The increased complexity of software systems together with the lack of tools and technique to support their development led to the so-called "software crisis". Different views about the problem originated diverse approaches to a possible solution, although it is now generally accepted that a "silver bullet" does not exist.
The formal methods view considers mathematical reasoning as fundamental to fulfill the most important property of software systems: correctness. However, since correctness proofs are generally difficult and expensive, only critical applications are regarded as potential targets for their use. Developments in tool support such as proof assistants, model checkers and abstract interpreters allows for reducing this cost and making proofs affordable to a wider range of applications. Nevertheless, the effectiveness of a tool is highly dependent of the underlying theory.
This dissertation follows a calculational proof style in which equality plays the fundamental role. Instead of the traditional logical approach, fork algebras, an extension of relation algebras, are used. In this setting, Galois connections are important because they reinforce the calculational nature of the algebraic approach, bringing additional structure to the calculus. Moreover, Galois connections enjoy several valuable proper- ties and allow for transformation in the domain of problems. In this dissertation, it is shown how fork algebras and Galois connections can be integrated together with the indirect equality principle. This combination offers a very powerful, generic device to tackle the complexity of proofs in program verification. This power is enhanced with the design of an innovative proof assistant prototype, the Galculator.
