The reverse of a string, introduced informally above, can be defined more precisely by the recursive rules aR = a, (wa)R = awR, for all a ∈ Σ, w ∈ Σ*. Use this to prove that (uv)R = vRuR, for all u, v ∈ Σ+.

Johnson enterprises uses a computer to handle its sales invoices. lately, business has been so good that it takes an extra 3 hours per night, plus every third saturday, to keep up with the volume of sales invoices. management is considering updating its computer with a faster model that would eliminate all of the overtime processing.