User:Jshflynn/Length of Concatenation

Theorem
Let $\Sigma$ be an alphabet.

Let $x$ and $y$ be words over $\Sigma$ and $\circ$ denote concatenation.

Then $\operatorname{len}\left({x \circ y}\right) = \operatorname{len}(x) + \operatorname{len}(y)$.

Proof
If $x=\lambda$ or $y=\lambda$ then the result follows immediately from the definition of concatenation with the empty word and the definition of word length.

Otherwise, from the definition of concatenation:


 * $x \circ y$ is a mapping from $[1..\operatorname{len}(x) + \operatorname{len}(y)]$ to $\Sigma$.

From the definition of the length of a word:


 * $\operatorname{len}(x \circ y)$

is the cardinality of the domain of $x \circ y$ when viewed as a sequence.

Hence $\operatorname{len}(x \circ y) = \operatorname{len}(x) + \operatorname{len}(y)$.