Towards Formal Proofs of Feedback Control Theory

Control theory can establish properties of systems which hold with all signals within the system and hence cannot be proven by simulation. The most basic of such property is the stability of a control subsystem or the overall system. Other examples are statements on robust control performance in the face of dynamical uncertainties and disturbances in sensing and actuation. Until now these theories were developed and checked for their correctness by control scientist manually using their mathematical knowledge. With the emergence of formal methods, there is now the possibility to derive and prove robust control theory by symbolic computation on computers. There is a demand for this approach from industry for the verification of practical control systems with concrete numerical values where the applicability of a control theorem is specialised to an application with given numerical boundaries of parameter variations. The paper gives an overview of the challenges of the area and illustrates them on a computer-based formal proof of the Small-gain theorem and conclusions are drawn from these initial experiences

Automation for interactive proof: First prototype

AbstractInteractive theorem provers require too much effort from their users. We have been developing a system in which Isabelle users obtain automatic support from automatic theorem provers (ATPs) such as Vampire and SPASS. An ATP is invoked at suitable points in the interactive session, and any proof found is given to the user in a window displaying an Isar proof script. There are numerous differences between Isabelle (polymorphic higher-order logic with type classes, natural deduction rule format) and classical ATPs (first-order, untyped, and clause form). Many of these differences have been bridged, and a working prototype that uses background processes already provides much of the desired functionality

ProofPeer - A Cloud-based Interactive Theorem Proving System

ProofPeer strives to be a system for cloud-based interactive theorem proving. After illustrating why such a system is needed, the paper presents some of the design challenges that ProofPeer needs to meet to succeed. Contexts are presented as a solution to the problem of sharing proof state among the users of ProofPeer. Chronicles are introduced as a way to organize and version contexts

From LCF to Isabelle/HOL

Interactive theorem provers have developed dramatically over the past four decades, from primitive beginnings to today's powerful systems. Here, we focus on Isabelle/HOL and its distinctive strengths. They include automatic proof search, borrowing techniques from the world of first order theorem proving, but also the automatic search for counterexamples. They include a highly readable structured language of proofs and a unique interactive development environment for editing live proof documents. Everything rests on the foundation conceived by Robin Milner for Edinburgh LCF: a proof kernel, using abstract types to ensure soundness and eliminate the need to store proofs. Compared with the research prototypes of the 1970s, Isabelle is a practical and versatile tool. It is used by system designers, mathematicians and many others