Coalgebraic Theories of Sequences in PVS

This paper explains the setting of an extensive formalisation of the theory of sequences ((nite and innnite lists of elements of some data type) in the Prototype Veriication System pvs. This formalisation is based on the characterisation of sequences as a nal coalgebra, which is used as an axiom. The resulting theories comprise standard operations on sequences like composition (or concatenation), ltering, attening, and their properties. They also involve the preex ordering and proofs that sequences form an algebraic complete partial order. The nality axiom gives rise to various reasoning principles, like bisimulation, simulation, invariance, and induction for admissible predicates. Most of the proofs of equality statements are based on bisimulations, and most of the proofs of preex order statements use simulations. Some signiicant aspects of these theories are described in detail. This coalgebraic formalisation of sequences is presented as a concrete example that shows the importance and usefulness of coalgebraic modeling and reasoning. Hopefully, it will help to convey the view that coalgebraic data types should form an intrinsic part of (future) languages for programming and reasoning. Therefore, some suggestions for an appropriate syntax for coalgebraic datatypes are included. The use of sequences as a nal coalgebra is demonstrated in two (standard) applications: a reenement result for automata involving sequences of actions, and a coalgebraic deenition plus correctness proof for an insert operation on ordered sequences.