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=== In this part ===

We will see how to create proof obligations to discharge for the machines relatively to our extensions for Probabilistic Reasoning and after having statically checked the machines. The latter operation is the first part of the proof obligation generation as the proof obligation generator takes statically checked files in input. One will notice that the provided architecture for static checking is really similar to the one for proof obligation generation. Thus, it can be useful for the reader to understand well the previous part of this tutorial, as we will not repeat all the ideas shared by the both processes.
The question here is "'''What needs to be mathematically proved with those newly added elements in hands?'''".

We will study here, the case of the ''BFN'' proof obligation, which is described in the paper. This PO overrides the FIN proof obligation. Thus we will see here, how to :
* remove the FIN PO which is generated by default,
* add our new ''BFN'' PO.
 
=== Principles ===

1. To extend the proof obligation generator (POG) in order to add some proof obligations that one has to discharge, one has to '''define a proof obligation processor module''' using the extension point : <tt>org.eventb.core.pogModuleTypes</tt>.<br>
2. Then , one has to '''set up a configuration''' involving those modules and giving them a hierachy. This is done exactly the same way as for creating a static checker configuration.<br>
3. Finally, it is needed to '''add this POG configuration''' to the default one, so the proof obligation generation can be performed.<br>


We want here to show how to generate one proof obligation. We will add the PO named ''BFN'' to ensure that the bound is a natural number or finite. It will be generated once for all for the machine taken into account. Moreover, this PO overrides the default FIN proof obligation which is generated if a convergent event (with the associated variant) is present in the model. If a probabilistic event is in the machine, we want to create our ''BFN'' PO, thus we have to remove the FIN PO.

In step 1, we will explain how to create our PO ''BFN'' using informations in the state repository that we will add in step 2, so in step 3 we could create a filter to remove the PO ''FIN'' if our machine contains a probabilistic event.

=== Step 1 : Adding POG modules ===

As we know that the POG takes its input from the static checker, the presence of a statically checked bound (ISCBound) in the statically checked model, means that one aims to prove the probabilistic convergence of this model. Thus, this information shall be shared throught our hierachy of POG modules, as it triggers the operations they could perform.

We will anticipate using this information (stored in a <tt>IPOGState</tt>) to create the ''BFN'' proof obligation :

From the extension point <tt>org.eventb.core.pogModuleTypes</tt>, create a <tt>processorModule</tt> extension to implement our first PO generation process using a POG processor :<br>
As for a static checker module,<br>
1. give the module an id (here fwdMachineBoundModule),<br>
2. a human readable name (here "Machine POG Forward Bound Module"),<br>
3. register a parent in the hierarchy of modules (here we used the machine POG module of the Event-B POG : <tt>org.eventb.core.machineModule</tt>),<br>
4. create a class for this module.(here we created the class <tt>fr.systerel.rodinextension.sample.pog.modules.FwdMachineBoundModule</tt>).

The above module should share (this is done by <tt>repository.setState()</tt>), at its initialisation, an <tt>IMachineBoundInfo</tt> state that we will implement in step 2.

 	@Override
 	public void initModule(IRodinElement element,IPOGStateRepository repository, IProgressMonitor monitor) throws CoreException {
 		repository.setState(createMachineBoundInfo(element, repository));
 	}
 
 	private IMachineBoundInfo createMachineBoundInfo(IRodinElement element,	IPOGStateRepository repository) throws CoreException {
 		final IRodinFile machineFile = (IRodinFile) element;
 		final ISCMachineRoot root = (ISCMachineRoot) machineFile.getRoot();
 		final ISCBound[] bounds = root.getChildrenOfType(ISCBound.ELEMENT_TYPE);
 		if (bounds.length != 1) {
 			return new MachineBoundInfo();
 		}
 		final ISCBound scBound = bounds[0];
 		final ITypeEnvironment typeEnv = repository.getTypeEnvironment();
 		final Expression expr = scBound.getExpression(typeEnv.getFormulaFactory(), typeEnv);
 		return new MachineBoundInfo(expr, scBound);
 	}
Where <tt>MachineBoundInfo</tt> will be our class representing the state for the bound of the traversed machine.

To use a registered state of the repository, one can use 
 repository.getState(IStateType<? extends IPOGState> stateType)

As we suppose the MachineBoundInfo to be a state available after our module is initialized, we will here use :
 final IMachineBoundInfo machineBoundInfo = (IMachineBoundInfo) repository.getState(IMachineBoundInfo.STATE_TYPE);

Sub-modules of our module <tt>fwdMachineBoundFinitenessModule</tt> can use this state freely from the repository using the above invocation.
What we want to do is creating a ''BFN'' PO if the bound expression is not trivially finite. A trivially finite expression is an integer expression or derived from a boolean type.

Here is the code that makes those checkings :

 private boolean mustProveFinite(Expression expr, FormulaFactory ff) {
 		final Type type = expr.getType();
 		if (type.equals(ff.makeIntegerType()))
 			return false;
 		if (derivedFromBoolean(type, ff))
 			return false;
 		return true;
 	}

 private boolean derivedFromBoolean(Type type, FormulaFactory ff) {
 	if (type.equals(ff.makeBooleanType()))
 		return true;
 	final Type baseType = type.getBaseType();
 	if (baseType != null)
 		return derivedFromBoolean(baseType, ff);
 	if (type instanceof ProductType) {
 		final ProductType productType = (ProductType) type;
 		return derivedFromBoolean(productType.getLeft(), ff) && derivedFromBoolean(productType.getRight(), ff);
 	}
 	return false;
 }

Here is the corresponding code that generates the PO ''BFN'' put into the process() method of our module :

 		final IMachineBoundInfo machineBoundInfo = (IMachineBoundInfo) repository.getState(IMachineBoundInfo.STATE_TYPE);
  		final ISCBound scBound = machineBoundInfo.getBound();
 		final Expression expr = machineBoundInfo.getExpression();
 		final FormulaFactory ff = repository.getFormulaFactory();
 		final IPOGSource[] sources = new IPOGSource[] { makeSource(IPOSource.DEFAULT_ROLE, scBound.getSource()) };
 		final IPORoot target = repository.getTarget();
 		final IMachineHypothesisManager machineHypothesisManager = (IMachineHypothesisManager) repository.getState(IMachineHypothesisManager.STATE_TYPE);
 
 		// if the finitness of bound is not trivial
 		// we generate the PO
 		if (mustProveFinite(expr, ff)) {
 			final Predicate finPredicate = ff.makeSimplePredicate(Formula.KFINITE, expr, null);
 			createPO(target, "BFN",
 					POGProcessorModule.makeNature("Finiteness of bound"),
 					machineHypothesisManager.getFullHypothesis(),
 					makePredicate(finPredicate, scBound.getSource()), sources,
 					machineHypothesisManager.machineIsAccurate(), monitor);

Add this module to the configuration created for the static checker by creating an extension <tt>pogModule</tt>.

=== Step 2 : creating the support for sharing bound informations among POG sub-modules ===

We will here create the extension to store the informations about the statically checked bound which we want available through sub-modules. 
To do this : add the <tt>org.eventb.core.pogStateTypes</tt> extension point to our plugin.
Then create an extension <tt>stateType</tt> :<br>
- id : machineBoundInfo<br>
- name : POG Machine Bound Info<br>
- class : a new class that will implement the interface described below (here <tt>MachineBoundInfo</tt>).<br>

We want three methods to be available in this interface :
* <tt>getExpression()</tt> to retrieve the expression of the bound,
* <tt>getBound()</tt> to retrieve the statically checked bound,
* <tt>hasMachineBound()</tt> telling if the currently processed machine has a bound or not.

Here is the interface <tt>IMachineBoundInfo</tt> one has to create: 

 public interface IMachineBoundInfo extends IPOGState {
 
 	final static IStateType<IMachineBoundInfo> STATE_TYPE = POGCore.getToolStateType(QualProbPlugin.PLUGIN_ID + ".machineBoundInfo");
 	
 	/**
 	 * Returns the parsed and type-checked bound expression, or <code>null</code> 
 	 * if the machine does not have a bound.
 	 * 
 	 * @return the parsed and type-checked bound expression, or <code>null</code> 
 	 * 		if the machine does not have a bound
 	 */
 	Expression getExpression();
 	
 	/**
 	 * Returns a handle to the bound, or <code>null</code> if the machine does not have a bound.
 	 * 
 	 * @return a handle to the bound, or <code>null</code> if the machine does not have a bound
 	 */
 	ISCBound getBound();
 	
 	/**
 	 * Returns whether the machine has a bound.
 	 * 
 	 * @return whether the machine has a bound
 	 */
 	boolean machineHasBound();
 
 }

and here is its implementation class :

 public class MachineBoundInfo implements IMachineBoundInfo {
 
 	private final Expression boundExpression;
 	private final ISCBound bound;
  	private boolean immutable;
 
 	/**
 	 * Constructor
 	 */
 	public MachineBoundInfo(final Expression expression, final ISCBound bound) {
 		this.boundExpression = expression;
 		this.bound = bound;
 		immutable = false;
 	}
 	
 	/**
 	 * Constructor with no bound attached
 	 */
 	public MachineBoundInfo() {
 		this.boundExpression = null;
 		this.bound = null;
 		immutable = false;
 	}
 
 	@Override
 	public String toString() {
 		return boundExpression == null ? "null" : boundExpression.toString();
 	}
  	
 	public Expression getExpression() {
 		return boundExpression;
 	}
 
 	public ISCBound getBound() {
 		return bound;
 	}
 
 	public IStateType<?> getStateType() {
 		return IMachineBoundInfo.STATE_TYPE;
 	}
 
 	public boolean machineHasBound() {
 		return boundExpression != null;
 	}
 
 	@Override
 	public void makeImmutable() {
 		immutable = true;
 	}
 
 	@Override
 	public boolean isImmutable() {
 		return immutable;
 	}
 
 }

=== Step 3 : Removing a PO ===

To remove a PO, one has to create a filter module. This can be done the same way as for the static checker. After a small search in the package <tt>org.eventb.internal.core.pog.modules</tt>, we identify that the module responsible of creating the ''FIN'' PO is actually <tt>FwdMachineVariantModule</tt>. The goal is here to register our filter as a submodule of <tt>FwdMachineVariantModule</tt> and prevent it to create the ''FIN'' PO.

The code is really simple... First, one has to check if the model contains a probabilistic event, which means that we want to override the ''FIN'' PO that will be created by default, 
and then, one has to search from the generated PO if one corresponds to ''FIN'' by searching inside PO names and reject it.

1. give the module an id (here <tt>finPORejectingModule</tt>),<br>
2. a human readable name (here "Machine POG Filter FIN PO Rejecting Module"),<br>
3. register a parent in the hierarchy of modules (here we used the variant POG module of the Event-B POG that creates the POG we want to suppress: <tt>org.eventb.core.fwdMachineVariantModule</tt>),<br>
4. create a class for this module.(here we created the class <tt>fr.systerel.rodinextension.sample.pog.modules.FinPORejectingModule</tt>).

Here we just retrieve the bound informations in the <tt>initModule()</tt>, so we can check in the <tt>accept()</tt> method that the current machine aims to be proved against probabilistic convergence, and remove the ''FIN'' PO which is about to be created. Here is what the code might look like :

 public class FinPORejectingFilterModule extends POGFilterModule {
 
 	private static final IModuleType<FinPORejectingFilterModule> MODULE_TYPE = POGCore.getModuleType(QualProbPlugin.PLUGIN_ID + ".finPORejectingModule");
 	private IMachineBoundInfo boundInfo;
 
 	@Override
 	public IModuleType<?> getModuleType() {
 		return MODULE_TYPE;
 	}
 
 	@Override
 	public boolean accept(String poName, IProgressMonitor monitor) throws CoreException {
 		if (! boundInfo.machineHasBound()) {
 			return true;
 		}
 		final boolean rejectedFIN = poName.equals("FIN");
 		if (QualProbPlugin.DEBUG) {
 			System.out.println("PO " + poName + " is "+ (rejectedFIN ? "" : "not ") + "filtered out.");
 		}
 		return !rejectedFIN;
 	}
 
 	@Override
 	public void initModule(IPOGStateRepository repository, IProgressMonitor monitor) throws CoreException {
 		boundInfo = (IMachineBoundInfo) repository.getState(IMachineBoundInfo.STATE_TYPE);
 	}
 
 	@Override
 	public void endModule(IPOGStateRepository repository, IProgressMonitor monitor) throws CoreException {
 		boundInfo = null;
 	}
 
 }

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