Structured Types

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Modelling Structured Types

The Event-B mathematical language currently does not support a syntax for the direct definition of structured types such as records or class structures. Nevertheless it is possible to model structured types using projection functions to represent the fields/ attributes. For example, suppose we wish to model a structured type C with fields e and f (with type E and F respectively). Let us use the following syntax for this (not part of Event-B):

 \textbf{STRUCT}~~~~ C &::&  e\in E\\
    &&  f \in F

We can model this structure in Event-B by introducing (in a context) a set C and two functions e and f as constants as follows:

 \textbf{SETS}~~ C\\
\textbf{CONSTANTS}~~ e,~ f\\
   e \in  C \tfun E\\
   f \in  C \tfun F\\

Names in the proof tree: Predicate Prover

Names in the preferences: PP restricted, PP after lasso, PP unrestricted

Input: In the configuration "restricted" all selected hypotheses and the goal are passed to New PP. In the configuration "after lasso" a lasso operation is applied to the selected hypotheses and the goal and the result is passed to New PP. The lasso operation selects any unselected hypothesis that has a common symbol with the goal or a hypothesis that was selected before. In the configuration "unrestricted" all the available hypotheses are passed to New PP.

How the Prover Proceeds: First, all function and predicate symbols that are different from "\in" and not related to arithmetic are translated away. For example A \subseteq B is translated to \forall x\cdot x \in A \limp x \in B. Then New PP translates the proof obligation to CNF (conjunctive normal form) and applies a combination of unit resolution and the Davis Putnam algorithm.

Some Strengths:

  • New PP outputs a set of "used hypotheses". If an unused hypotheses changes, the old proof can be reused.
  • New PP has limited support for equational reasoning.