## Section: New Results

### Proof theory

Guillaume Burel, Gilles Dowek and Ying Jiang have introduced a general framework to prove the decidability of reachability and provability problems. This framework uses an analogy between the objects recognized by an automaton and cut-free proofs. Various aspects of this work have been published at FroCoS [19] , LPAR [21] , and another paper is in preparation.

Gilles Dowek's paper on the definition of the classical connectives and quantifiers has been published [30] .

Arnaud Spiwack gave a predicative shallow embedding of a weak version of system ${U}^{-}$ in dependent type theory, for Hurkens's paradox to hold. He also showed that a variety of incarnations of Hurkens's paradox are straightforward instantiations of this encoding, greatly simplifying existing proofs.

Arnaud Spiwack developped a topos-theoretic methodology to reason equationally on circuit languages. Results that hold for combinational circuits are lifted to sequential circuits thanks to a transfer principle. This approach allows, in particular, to simplify reasoning about more complex temporal gates than the unit delay. These results aim at enriching the compiler of the Faust audio signal processing programming language, which features such complex temporal gates.

For the sake of reliability, the kernels of Interactive Theorem Provers (ITPs) are kept relatively small in general. On top of the kernel, additional symbols and inference rules are defined. Some dependency analysis of symbols of HOL Light indicates that the depth of dependency could be reduced by introducing a few more symbols to the kernel. Shuai Wang showed that extending the kernel of HOL Light is a successful attempt to reduce proof size and speed up proof-checking. More specifically, symbols and inference rules of universal quantification and implication were added to the kernel. This approach has been proved to give equivalent proof-checking results with the size of the proof files reduced to 24% on average and a speedup of 38% for proof-checking overall.