# Function as compatible transformation

Function as compatible transformation is a method we can promote the traditional real function into arithmetic expression space, and then we can extended the traditional calculus into a bigger theoretical space. Compatible here means compatible condition, so we can treat the traditional real function as a projected image from a higher dimension space of arithmetical expression space to real number.

## Formal definition

For a given function $\displaystyle{ k }$ over $\displaystyle{ F }$, we say a function $\displaystyle{ l }$ over arithmetical expression space $\displaystyle{ H }$ is a promotion of $\displaystyle{ k }$, if and only if below diagram commute.

where $\displaystyle{ \nu }$ is the evaluation function of arithmetical expression.

## Different implementations by trace

For a given function $\displaystyle{ k }$ over $\displaystyle{ F }$ and $\displaystyle{ l }$ is a promotion of $\displaystyle{ k }$, there are many different implementations, some of them can draw a path on arithmetical expression space.

For example we can implement $\displaystyle{ l }$ by a path along the additional axis, or multiplicative axis, etc.