(*<*)theory Overloading0 = Main:(*>*)

text{* We start with a concept that is required for type classes but already

useful on its own: \emph{overloading}. Isabelle allows overloading: a

constant may have multiple definitions at non-overlapping types. *}

subsubsection{*An initial example*}

text{*

If we want to introduce the notion of an \emph{inverse} for arbitrary types we

give it a polymorphic type *}

consts inverse :: "'a \<Rightarrow> 'a"

text{*\noindent

and provide different definitions at different instances:

*}

defs (overloaded)

inverse_bool: "inverse(b::bool) \<equiv> \<not> b"

inverse_set:  "inverse(A::'a set) \<equiv> -A"

inverse_pair: "inverse(p) \<equiv> (inverse(fst p), inverse(snd p))"

text{*\noindent

Isabelle will not complain because the three definitions do not overlap: no

two of the three types @{typ bool}, @{typ"'a set"} and @{typ"'a \<times> 'b"} have a

common instance. What is more, the recursion in @{thm[source]inverse_pair} is

benign because the type of @{term[source]inverse} becomes smaller: on the

left it is @{typ"'a \<times> 'b \<Rightarrow> 'a \<times> 'b"} but on the right @{typ"'a \<Rightarrow> 'a"} and

@{typ"'b \<Rightarrow> 'b"}. The annotation @{text"(overloaded)"} tells Isabelle that

the definitions do intentionally define @{term[source]inverse} only at

instances of its declared type @{typ"'a \<Rightarrow> 'a"} --- this merely supresses

warnings to that effect.

However, there is nothing to prevent the user from forming terms such as

@{term[source]"inverse []"} and proving theorems as @{prop[source]"inverse []

= inverse []"}, although we never defined inverse on lists. We hasten to say

that there is nothing wrong with such terms and theorems. But it would be

nice if we could prevent their formation, simply because it is very likely

that the user did not mean to write what he did. Thus he had better not waste

his time pursuing it further. This requires the use of type classes.

*}

(*<*)end(*>*)