The recursive definition of a string:

A string is either 
- empty (no characters)
- a pair (x,s) where x is in Sigma and s is a string over Sigma

Write epsilon for the empty string

len(epsilon) = 0
len((x,s)) = 1 + len(s)

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Back to the array-like definition. How do we implement concatenation?
We need to define the "index deference function" for the concatenation
of two strings (w and v, say) to give the concatenated string c.

Example concatenation: 
"black" "dog" --> "blackdog"

w_0 ... w_{n-1} v_0 ... v_{k-1}

c(i) = { w(i)   if i < n
       { v(i-n) if i >= n