This paper proposes new derivations of three well-known sorting algorithms, in their functional formulation. The approach we use is based on three main ingredients: first, the algorithms are derived from a simpler algorithm, i.e. the specification is already a solution to the problem (in this sense our derivations are program transformations). Secondly, a mixture of inductive and coinductive arguments are used in a uniform, algebraic style in our reasoning. Finally, the approach uses structural invariants so as to strengthen the equational reasoning with logical arguments that cannot be captured in the algebraic framework.