## Section: New Results

### Proof and type theory modulo rewriting

Ali Assaf defined a sound and complete embedding of the cumulative universe
hierarchy of the *calculus of inductive constructions* (CIC) in the
$\lambda \Pi $-calculus modulo rewriting [18] .
By reformulating universes in the Tarski style, he showed that we can make
cumulativity explicit without losing any typing power. This result
refines the translation used by Coqine, which was unsound because it
collapsed the universe hierarchy to a single type universe. It also sheds
some light on the metatheory of Coq and its connection to Martin-Löf’s
intuitionistic type theory. This work was presented at the TYPES meeting
in Paris.

Frédéric Gilbert and Olivier Hermant defined new encodings from classical
to intuitionistic first-order logic. These encodings, based on the
introduction of double negations in formulas, are tuned to satisfy two purposes jointly:
basing their specifications on the definition of *classical connectives*
inside intuitionistic logic – which is the property of *morphisms*,
and reducing their impact on the shape and size of formulas,
by limiting as much as possible the number of negations
introduced. This paper has been submitted.

Raphael Cauderlier and Catherine Dubois defined a shallow embedding of an object calculus (formalized by Abadi and Cardelli), in the $\lambda \Pi $-calculus modulo rewriting . The main result concerns the encoding of subtyping. This encoding shows that rewriting is an effective help for handling of subtyping proofs. The implementation in Dedukti, Sigmaid . This work has been presented at the TYPES 2014 meeting in Paris. A paper has been submitted.

Ali Assaf, Olivier Hermant and Ronan Saillard defined a rewrite system such that all strongly normalizable proof term can be typed in Natural Deduction modulo this rewrite system. This work is inspired by Statman's work [49] , and can be understood as an encoding of intersection types.

Guillaume Burel showed how to get rewriting systems that admit cut
by using standard saturation techniques from automated theorem
proving, namely ordered resolution with selection, and
superposition. This work relies on a view of proposition rewriting
rules as oriented clauses, like term rewriting rules can be seen as
oriented equations. This also lead to introduce an extension of
deduction modulo with *conditional* term rewriting rules. This
work was presented at the RTA-TLCA conference in
Vienna [15] .

Gilles Dowek, has generalized the notion of super-consistency to the lambda-Pi-calculus modulo theory and proved this way the termination of the embedding of various formulations of Simple Type Theory and of the Calculus of Constructions in the Lambda-Pi calculs modulo theory.

Gilles Dowek and Alejandro Díaz-Caro have finished their work on the extension of Simply Typed Lambda-Calculus with Type Isomorphisms. This work has been presented at the Types meeting and recently accepted for publication in the Theoretical Computer Science journal [26] .

Gilles Dowek and Ying Jiang have given a new proof of the decidability of reachability in alternating pushdown systems, based on a cut-elimination theorem.

Vaston Costa presented to the group a new structure to represent proofs through references rather than copy. The structure, called Mimp-graph, was initially developed for minimal propositional logic but the results have been extended to first-order logic. Mimp-graph preserves the ability to represent any Natural Deduction proof and its minimal formula representation is a key feature of the mimp-graph structure, it is easy to distinguish maximal formulas and an upper bound in the length of the reduction sequence to obtain a normal proof. Thus a normalization theorem can be proved by counting the number of maximal formulas in the original derivation. The strong normalization follow as a direct consequence of such normalization, since that any reduction decreases the corresponding measures of derivation complexity. Sharing for inference rules is performed during the process of construction of the graph. This feature is very important, since we intend to use this graph in automatic theorem provers.