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# Event-B Theories

ProB has (limited) support for theories.

Currently supported are (examples refer to the theory project below):

• recursive datatypes (e.g. the List datatype)
• operators defined by direct definitions (e.g. operators in the BoolOps theory) or recursive definitions (e.g. operators in the List theory)
• special annotated operators (see below)

Axiomatically defined operators are not supported without additional annotations.

An example project with theories: media:theories2.zip

The project contains the following theories:

SUMandPRODUCT
Contains two operators SUM and PRODUCT which take a set of the type POW(T**INT) as argument (with T being a type variable) and return the sum (resp.) product of all element's integer value.

The operators are annotated such that ProB uses an extra implementation.

Seq
The theory of sequences provides operators for sequences that are defined by direct definitions, thus supported by ProB.
Real (unsupported)
A theory of real numbers, currently unsupported by ProB.
Natural
A theory of inductive naturals (defined by a constant zero and a successor function).

The mkinat operator is annotated such that ProB uses an explicit implementation.

List
A theory of lists that are either empty or have a head and a tail
FixPoint (not really supported)
The theory is defined by direct definitions but they usually get so complex that ProB cannot cope with them.
closure
The operator for transitive closures is supported by ProB.

The operator is annotated such that ProB uses an extra implementation.

Card (contains no operators or data types)
BoolOps
Operators on Booleans (e.g. AND, OR) are defined by direct definitions and as such supported by ProB.
BinaryTree
Binary Trees are supported by ProB.

## Tagging operators

ProB has some extra support for certain operators. ProB expects an annotation to an operator that provides the information that it should use a specific implementation for an operator. Such tags are given in a .ptm file (ProB Theory Mapping). The file must have the same name as the theory.

For each annotated operator, the file contains a line of the form

```operator Name internal {Tag}
```

where Name is the name of the operator in the theory and Tag is a ProB internal name.

Currently are the following tags supported (with T being an arbitrary type):

Tag Description Expected type Return type
closure1 the transitive closure POW(T**T) POW(T**T)
SIGMA the sum of a set POW(T**INT) INT
PI the product of a set POW(T**INT) INT
mu returns the element of a singleton set POW(T) T
choose returns (deterministically) one element of a non-emtpy set POW(T) T
mkinat(zero,succ) returns an inductive natural number where zero and succ are the two operators of a natural number datatype with zero having no args and succ having one arg (an inductive natural) INT Inductive Nat