Summation - Formal Definition

Formal Definition

If the iterated function notation is defined e.g. and is considered a more primitive notation then summation can be defined in terms of iterated functions as:

$\left\{b+1,\sum_{i=a}^b g(i)\right\} \equiv \left( \{i,x\} \rightarrow \{ i+1 ,x+g(i) \}\right)^{b-a+1} \{a,0\}$

Where the curly braces define a 2-tuple and the right arrow is a function definition taking a 2-tuple to 2-tuple. The function is applied b-a+1 times on the tuple {a,0}.

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