# [isabelle-dev] Partial functions

Manuel Eberl eberlm at in.tum.de
Wed May 22 13:50:45 CEST 2013

```Hallo,

partial function in Isabelle are usually modelled as 'a ⇀ 'b, which is
an abbreviation for 'a ⇒ 'b option. There is syntax such as "[1 ↦ 2, 3 ↦
4]", meaning, basically, "λx. if x = 1 then Some 2 else if x = 3 then
Some 4 else None". There is also update syntax for such function, such
as "f(1 ↦ 3)", meaning "λx. if x = 1 then 3 else f x".

You could therefore write something like
definition f :: "nat ⇀ nat" where "f = [1 ↦ 4, 2 ↦ 4, 3 ↦ 5]"

You can find out the domain/range of such a partial function using dom
and ran. For instance, "dom f" is defined as {a. f a ≠ None}.

If you want to apply such a function, you can either do "case f x of
None ⇒ … | Some y ⇒ …", or you can simply write "the (f x)". "the" takes
an option value and extracts the "y" out of a "Some y" or returns
"undefined" if the option value is "None".

Cheers,
Manuel

On 22/05/13 13:35, Roger H. wrote:
>
> Hallo,
>
>
> i want to create a datatype that allows me to write functions from a
> nat subset to another nat subset.
> for example i wanna be able to write:
>
>
> definition f: {1,2,3} => {4,5}
> 1 -->4, 2-->4, 3-->5
>
>
> or another one:
>
>
> definition g : {6,8} => {2,3,4}
>
>
>
> So the thing i want to somehow parametrize is "which subset of the nat
> im using each time as domain and range" ,
>
>
> I thought about creating a new datatype : 'a nat
> The problem with this is that 'a is instantiated with datatypes, and
> not sets like {1,2,3}.
>
>
>
>
> 1. Everytime i want declare a new function, i have to declare by
> "typedef" the nat subsets i want as domain and range
>
>
>
>
> 2. definition f : "nat => nat" where
> "f x = (if x : {1,2,3} then .... else undefined)
>
>
> This second approach is bad, cause i dont want the domain to be
> decided as late as the second line of the declaration, cause after
> this i want to be able to program a selector "domain f"
> that returns me the domain of f, thats why i want the domain of f to
> be somehow incapsulated (parametrized) in the first line "f: nat =>nat
> " so that i can use it later.
>
>
>
> What would you do in this situation?
>
>
> Many thanks!
>
>
>
>
>
>
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```