Function as compatible transformation

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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 [math]\displaystyle{ k }[/math] over [math]\displaystyle{ F }[/math], we say a function [math]\displaystyle{ l }[/math] over arithmetical expression space [math]\displaystyle{ H }[/math] is a promotion of [math]\displaystyle{ k }[/math], if and only if below diagram commute.

Function-diagram.png

where [math]\displaystyle{ \nu }[/math] is the evaluation function of arithmetical expression.

Different implementations by trace

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

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