Difference between pages "Rodin Proving Perspective" and "The Use of Theories in Code Generation"

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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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= Defining Translations Using The Theory Plug-in =
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The theory plug-in is used to add mathematical extensions to Rodin. The theories are created, and deployed, and can then be used in any models in the workspace. When dealing with implementation level models, such as in Tasking Event-B, we need to consider how to translate newly added types and operators into code. We have augmented the theory interface with a Translation Rules section. This enables a user to define translation rules that map Event-B formulas to code.
== Overview ==
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== Translation Rules==
The Proving Perspective is made of a number of windows (views): the Proof Tree, the Goal, the Selected Hypotheses, the Proof Control, the Proof Information, and the Search Hypotheses. In subsequent sections, we study each of these windows. Below is a screenshot of the proving perspective:
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<div id="fig:Translation Rules">
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<br/>
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[[Image:TheoryCGRules.png|center||caption text]]
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<center>'''Figure 1''': Translation Rules</center>
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<br/>
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</div>
  
[[Image:ProvPers.png|center]]
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Figure 1 shows the interface, and some translations rules of the mapping to Ada.
  
== Loading a Proof ==
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The theory is given a name, and may import some other theories. Type parameters can be added, and we use them here to type the meta-variables. The meta-variable ''a'' is restricted to be an integer type, but meta-variable ''c'' can be any type. Meta-variables are used in the translator rules for pattern matching.
In order to load a proof, switch to the Proving Perspective, select the project from the Event-B Explorer, select and expand the component (context or machine), finally select the proof obligation of interest. The corresponding proof will be loaded.
 
  
== The Proof Tree ==
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Translator rules are templates, which are used in pattern matching. Event-B formulas are defined on the left hand side of the rule, and the code to be output (as text) appears on the right hand side of the matching rule. During translation an abstract syntax tree (AST) representation of the formula is used. The theory plug-in attempts to match the formulas in the rules with each syntactic element of the AST. As it does so it builds the textual output as a string, until the whole AST has been successfully matched. When a complete tree is matched, the target code is returned. If the AST is not matched, a warning is issued, and a string representation of the original formula is returned.
The proof tree describe each individual proof step in the proof. The proof tree can be seen in its corresponding window as seen in the following screenshot:
 
  
[[Image:ProTree.png|center]]
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== Type Rules for Code Generation ==
  
Each line in the 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. Each node is labelled with a comment (description) explaining how it can be discharged. By selecting a node in the proof tree, the corresponding sequent is loaded: the hypotheses of the sequent are loaded to the Selected Hypotheses window, and the goal of the sequent is loaded to the Goal window.
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The type rules section, shown in Figure 1, is where the relationship is defined, between Event-B types and the type system of the implementation.
  
=== Decoration===
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= Adding New (implementation-level) Types =
The leaves of the tree are decorated with three kinds of logos:
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When we are working at abstraction levels close to the implementation level, we may make an implementation decision which requires the introduction of a new type to the development. We give an example of our approach, where we add a new array type, shown in Figure 2, and then define its translation to code.
  
* a green logo with a "'''<math>\surd </math>'''" in it means that this leaf is discharged,
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== An Array Type Definition ==
* a brown logo with a "'''?'''" in it means that this leaf is not discharged,
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<div id="fig:Extension with an Array Type">
* a blue logo with a "'''R'''" in it means that this leaf has been reviewed.
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<br/>
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[[Image:ArrayDef.png|center||caption text]]
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<center>'''Figure 2''': Array Definition</center>
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<br/>
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</div>
  
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: the node will become "brown" again (undischarged) and you can now try to discharge it.
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The array operator notation is defined in the expression array(s: P(T)); and the semantics is defined in the direct definition. arrayN constrains the arrays to be of fixed length. Array lookup, update, and constructor operators are subsequently defined. In the next step we need to define any translations required to implement the array in code.
  
=== Navigation within the Proof Tree===
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== Translation Rules ==
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
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<div id="Translation Rules for the Array Type">
* the "'''+'''" button allows you to fully expand the proof tree
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<br/>
* the "'''-'''" allows you to fully collapse the tree: only the root stays visible.
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[[Image:ArrayTrans.png|center||caption text]]
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<center>'''Figure 3''': Translation Rules for the Array Type</center>
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<br/>
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</div>
  
=== Manipulating the Proof Tree===
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Figure 3 shows the Ada translation; beginning with the meta-variable definitions that are used for pattern matching in the translation rules. Each of the operators; ''newArray'', and ''update'', and an expression using the ''lookup'' operator, are mapped to their implementations on the right hand side of the rule. The ''Type Rules'' section describes the implementation's description of the ''arrayN'' type.
 
 
==== Hiding ====
 
The little triangle (with a "+" or "-" inside) 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 brown decorated. This allows you to resume the proof from that node. After selecting a sequent in the proof tree, pruning can be performed by right-clicking and then selecting "Prune".
 
 
 
Note that after pruning, the post-tactic is not applied to the new current sequent: if needed you have to press the "post-tactic" button in the Proof Control window. 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:
 
[[Image:GoalHyp.png|center]]
 
 
 
A selected hypothesis can be deselected (and as a result becomes hidden) by first clicking in the box (check 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:
 
 
 
[[Image:GoalHypSelect.png|center]]
 
 
 
Here is the result:
 
 
 
[[Image:GoalHypSelectRes.png|center]]
 
 
 
Notice that the deselected hypotheses are not lost: you can get them back by means of the Search Hypotheses window. The other two buttons next to the red ('''-''') button allow the user (in the order of their appearance from left to right) to select all hypotheses as well as inverse the current selection.
 
 
 
The ('''ct''') button next to the goal allows you to do a proof by contradiction: by pressing it, the negation of the goal becomes a selected hypothesis whereas the goal becomes "'''⊥'''".
 
 
 
The ('''ct''') button next to a selected hypothesis allows you to do another kind of proof by contradiction: by pressing it, the negation of the concerned hypothesis becomes the goal whereas the negated goal becomes an hypothesis.
 
 
 
=== Applying Proof Rules ===
 
The red hyperlinks as well as the ('''ct''') buttons (also occasionally other green filled button next to it e.g., AND introduction in goal) allows the user to carry out interactive proofs. They are used to invoke proof rules (rewrite rules as well as inference rules).
 
 
 
[[Image:ApplyRewRule.png|center]]
 
 
 
==== Rewrite Rules ====
 
Rewrite rules are one-directional equalities (and equivalences) that can be used to simplify formulas (the goal or a single hypothesis). A rewrite rule is applied from left to right either in the goal or in one of the selected hypotheses, when its ''side condition'' holds.
 
 
 
A rewrite rule is applied either automatically ('''A''') or manually ('''M'''):
 
* automatically, when post tactics are enabled. These rules are equivalence simplification laws.
 
* manually, through an interactive command. These rules gathers non equivalence laws, definition laws, distributivity laws and derived laws.
 
 
 
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:
 
 
 
<center><math>  \True  = \False  \;\;\defi\;\;  \bfalse </math></center>
 
 
 
should also produce the rule:
 
 
 
<center><math>  \False  = \True  \;\;\defi\;\;  \bfalse </math></center>
 
 
 
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 (see [[Proof 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.
 

Revision as of 15:51, 15 May 2012

Defining Translations Using The Theory Plug-in

The theory plug-in is used to add mathematical extensions to Rodin. The theories are created, and deployed, and can then be used in any models in the workspace. When dealing with implementation level models, such as in Tasking Event-B, we need to consider how to translate newly added types and operators into code. We have augmented the theory interface with a Translation Rules section. This enables a user to define translation rules that map Event-B formulas to code.

Translation Rules


caption text
Figure 1: Translation Rules


Figure 1 shows the interface, and some translations rules of the mapping to Ada.

The theory is given a name, and may import some other theories. Type parameters can be added, and we use them here to type the meta-variables. The meta-variable a is restricted to be an integer type, but meta-variable c can be any type. Meta-variables are used in the translator rules for pattern matching.

Translator rules are templates, which are used in pattern matching. Event-B formulas are defined on the left hand side of the rule, and the code to be output (as text) appears on the right hand side of the matching rule. During translation an abstract syntax tree (AST) representation of the formula is used. The theory plug-in attempts to match the formulas in the rules with each syntactic element of the AST. As it does so it builds the textual output as a string, until the whole AST has been successfully matched. When a complete tree is matched, the target code is returned. If the AST is not matched, a warning is issued, and a string representation of the original formula is returned.

Type Rules for Code Generation

The type rules section, shown in Figure 1, is where the relationship is defined, between Event-B types and the type system of the implementation.

Adding New (implementation-level) Types

When we are working at abstraction levels close to the implementation level, we may make an implementation decision which requires the introduction of a new type to the development. We give an example of our approach, where we add a new array type, shown in Figure 2, and then define its translation to code.

An Array Type Definition


caption text
Figure 2: Array Definition


The array operator notation is defined in the expression array(s: P(T)); and the semantics is defined in the direct definition. arrayN constrains the arrays to be of fixed length. Array lookup, update, and constructor operators are subsequently defined. In the next step we need to define any translations required to implement the array in code.

Translation Rules


caption text
Figure 3: Translation Rules for the Array Type


Figure 3 shows the Ada translation; beginning with the meta-variable definitions that are used for pattern matching in the translation rules. Each of the operators; newArray, and update, and an expression using the lookup operator, are mapped to their implementations on the right hand side of the rule. The Type Rules section describes the implementation's description of the arrayN type.