Conclusions

• Every AXIOM expression has a unique and precise type
• AXIOM has an inference mechanism to help the user in the desk-calculator mode
• This mechanism uses coercions between types.
• A given mathematical object may have several implementations in AXIOM which one with is different type (for instance multivariate polynomials can be viewed recursively univariate or distributed).
• Types control internal representations and available methods.
• The AXIOM programming language offers types of types which are extremely convenient for implementing algebraic structures.
• Types are organized in a hierarchical manner.
• Types of types are called categories. Other types are called domains.
• An AXIOM function can return a type (or admit a type among its arguments). Such a function is called a constructor.