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Formalisation tools for classical analysis : a case study in control theory

Damien Rouhling 1, 2
1 MARELLE - Mathematical, Reasoning and Software
CRISAM - Inria Sophia Antipolis - Méditerranée
2 STAMP - Sûreté du logiciel et Preuves Mathématiques Formalisées
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : In this thesis, we put a library for analysis in the Coq proof assistant to the test through a case study in control theory. We formalise a proof of stability for the inverted pendulum, a standard example in control theory. Controlling the inverted pendulum is challenging because of its non-linearity, so that this system is often used as a benchmark for new control techniques. Through this case study, we identify issues in the tools that are currently available for the formalisation of classical analysis and we develop new ones in order to achieve our formalisation goal. In particular, we try to imitate the pen-and-paper proof style thanks to new notations and inference mechanisms. This is an essential step to make formal proofs more accessible to mathematicians. We then develop a new library for classical analysis in Coq that integrates these new tools and tries to palliate the limitations of the library we tested, especially in the domain of asymptotic reasoning. We also experiment with this new library on the same formal proof and draw lessons on its strengths and weaknesses. Finally, we sketch a new methodology in order to address the limitations of our library in the particular domain of computation. We exploit a technique called refinement to refactor the methodology of proof by reflection, a technique that automates proofs through computation and also reduces the size of proof terms. We implement this methodology on the example of arithmetic reasoning in rings and discuss how this work could be used to generalise existing tools.
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Damien Rouhling. Formalisation tools for classical analysis : a case study in control theory. Logic in Computer Science [cs.LO]. Université Côte d'Azur, 2019. English. ⟨NNT : 2019AZUR4058⟩. ⟨tel-02333396v2⟩

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