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Concepts
This section of the documentation covers the use of Objects, Libraries and References, Notations, Styles and Types and how each is created, edited and used in the Sym application.
Types
Every term in every expression in a Sym document has a type assigned to it. Every object definition has a type definition as one of its components. Since type definitions can include things like type variables, Sym uses a type inferencing process to determine the explicit types assigned to each term in every expression.
Subtypes
To say that S is a subtype of T means that any term of type S can appear in an expression where a term of type T is expected. Including subtypes in the type system allows, for example, putting an Integer in an expression that is defined to accept a Real.

Concepts
This section of the documentation covers the use of Objects, Libraries and References, Notations, Styles and Types and how each is created, edited and used in the Sym application.
- Objects
  An object is the basic building block of all of the symbolic content contained in a document - every term in every symbolic expression in Sym is an instance of an object. Many of the objects used in Sym have an object definition that specifies how the object looks and behaves, but it's also possible to create objects that don't have object definitions. All objects are identified by their name and object names are always defined in a specific namespace. All object definitions contain a type definition. Optionally, objects can also contain Properties, Clauses, Renderings, and a Description.
- Library Locations and References
  Locations are used to manage groups of libraries and documents. The purpose of locations is to make documents portable from one machine to another. To do that, locations can be online as well as on your local machine. If a document references online libraries, the required libraries will be downloaded automatically when the document is opened.
- Notations and Rendering
  The purpose of notations is to make it possible to easily change the way expressions are displayed. Notations are assigned to Document Modules, Unit Test Modules and Object Modules. The notation assigned determines which renderings are selected for each expression in the module.
- Styles
  Each notation refers to a stylesheet. Stylesheets define Styles. Styles are used in two places. First, they can be used to style text. Second, they can be used to style parts of an expression in a rendering. You can create new style and new stylesheets.
- Types
  Every term in every expression in a Sym document has a type assigned to it. Every object definition has a type definition as one of its components. Since type definitions can include things like type variables, Sym uses a type inferencing process to determine the explicit types assigned to each term in every expression.
  - Type Definitions
    Every object definition contains a type definition. The type definition defines the allowed types for the object and its parameters. It also determines the number of parameters that the object is allowed to have.
  - Conditions in Type Definitions
    Conditions in type definitions are used to further restrict the types that are valid for a type variable in a type definition. A type definition with conditions might look like Type: U → U with (condition)
  - Subtypes
    To say that S is a subtype of T means that any term of type S can appear in an expression where a term of type T is expected. Including subtypes in the type system allows, for example, putting an Integer in an expression that is defined to accept a Real.
  - The Object (Top) and Null (Bottom) Types
    Type systems with subtyping typically include a Top and Bottom type. The Top type has the property that for any type S, S is a subtype of Top and the Bottom type has the property that for any type S, Bottom is a subtype of S. In the type system for Sym the Top type is named Object and the Bottom type is named Null and the type relationships between these two types and any other type is assumed.
  - Built in Types
  - Type Functions
    A type function is a type (is allowed to appear in a type definition) that has one or more parameters. An example of a type function is Set[Real].
  - The Difference Between Types and Sets
    The type system in the Sym application does not assume that types are sets. Types are just names in the Sym type system. The relationship between two types is defined by declaring one type to be a subtype of another. The type system consists of a list of types and a list of subtype relationships between the types.

Subtypes

To say that S is a subtype of T means that any term of type S can appear in an expression where a term of type T is expected. Including subtypes in the type system allows, for example, putting an Integer in an expression that is defined to accept a Real.

In the Sym application, subtype relationships between types are declared explicitly in the Subtypes function in the Sym namespace. The Subtypes function has to contain clauses of the form "A subtype B".

The subtypes defined by the Subtypes function have to form a directed acyclic graph. The subtypes validator is run at the beginning of each build to make sure that there are no cycles in the list of subtypes returned by the Subtypes function (a build is also run when a file is opened). If any cycles (for example, A subtype B, B subtype C, C subtype A) are found, some of the clauses will be disabled to eliminate the cycles and an error message will be shown in the Error Manager.

The subtype validator will also test for clauses of the form "A subtype A". This clause is disabled and a warning message is generated because "A subtype A" is true by definition.

The type inferencing process refers to the Subtypes function when it needs to determine if one type is a subtype of another.

The list of type relationships provided by the Subtypes function can be extended in newly created libraries and in individual documents if desired.