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##### PI.R2 - 2016

New Software and Platforms
Partnerships and Cooperations
Bibliography

## Section: New Results

### Formalisation work

Participants : Jean-Jacques Lévy, Daniel de Rauglaudre.

#### Proofs of algorithms on graphs

Jean-Jacques Lévy and Chen Ran (a PhD student of the Institute of Software, Beijing, visiting the Toccata team) pursue their work about formal proofs of algorithms. Their goal is to provide proofs of algorithms which ought to be both checked by computer and easily human readable. If these kinds of proofs exist for algorithms on inductive structures or recursive algorithms on arrays, they seem less easy to design for combinatorial structures such as graphs. In 2016, they completed proofs for algorithms computing the strongly connected components in graphs. There are mainly two algorithms: one by Kosaraju (1978) working in two phases (some formal proofs of it have already been achieved by Pottier with Coq-classic and by Théry and Gonthier with Coq-ssreflect), one by Tarjan (1972) working in a single pass.

Their proofs use a first-order logic with definitions of inductive predicates. This logic is the one defined in Why3 (research-team Toccata, Saclay). They widely use automatic provers interfaced by Why3. A very minor part of these proofs is also achieved in Coq. The difficulty of this approach is to combine automatic provers and intuitive design.

Part of this work (Tarjan 1972) is presented at JFLA 2017 in Gourette [30] A more comprehensive version is under submission to another conference [34]. Scripts of proofs can be found at http://jeanjacqueslevy.net/why3.

#### Formalization of theorems in Coq

This section reports on formalisation work by Daniel de Rauglaudre.

##### Puiseux' Theorem

Puiseux' theorem states that the set of Puiseux series (series with rational powers) is an algebraically closed field, i.e. every non-constant polynomial with Puiseux series coefficients admits a zero. This theorem was formalized in Coq a couple of years ago, but it depended on five ad hoc axioms. This year, all these axioms have been grouped together into the only axiom LPO (Limited Principle of Omniscience), stating that for each sequence of booleans, we can decide whether it is always false or if there is at least one true element. This formalized theorem now depends only on this axiom.