated contexts). We make use of the Event-B EMF extension mechanism <ref name = "EBEMF">[[EMF framework for Event-B]]</ref>, which allows addition of new constructs to a model. The machines in the Tasking Development are extended with the constructs shown in the table, and may be viewed as keywords in a textual representation of the language. With extensions added, a Tasking Development can be translated to a common language model for mapping to implementation source code. There is also a translator that constructs new machines/contexts modelling the implementation, and these should refine/extend the existing elements of the Event-B project.
 
  
−
==== Tasking Machines ====
+
In order to load a proof, enter the Proof Obligation window, select the project, select and expand the component, finally select the proof obligation: the corresponding proof will be loaded. As a consequence:
−
The following constructs relate only to Tasking Machines, and provide implementation details. Timing of periodic tasks is not modelled formally. Tasking Machines are related to the concept of an Ada task. These can be implemented in Ada using tasks, in C using the pthread library C, or in Java using threads.
 
  
−
* Tasking Machines may be characterised by the following types:
+
* the proof tree is loaded in the Proof Tree window. As we shall see in section [[The_Proving_Perspective_(Rodin_User_Manual)#The_Proof_Tree|6.2]], each node of the proof tree is associated with a sequent.
−
** AutoTasks - Singleton Tasks.
+
* In case the proof tree has some "pending" nodes (whose sequents are not discharged yet) then the sequent corresponding to the first pending node is decomposed: its goal is loaded in the Goal window (section [[The_Proving_Perspective_(Rodin_User_Manual)#Goal and Selected Hypotheses|6.3]]), whereas parts of its hypotheses (the "selected" ones) are loaded in the Selected Hypotheses window (section [[The_Proving_Perspective_(Rodin_User_Manual)#Goal and Selected Hypotheses|6.3]]).
−
** Declared tasks - (Not currently used) A task template relating to an Ada ''tasktype'' declaration.
+
* In case the proof tree has no pending node, then the sequent of the root node is loaded as explained previously.
−
** TaskType - Defines the scheduling, cycle and lifetime of a task. i.e. one-shot periodic or triggered. The period of a task is specified in milliseconds.
 
−
** Priority - An integer value is supplied, the task with the highest value priority takes precedence when being scheduled. For the demonstrator tool the default priority is 5.
 
  
−
''Auto Tasks'' are tasks that will be declared and defined in the ''Main'' procedure of the implementation. The effect of this is that the ''Auto Tasks'' are created when the program first loads, and then activated (made ready to run) before the ''Main'' procedure body runs.
+
== The Proof Tree ==
  
−
==== Shared Machines ====
+
=== Description of the Proof Tree ===
−
A Shared Machine corresponds to the concept of a protected resource, such as a monitor. They may be implemented in Ada as a Protected Object, in C using mutex locking, or in Java as a monitor.
 
  
−
* Applied to the Shared Machine we have:
+
The proof tree can be seen in the corresponding window as shown in the following screen shot:
−
** A SharedMachine ''keyword'' that identifies a machine as a Shared Machine.
 
  
−
==== The Environ Machine ====
+
[[Image:um-0095.png|center]]
−
An Environ machine is a model of the environment. It can be used to generate code for use in a simulation, or discarded (from the IL1 model) in the case that a simulated environment is not required.
 
  
−
* Applied to the Environ Machine we have:
+
Each line in the proof tree corresponds to a node which is a sequent. A line is right shifted when the corresponding node is a direct descendant of the node of the previous line. Here is an illustration of the previous tree:
−
** An Environ Machine ''keyword'' that identifies a machine as an Environ Machine.
 
  
−
=== Implementation Specifics ===
+
[[Image:Tree.png|center]]
−
At the stage where we are considering how to implement the Event-B development we need to think about controlling the flow of execution, and how events should be implemented. The following section describes the constructs that we have introduced to facilitate this.
 
−
==== Control Constructs ====
 
−
Each Tasking Machine has a ''task body'' which contains the flow control (algorithmic constructs).  
 
  
−
* We have the following constructs available in the Tasking Machine body:
+
Each node is labelled with a comment explaining how it can be discharged. By selecting a node in the proof tree, the corresponding sequent is decomposed and loaded in the Goal and Selected Hypotheses windows as explained in section [[The_Proving_Perspective_(Rodin_User_Manual)#Loading a Proof|6.1]].
−
** Sequence - for imposing an order on events.
 
−
** Branch - choice between a number of mutually exclusive events.
 
−
** Loop - event repetition while it's guard remains true.
 
−
** EventWrapper - a wrapper for the Event-B element (soon to be redundant).
 
−
** Synch Events - synchronizes two events.
 
−
** Text Output - writes textual output to the screen.
 
  
−
Synchronization corresponds to:
+
=== Decoration ===
−
* a subroutine call from task to shared machine, or,
 
−
* sensing or actuating of environment variables.
 
  
−
In the case of a subroutine call the subroutine is an atomic (with respect to an external viewer) update to state. The updates in the protected resource are implemented by a procedure call to a protected object, and tasks do no share state.  The synchronization construct also provides the means to specify parameter passing, both in and out of the task.
+
The leaves of the tree are decorated with three kinds of logos:
  
−
In the case of a sensing or actuating event, the updates of the action correspond to reads of monitored variables, and writes to controlled variables of the environment.
+
* a green logo with a "<math>\surd </math>" in it means that this leaf is discharged,
 +
* a red logo with a "?" in it means that this leaf is not discharged,
 +
* a blue logo with a "R" in it means that this leaf has been reviewed.
  
−
Event wrappers:
+
Internal nodes in the proof tree are decorated in the same (but lighter) way. Note that a "reviewed" leaf is one that is not discharged yet by the prover. Instead, it has been "seen" by the user who decided to have it discharged later. Marking nodes as "reviewed" is very convenient in order to perform an interactive proof in a gradual fashion. In order to discharge a "reviewed" node, select it and prune the tree at that node (section [[The_Proving_Perspective_(Rodin_User_Manual)#Pruning|6.2.5]]): the node will become "red" again (undischarged) and you can now try to discharge it.
−
* The event synchronization construct is contained in an event wrapper. The wrapper may also contain a single event (we re-use the synchronization construct, but do not use it for synchronizing). The event may belong to the Tasking Machine, a Shared Machine that is visible to the task, or the Environ machine. Single events in a wrapper correspond to a subroutine call in an implementation.
 
  
−
When Editing the EMF model the constructs have the following names:
+
=== Navigation within the Proof Tree ===
  
−
<center>
+
On top of the proof tree window one can see three buttons:
−
{| border="1"
 
−
!Construct
 
−
!EMF name
 
−
|-
 
−
|Sequence
 
−
|Seq
 
−
|-
 
−
|Branch
 
−
|Branch
 
−
|-
 
−
|Loop
 
−
|Do
 
−
|-
 
−
|Text Output
 
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|Output
 
−
|-
 
−
|EventWrapper
 
−
|EventWrapper
 
−
|-
 
−
|SynchEvents
 
−
|SynchEvents
 
−
|-
 
−
|}
 
−
</center>
 
  
−
== The Abstract Syntax ==
+
* the "G" buttons allows you to see the goal of the sequent corresponding to the node
 +
* the "+" button allows you to fully expand the proof tree
 +
* the "-" allows you to fully collapse the tree: only the root stays visible.
  
−
The following BNF syntax relates to the EMF model that developers will use in the tutorial. The task body is used to specify the ordering of events.  The use of the EMF editor should be seen as a short term solution, in the longer term a suitable form-based or text-based editor is envisaged. Text, Variable and Event nodes are terminals in the syntax, they relate to Event-B elements.
+
=== Hiding ===
  
−
<pre>
+
The little triangle next to each node in the proof tree allows you to expand or collapse the subtree starting at that node.
−
    TaskBody ::= Seq | Branch | Do | EventWrapper | Output
 
  
−
    Seq  ::= TaskBody TaskBody
+
=== Pruning ===
  
−
    Branch ::=  Body [SubBranch] Else
+
The proof tree can be pruned from a node: it means that the subtree starting at that node is eliminated. The node in question becomes a leaf and is red decorated. This allows you to resume the proof from that node. After selecting a sequent in the proof tree, pruning can be performed in two ways:
  
−
    SubBranch ::=  Body [SubBranch]
+
* by right-clicking and then selecting "Prune",
 +
* by pressing the "Scissors" button in the proof control window (section [[The_Proving_Perspective_(Rodin_User_Manual)#The Proof Control Window|6.4]]).
  
−
    Else ::=  EventWrapper
+
Note that after pruning, the post-tactic (section [[The_Proving_Perspective_(Rodin_User_Manual)#The Automatic Post-tactic|6.8]]) is not applied to the new current sequent: if needed you have to press the "post-tactic" button in the Proof Control window (section [[The_Proving_Perspective_(Rodin_User_Manual)#The Proof Control Window|6.4]]). This happens in particular when you want to redo a proof from the beginning: you prune the proof tree from the root node and then you have to press the "post-tactic" button in order to be in exactly the same situation as the one delivered automatically initially.
  
−
    Body ::=  EventWrapper
+
When you want to redo a proof from a certain node, it might be advisable to do it after copying the tree so that in case your new proof fails you can still resume the previous situation by pasting the copied version (see next section).
  
−
    Do ::= Body [Finally]
+
=== Copy/Paste ===
  
−
    Finally ::=  EventWrapper
+
By selecting a node in the proof tree and then clicking on the right key of the mouse, you can copy the part of the proof tree starting at that sequent: it can later be pasted in the same way. This allows you to reuse part of a proof tree in the same (or even another) proof.
  
−
    Output ::= Text Variable
+
== Goal and Selected Hypotheses ==
  
−
    EventWrapper ::=  SynchEvents
+
The "Goal" and "Selected Hypotheses" windows display the current sequent you have to prove at a given moment in the proof. Here is an example:
  
−
    SynchEvents ::=  Local Remote
+
[[Image:um-0098.png|center]]
  
−
    Local ::=  Event
+
A selected hypothesis can be deselected by first clicking in the box situated next to it (you can click on several boxes) and then by pressing the red (-) button at the top of the selected hypothesis window:
  
−
    Remote ::=  Event
+
[[Image:um-0099.png|center]]
−
</pre>
 
  
−
==== Implementing Events ====
+
Here is the result:
−
An event's role in the implementation is identified using the following extensions which are added to the event. Events used in task bodies are 'references' that make use of existing event definitions from the abstract development. The events are extended. to assist with translation, with a keyword indicating their role in the implementation.
 
  
−
* Event implementation role.
+
[[Image:um-0100.png|center]]
−
** Branch - In essence a task's event is split in the implementation; guards are mapped to branch conditions and actions are mapped to the branch body. If the branch refers to a Shared Machine event (procedureDef) then this is mapped to a simple procedure call.
 
−
** Loop - The task's event guard maps to the loop condition and actions to to loop body. If the loop refers to a Shared Machine event then it is mapped to a simple procedure call.
 
−
** ProcedureSynch - This usually indicates to the translator that the event maps to a subroutine, but an event in a task may not require a subroutine implementation if its role is simply to provide parameters for a procedure call.
 
−
** ProcedureDef - Identifies an event that maps to a (potentially blocking) subroutine definition. Event guards are implemented as a conditional wait; in Ada this is an entry barrier, and in C may use a pthread condition variable .
 
−
** Sensing - Identifies an event that maps to a read from the environment. If the environment is simulated without address variables then the sensing event is similar to a ProcedureSynch event, in that it has an update action that models assignment of a return value from a subroutine call. The event parameters act like the ''actualIn'' parameters of a ProcedureSynch event. On the other hand, if the event has addressed variables associated with its event parameters, then they map to direct reads from memory mapped variables in the generated code.
 
−
** Actuating - Identifies an event that maps to a write to the environment. If the environment is simulated without address variables then the actuating event has no update action, the parameters act like ''actualOut'' parameters of a ProcedureSynch event. If a sensing event has addressed variables associated with its parameters then they map to direct writes to memory mapped variables in the generated code.  
 
  
−
Sensing (and actuating) can be viewed as a kind of synchronisation. Synchronisation between tasks and shared objects are represented as subroutine calls. The sensing/actuating synchronisations only occur between tasks and the environment. In implementable code, when an subroutine is defined, its formal parameters are replaced by actual parameter values at run-time. To assist the code generator we extend the Event-B parameters. We identify formal and actual parameters in the implementation, and add the following keywords to the event parameters, as follows:
+
Notice that the deselected hypotheses are not lost: you can get them back by means of the Searched Hypotheses window (section [[The_Proving_Perspective_(Rodin_User_Manual)#srchhyp|6.7]]).
  
−
* Event parameter types - Note: formal parameters are place-holders in a subroutine; they are replaced by the actual parameters at call time.
+
The three other buttons next to the red (-) button allow you to do the reverse operation, namely keeping some hypotheses. The (ct) button next to the goal allows you to do a proof by contradiction: pressing it makes the negation of the goal being a selected hypothesis whereas the goal becomes "false". The (ct) button next to a selected hypothesis allows you to do another kind of proof by contradiction: pressing it makes the negation of the hypothesis the goal whereas the negated goal becomes an hypothesis.
−
** FormalIn or FormalOut - event parameters are extended with the ParameterType construct. Extension with formal parameters indicates a mapping to formal parameters in the implementation.
 
−
** ActualIn or ActualOut - Extension with an actual parameter indicates a mapping to an actual parameter in the implementation.
 
  
−
===== Addressed Variables =====
+
== Rewrite rules ==
−
When sensing monitored variables, or actuating controlled variables in the environment, we may wish to use explicit memory addresses for use in the final implementation, or perhaps in the environment simulation too. We can link a task's event parameters, and an Environ machines variables, with specific addresses and use these in the generated code.
 
  
−
== References ==
+
Rewrite rules are applied either automatically (A) or manually (M).
  
−
<references/>
+
A rule may be applied:
 +
* Automatically (A), when post tactics are enabled. These rules are equivalence simplification laws.
 +
* Manually (M), through an interactive command. These rules gathers non equivalence laws, definition laws, distributivity laws and derived laws.
  
 +
Rewrite rules are applied from left to right either in the goal or in the selected hypotheses, when their ''side condition'' hold.
  
−
[[Category:User documentation]]
+
Each rule is named after the following elements:
 +
 
 +
* The law category: simplification law (SIMP), definition law (DEF), distributivity law (DISTRI), or else derived law (DERIV).
 +
* Particularity on terminal elements of the left part of the rule (optional): special element (SPECIAL) such as the empty-set, type expression (TYPE), same element occurring more then once (MULTI), literal (LIT). A type expression is either a basic type (<math>\intg, \Bool</math>, any carrier set), or <math>\pow</math>(type expression), or type expression<math>\cprod</math>type expression.
 +
* One or more elements describing from top to down the left part of the rule, eg. predicate AND, expression BUNION.
 +
* Detail to localize those elements (optional): left (L), right (R).
 +
 
 +
Rewrite rules having an equivalence operator in their left part may also describe other rules. eg: the rule:
 +
 
 +
<math>  \True  = \False  \;\;\defi\;\;  \bfalse </math>
 +
 
 +
should also produce the rule:
 +
 
 +
<math>  \False  = \True  \;\;\defi\;\;  \bfalse </math>
 +
 
 +
 
 +
For associative operators in connection with distributive laws as in:
 +
 
 +
<center><math> P \land (Q \lor \ldots \lor R) </math></center>
 +
 
 +
it has been decided to put the "button" on the first associative/commutative operator (here <math>\lor </math>). Pressing that button will generate a menu: the first option of this menu will be to distribute all associative/commutative operators, the second option will be to distribute only the first associative/commutative operator. In the following presentation, to simplify matters, we write associative/commutative operators with two parameters only, but it must always be understood implicitly that we have a sequence of them. For instance, we shall never write <math> Q \lor \ldots \lor R </math> but <math> Q \lor R </math> instead. Rules are sorted according to their purpose.
 +
 
 +
Rules marked with a star in the first column are implemented in the current prover.  Rules without a star are planned for implementation.
 +
 
 +
Rewrite rules are split into:
 +
 
 +
* [[Set Rewrite Rules]]
 +
* [[Relation Rewrite Rules]]
 +
* [[Arithmetic Rewrite Rules]]
 +
 
 +
They are also available in a single large page [[All Rewrite Rules]].
 +
 
 +
== Inference Rules ==
 +
 
 +
Inference rules are applied either automatically (A) or manually (M).
 +
 
 +
 
 +
Inference rules applied automatically are applied at the end of each proof step. They have the following possible effects:
 +
 
 +
* they discharge the goal,
 +
* they simplify the goal and add a selected hypothesis,
 +
* they simplify the goal by decomposing it into several simpler goals,
 +
* they simplify a selected hypothesis,
 +
* they simplify a selected hypothesis by decomposing it into several simpler selected hypotheses.
 +
 
 +
<br />
 +
 
 +
Inference rules applied manually are used to perform an interactive proof. They can be invoked by pressing "buttons" which corresponds to emphasized (red) operators in the goal or the hypotheses. A menu is proposed when there are several options.
 +
 
 +
 
 +
See [[Inference Rules]] list
 +
 
 +
== The Proof Control Window ==
 +
 
 +
The Proof Control window contains the buttons which you can use to perform an interactive proof. Next is a screen shot where you can see successively from top to bottom:
 +
 
 +
* some selected hypotheses,
 +
* the goal,
 +
* the "Proof Control" window,
 +
* a small editing area within which you can enter parameters used by some buttons of the Proof Control window
 +
* the smiley (section [[The_Proving_Perspective_(Rodin_User_Manual)#The Simley|6.5]])
 +
 
 +
[[Image:um-0101.png|center]]
 +
 
 +
The Proof Control window offers a number of buttons which we succinctly describe from left to right:
 +
 
 +
* (p0): the prover PP attempts to prove the goal (other cases in the list)
 +
* (R) review: In an attempt by the user to carry out proofs in a gradual fashion, they might decide to postpone the task of discharging some interactive proofs for a later stage.  In this case they have the possibility to mark these proofs as reviewed (by choosing the proof node and pressing the blue button with a “R” letter on the top-left corner of the Proof Control view).  This means that by visually checking this proof, the user convinced that they can discharge it later but they do not want to do it right now.
 +
* (dc) proof by cases: the goal is proved first under the predicate written in the editing area and then under its negation,
 +
* (ah) lemma: the predicate in the editing area is proved and then added as a new selected hypothesis,
 +
* (ae) abstract expression: the expression in the editing area is given a fresh name,
 +
* the auto-prover attempts to discharge the goal. The auto-prover is the one which is applied automatically on all proof obligations (as generated automatically by the proof obligation generator after a "save") without any intervention of the user. With this button, you can call yourself the auto-prover within an interactive proof.
 +
* the post-tactic is executed (see section [[The_Proving_Perspective_(Rodin_User_Manual)#The Automatic Post-tactic|6.8]]),
 +
* lasso: load in the Selected Hypotheses window those unseen hypotheses containing identifiers which are common with identifiers in the goal and selected hypotheses,
 +
* backtrack form the current node (i.e., prune its parent),
 +
* scissors: prune the proof tree from the node selected in the proof tree,
 +
* show (in the Search Hypotheses window) hypotheses containing the character string as in the editing area,
 +
* '''Cache Hypotheses Button''': By pressing ''"Cache Hypotheses"'' button the tool displays the ''"Cache Hypotheses"'' view. This view displays all hypotheses that are related to the current goal. Some of these hypotheses might not been chosen as current active hypotheses (this is the set of hypotheses that considered by the prover to discharge the current goal). By opening the cached hypotheses view the user can manually select and add some of them to the current active hypotheses view.
 +
* load the previous non-discharged proof obligation,
 +
* load the next undischarged proof obligation,
 +
* (i) show information corresponding to the current proof obligation in the corresponding window. This information correspond to the elements that took directly part in the proof obligation generation (events, invariant, etc.),
 +
* goto the next pending node of the current proof tree,
 +
* load the next reviewed node of the current proof tree.
 +
 
 +
== The Smiley ==
 +
 
 +
The smiley can take three different colors: (1) red, meaning that the proof tree contains one or more non-discharged sequents, (2) blue, meaning that all non-discharged sequents of the proof tree have been reviewed, (3) green, meaning that all sequents of the proof tree are discharged.
 +
 
 +
== The Operator "Buttons" ==
 +
 
 +
In the goal and in the selected, searched, or cache hypotheses some operators are colored in red. It means that they are "buttons" you can press. When doing so, the meaning (sometimes several) is shown in a menu where you can select various options. The operation performed by these options is described in sections [[The_Proving_Perspective_(Rodin_User_Manual)#Interactive Rewrite Rules|6.9.1]] and [[The_Proving_Perspective_(Rodin_User_Manual)#Interactive Inference Rules|6.9.2]].
 +
 
 +
== The Search Hypotheses Window ==
 +
 
 +
By typing a string in the '''Proof Control''' window and pressing the '''Search Hypotheses''' button a window is provided which contains the hypotheses having a character string in common with the one entered by the user in the editing area. For example, if we search for hypotheses involving the character string "cr", then after pressing the '''Search Hypothesis''' button on the proof control window, we obtain the following:
 +
 
 +
[[Image:um-0102.png|center]]
 +
 
 +
In the next step any of these hypotheses can be selected and then by pressing the '''(+)button''' they will be moved to the '''Selected Hypotheses''' window . Adding these hypotheses to the set '''Selected Hypotheses''' means that they will be visible to the prover. Their visibility means that they can be used during the next interactive proof phase. Accordingly by selecting any numbers of hypotheses and pressing the '''(-)button''' they will be removed from the '''Search Hypotheses''' window. The '''(-)button''' also appears above the selected hypotheses, and allow the user to remove and hypothesis for '''Selected Hypotheses''' window. The other button which is situated in the left-hand-side of all hypotheses is the '''(ct) Button'''. By pressing the '''(ct) Button''' the negation of the corresponding hypothesis is taken as a new hypothesis. For this purpose we have to prove it first and therefore it appears as a new goal which the user discharge it first then proceed with the previous goal.
 +
 
 +
== The Automatic Post-tactic ==
 +
 
 +
In this section, we present the various rewrite or inference rules which are applied automatically as a systematic post-tactic after each proof step. Note that the post-tactic can be disabled by using the "'''P'''<math>\! \! \! \! /</math>" button situated on the right of the proof control window.
 +
 
 +
The post-tactic is made of two different rules: rewrite rules, which are applied on any sub-formula of the goal or selected hypotheses (section [[The_Proving_Perspective_(Rodin_User_Manual)#Rewrite Rules|6.8.1]]) and inference rules which are applied on the current sequent (section [[The_Proving_Perspective_(Rodin_User_Manual)#Automatic Inference Rules|6.8.2]]).
 +
 
 +
 
 +
 
 +
 
 +
 
 +
=== Preferences for the Post-tactic ===
 +
 
 +
The post-tactic can be configured by means of a preference page which can be obtained as follows: press the "Window" button on the top tooolbar. On the coming menu, press the "Preferences" button. On the coming menu, press the "Event-B" menue, then the "Sequent Prover’, and finally the "Post-Tactic" button. This yilds the following window:
 +
 
 +
[[Image:um-0147.png|center]]
 +
 
 +
In the left part you can see the ordered sequence of individual tactics composing the post-tactic, whereas the right part contains further tactics you can incorporate in the left part. By selecting a tactic you can move it from on part to the other or change the order in the left part.
 +
 
 +
 
 +
[[Category:User documentation|The Proving Perspective]]
 +
[[Category:Rodin Platform|The Proving Perspective]]
 +
[[Category:User manual|The Proving Perspective]]

Revision as of 11:56, 1 March 2010

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The Proving Perspective is made of a number of windows: the proof tree, the goal, the selected hypotheses, the proof control, the proof information, and the searched hypotheses. In subsequent sections, we study these windows, but before that let us see how one can load a proof.

Loading a Proof

In order to load a proof, enter the Proof Obligation window, select the project, select and expand the component, finally select the proof obligation: the corresponding proof will be loaded. As a consequence:

  • the proof tree is loaded in the Proof Tree window. As we shall see in section 6.2, each node of the proof tree is associated with a sequent.
  • In case the proof tree has some "pending" nodes (whose sequents are not discharged yet) then the sequent corresponding to the first pending node is decomposed: its goal is loaded in the Goal window (section 6.3), whereas parts of its hypotheses (the "selected" ones) are loaded in the Selected Hypotheses window (section 6.3).
  • In case the proof tree has no pending node, then the sequent of the root node is loaded as explained previously.

The Proof Tree

Description of the Proof Tree

The proof tree can be seen in the corresponding window as shown in the following screen shot:

Um-0095.png

Each line in the proof tree corresponds to a node which is a sequent. A line is right shifted when the corresponding node is a direct descendant of the node of the previous line. Here is an illustration of the previous tree:

Tree.png

Each node is labelled with a comment explaining how it can be discharged. By selecting a node in the proof tree, the corresponding sequent is decomposed and loaded in the Goal and Selected Hypotheses windows as explained in section 6.1.

Decoration

The leaves of the tree are decorated with three kinds of logos:

  • a green logo with a "\surd " in it means that this leaf is discharged,
  • a red logo with a "?" in it means that this leaf is not discharged,
  • a blue logo with a "R" in it means that this leaf has been reviewed.

Internal nodes in the proof tree are decorated in the same (but lighter) way. Note that a "reviewed" leaf is one that is not discharged yet by the prover. Instead, it has been "seen" by the user who decided to have it discharged later. Marking nodes as "reviewed" is very convenient in order to perform an interactive proof in a gradual fashion. In order to discharge a "reviewed" node, select it and prune the tree at that node (section 6.2.5): the node will become "red" again (undischarged) and you can now try to discharge it.

Navigation within the Proof Tree

On top of the proof tree window one can see three buttons:

  • the "G" buttons allows you to see the goal of the sequent corresponding to the node
  • the "+" button allows you to fully expand the proof tree
  • the "-" allows you to fully collapse the tree: only the root stays visible.

Hiding

The little triangle next to each node in the proof tree allows you to expand or collapse the subtree starting at that node.

Pruning

The proof tree can be pruned from a node: it means that the subtree starting at that node is eliminated. The node in question becomes a leaf and is red decorated. This allows you to resume the proof from that node. After selecting a sequent in the proof tree, pruning can be performed in two ways:

  • by right-clicking and then selecting "Prune",
  • by pressing the "Scissors" button in the proof control window (section 6.4).

Note that after pruning, the post-tactic (section 6.8) is not applied to the new current sequent: if needed you have to press the "post-tactic" button in the Proof Control window (section 6.4). This happens in particular when you want to redo a proof from the beginning: you prune the proof tree from the root node and then you have to press the "post-tactic" button in order to be in exactly the same situation as the one delivered automatically initially.

When you want to redo a proof from a certain node, it might be advisable to do it after copying the tree so that in case your new proof fails you can still resume the previous situation by pasting the copied version (see next section).

Copy/Paste

By selecting a node in the proof tree and then clicking on the right key of the mouse, you can copy the part of the proof tree starting at that sequent: it can later be pasted in the same way. This allows you to reuse part of a proof tree in the same (or even another) proof.

Goal and Selected Hypotheses

The "Goal" and "Selected Hypotheses" windows display the current sequent you have to prove at a given moment in the proof. Here is an example:

Um-0098.png

A selected hypothesis can be deselected by first clicking in the box situated next to it (you can click on several boxes) and then by pressing the red (-) button at the top of the selected hypothesis window:

Um-0099.png

Here is the result:

Um-0100.png

Notice that the deselected hypotheses are not lost: you can get them back by means of the Searched Hypotheses window (section 6.7).

The three other buttons next to the red (-) button allow you to do the reverse operation, namely keeping some hypotheses. The (ct) button next to the goal allows you to do a proof by contradiction: pressing it makes the negation of the goal being a selected hypothesis whereas the goal becomes "false". The (ct) button next to a selected hypothesis allows you to do another kind of proof by contradiction: pressing it makes the negation of the hypothesis the goal whereas the negated goal becomes an hypothesis.

Rewrite rules

Rewrite rules are applied either automatically (A) or manually (M).

A rule may be applied:

  • Automatically (A), when post tactics are enabled. These rules are equivalence simplification laws.
  • Manually (M), through an interactive command. These rules gathers non equivalence laws, definition laws, distributivity laws and derived laws.

Rewrite rules are applied from left to right either in the goal or in the selected hypotheses, when their side condition hold.

Each rule is named after the following elements:

  • The law category: simplification law (SIMP), definition law (DEF), distributivity law (DISTRI), or else derived law (DERIV).
  • Particularity on terminal elements of the left part of the rule (optional): special element (SPECIAL) such as the empty-set, type expression (TYPE), same element occurring more then once (MULTI), literal (LIT). A type expression is either a basic type (\intg, \Bool, any carrier set), or \pow(type expression), or type expression\cprodtype expression.
  • One or more elements describing from top to down the left part of the rule, eg. predicate AND, expression BUNION.
  • Detail to localize those elements (optional): left (L), right (R).

Rewrite rules having an equivalence operator in their left part may also describe other rules. eg: the rule:

\True = \False \;\;\defi\;\; \bfalse

should also produce the rule:

\False = \True \;\;\defi\;\; \bfalse


For associative operators in connection with distributive laws as in:

P \land (Q \lor \ldots \lor R)

it has been decided to put the "button" on the first associative/commutative operator (here \lor ). Pressing that button will generate a menu: the first option of this menu will be to distribute all associative/commutative operators, the second option will be to distribute only the first associative/commutative operator. In the following presentation, to simplify matters, we write associative/commutative operators with two parameters only, but it must always be understood implicitly that we have a sequence of them. For instance, we shall never write Q \lor \ldots \lor R but Q \lor R instead. Rules are sorted according to their purpose.

Rules marked with a star in the first column are implemented in the current prover. Rules without a star are planned for implementation.

Rewrite rules are split into:

They are also available in a single large page All Rewrite Rules.

Inference Rules

Inference rules are applied either automatically (A) or manually (M).


Inference rules applied automatically are applied at the end of each proof step. They have the following possible effects:

  • they discharge the goal,
  • they simplify the goal and add a selected hypothesis,
  • they simplify the goal by decomposing it into several simpler goals,
  • they simplify a selected hypothesis,
  • they simplify a selected hypothesis by decomposing it into several simpler selected hypotheses.


Inference rules applied manually are used to perform an interactive proof. They can be invoked by pressing "buttons" which corresponds to emphasized (red) operators in the goal or the hypotheses. A menu is proposed when there are several options.


See Inference Rules list

The Proof Control Window

The Proof Control window contains the buttons which you can use to perform an interactive proof. Next is a screen shot where you can see successively from top to bottom:

  • some selected hypotheses,
  • the goal,
  • the "Proof Control" window,
  • a small editing area within which you can enter parameters used by some buttons of the Proof Control window
  • the smiley (section 6.5)
Um-0101.png

The Proof Control window offers a number of buttons which we succinctly describe from left to right:

  • (p0): the prover PP attempts to prove the goal (other cases in the list)
  • (R) review: In an attempt by the user to carry out proofs in a gradual fashion, they might decide to postpone the task of discharging some interactive proofs for a later stage. In this case they have the possibility to mark these proofs as reviewed (by choosing the proof node and pressing the blue button with a “R” letter on the top-left corner of the Proof Control view). This means that by visually checking this proof, the user convinced that they can discharge it later but they do not want to do it right now.
  • (dc) proof by cases: the goal is proved first under the predicate written in the editing area and then under its negation,
  • (ah) lemma: the predicate in the editing area is proved and then added as a new selected hypothesis,
  • (ae) abstract expression: the expression in the editing area is given a fresh name,
  • the auto-prover attempts to discharge the goal. The auto-prover is the one which is applied automatically on all proof obligations (as generated automatically by the proof obligation generator after a "save") without any intervention of the user. With this button, you can call yourself the auto-prover within an interactive proof.
  • the post-tactic is executed (see section 6.8),
  • lasso: load in the Selected Hypotheses window those unseen hypotheses containing identifiers which are common with identifiers in the goal and selected hypotheses,
  • backtrack form the current node (i.e., prune its parent),
  • scissors: prune the proof tree from the node selected in the proof tree,
  • show (in the Search Hypotheses window) hypotheses containing the character string as in the editing area,
  • Cache Hypotheses Button: By pressing "Cache Hypotheses" button the tool displays the "Cache Hypotheses" view. This view displays all hypotheses that are related to the current goal. Some of these hypotheses might not been chosen as current active hypotheses (this is the set of hypotheses that considered by the prover to discharge the current goal). By opening the cached hypotheses view the user can manually select and add some of them to the current active hypotheses view.
  • load the previous non-discharged proof obligation,
  • load the next undischarged proof obligation,
  • (i) show information corresponding to the current proof obligation in the corresponding window. This information correspond to the elements that took directly part in the proof obligation generation (events, invariant, etc.),
  • goto the next pending node of the current proof tree,
  • load the next reviewed node of the current proof tree.

The Smiley

The smiley can take three different colors: (1) red, meaning that the proof tree contains one or more non-discharged sequents, (2) blue, meaning that all non-discharged sequents of the proof tree have been reviewed, (3) green, meaning that all sequents of the proof tree are discharged.

The Operator "Buttons"

In the goal and in the selected, searched, or cache hypotheses some operators are colored in red. It means that they are "buttons" you can press. When doing so, the meaning (sometimes several) is shown in a menu where you can select various options. The operation performed by these options is described in sections 6.9.1 and 6.9.2.

The Search Hypotheses Window

By typing a string in the Proof Control window and pressing the Search Hypotheses button a window is provided which contains the hypotheses having a character string in common with the one entered by the user in the editing area. For example, if we search for hypotheses involving the character string "cr", then after pressing the Search Hypothesis button on the proof control window, we obtain the following:

Um-0102.png

In the next step any of these hypotheses can be selected and then by pressing the (+)button they will be moved to the Selected Hypotheses window . Adding these hypotheses to the set Selected Hypotheses means that they will be visible to the prover. Their visibility means that they can be used during the next interactive proof phase. Accordingly by selecting any numbers of hypotheses and pressing the (-)button they will be removed from the Search Hypotheses window. The (-)button also appears above the selected hypotheses, and allow the user to remove and hypothesis for Selected Hypotheses window. The other button which is situated in the left-hand-side of all hypotheses is the (ct) Button. By pressing the (ct) Button the negation of the corresponding hypothesis is taken as a new hypothesis. For this purpose we have to prove it first and therefore it appears as a new goal which the user discharge it first then proceed with the previous goal.

The Automatic Post-tactic

In this section, we present the various rewrite or inference rules which are applied automatically as a systematic post-tactic after each proof step. Note that the post-tactic can be disabled by using the "P\! \! \! \! /" button situated on the right of the proof control window.

The post-tactic is made of two different rules: rewrite rules, which are applied on any sub-formula of the goal or selected hypotheses (section 6.8.1) and inference rules which are applied on the current sequent (section 6.8.2).



Preferences for the Post-tactic

The post-tactic can be configured by means of a preference page which can be obtained as follows: press the "Window" button on the top tooolbar. On the coming menu, press the "Preferences" button. On the coming menu, press the "Event-B" menue, then the "Sequent Prover’, and finally the "Post-Tactic" button. This yilds the following window:

Um-0147.png

In the left part you can see the ordered sequence of individual tactics composing the post-tactic, whereas the right part contains further tactics you can incorporate in the left part. By selecting a tactic you can move it from on part to the other or change the order in the left part.

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