  Two constructive proofs on d-majorization and thermo-majorization are
provided. In the first part, we present a diagrammatic proof of the equivalence
between d-majorization and the existence of a proper stochastic matrix. We
explicitly construct the desired stochastic matrix by using a graphical
argument. In the second part, we present a constructive proof of the
equivalence between the Gibbs-preserving map and thermal operation in classical
systems. We construct the desired thermal operation and a heat bath which
emulates any Gibbs-preserving map with an arbitrary accuracy.
