## Section: New Results

### Coq as a functional programming language

Participants : Pierre Boutillier, Guillaume Claret, Lourdes Del Carmen González Huesca, Thibaut Girka, Hugo Herbelin, Pierre Letouzey, Matthias Puech, Yann Régis-Gianas, Matthieu Sozeau, Arnaud Spiwack.

#### Type classes and libraries

Type Classes are heavily used in the HoTT/Coq library (http://github.com/HoTT/coq ) started by the Univalent Foundations program at the IAS, to which Matthieu Sozeau participated. To ease the development of this sophisticated library, Matthieu Sozeau implemented a number of extensions to type class resolution to make it more predictable and efficient. These are now part of the Coq 8.5 release.

#### Dependent pattern-matching

The dissertation of Pierre Boutillier presents and formalises a new algorithm to compile dependent pattern-matching into a chain of Coq case analyses. It avoids the use of the “uniqueness of identity proofs” axiom in more cases than the former proposal by McBride and McKinna.

#### Incrementality in proof languages

Lourdes del Carmen González Huesca and Yann Régis-Gianas developed a new variant of the differential lambda calculus that has two main features: (i) it is deterministic ; (ii) it is based on a notion of a first-class changes. A paper is in preparation.

#### Proofs of programs in Coq

In collaboration with David Mentre (Mitsubishi Rennes), Thibaut Girka and Yann Régis-Gianas worked on a certified generator for correlating programs. A correlating program is a program that represents the semantic difference between two (close) versions of a program by performing a static scheduling of their instructions. Performing an abstract interpretation on the correlating program provides a representation of the semantic differences between the two versions of a program. A paper is written and should be submitted soon.

#### Typed tactic language

In collaboration with Beta Ziliani (MPI) and Thomas Refis (master 2 student at University Paris Diderot), Yann Régis-Gianas starts the development of the version 2 of Mtac, a tactic language for Coq. Mtac is a DSL embedded in the Coq proof assistant. Roughly speaking, it allows Coq to be used as a tactic language for itself. With this work, Mtac 2 now includes first class goals. A paper is in preparation.

#### Tactic engine

Arnaud Spiwack joined the team for two months (Sept—Oct 2014) to finalise the integration and documentation of his re-engineering of Coq's interactive proof engine for the v8.5 version. The new perspective taken by this new engine is to shift the primary focus from how tactics (proof instructions) can modify goals (proof obligations) to focus on the way tactics compose. By making sure that composition of tactics has good mathematical properties, the new engine makes it possible to combine tactics in a more predictable and more powerful way. This new engine is also notable for the introduction of an abstract interface for tactics and tactic composition which makes it easy to augment tactics with new capabilities. The most notable such features are so-called dependent subgoals, which makes more fine-grained proofs possible and significantly improves the support for dependent types; and backtracking which gives the possibility to deploy very modular proof-search components. During his two months in the team, Arnaud Spiwack also added support for tracing tactic execution (Info), again taking advantage of his modular design.

#### Effectful programming

Guillaume Claret and Yann Régis-Gianas developed a compiler from a subset of OCaml with effects to Coq. Possible effects are the exceptions, the global references and the non-termination. Guillaume Claret and Yann Régis-Gianas developed Pluto, a concurrent HTTP web server written in Gallina. They worked on techniques to certify such interactive programs, formalising the reasoning by use cases. Use cases are proven correct giving a scenario, a typed schema of interactions between a program and an environment, built using the tactic mode of Coq as a symbolic debugger.

#### Libraries

Sébastien Hinderer and Pierre Letouzey contributed an extended library of lists. Pierre Letouzey contributed an extended library about Peano numbers, that takes advantages of the “Numbers” modular framework done earlier.