Practical Subtyping for System F with Sized (Co-)Induction

We present a rich type system with subtyping for an extension of System F. Our type constructors include sum and product types, universal and existential quantifiers, inductive and coinductive types. The latter two size annotations allowing the preservation of size invariants. For example it is possible to derive the termination of the quicksort by showing that partitioning a list does not increase its size. The system deals with complex programs involving mixed induction and coinduction, or even mixed (co-)induction and polymorphism (as for Scott-encoded datatypes). One of the key ideas is to completely separate the induction on sizes from the notion of recursive programs. We use the size change principle to check that the proof is well-founded, not that the program terminates. Termination is obtained by a strong normalization proof. Another key idea is the use symbolic witnesses to handle quantifiers of all sorts. To demonstrate the practicality of our system, we provide an implementation that accepts all the examples discussed in the paper and much more

Computational contents of classical logic and extensional choice

We present here a logical system mini PML which is an extension of HOL with the Curry-Howard correspondence allowing both classical logic and the extensional axiom of choice for natural numbers and higher-order functionals on natural numbers

Isotopic piecewise affine approximation of algebraic or C1C^1 varieties

We propose a novel sufficient condition establishing that a piecewise affine variety has the same topology as a variety of the sphere $\mathbb{S}^n$ defined by positively homogeneous $C^1$ functions. This covers the case of $C^1$ varieties in the projective space $\mathbb{P}^n$. We prove that this condition is sufficient in the case of codimension one and arbitrary dimension. We describe an implementation working for homogeneous polynomials in arbitrary dimension and codimension and give experimental evidences that our condition might still be sufficient in codimension greater than one

Simple proof of the completeness theorem for second order classical and intuitionictic logic by reduction to first-order mono-sorted logic

International audienceWe present a simpler way than usual to deduce the completeness theorem for the second-oder classical logic from the first-order one. We also extend our method to the case of second-order intuitionistic logic

On a question of supports

We give a sufficient condition in order that $n$ closed connected subsets in the $n$-dimensional real projective space admit a common multitangent hyperplane

Asymptotically almost all \lambda-terms are strongly normalizing

We present quantitative analysis of various (syntactic and behavioral) properties of random \lambda-terms. Our main results are that asymptotically all the terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the \lambda-calculus into combinators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator

Krivine's realizability: from storage operators to the intentional axiom of choice

International audienceWe will give a survey of some results in realizability including: * basic notions and control operators associated to Pierce's law. * the definition of data-types and storage operators (which are essential). * the interaction of intuitionistic and classical proofs to extract algorithm (illustrated by Dickson's lemma). * the use of the system clock to realize the intentional axiom of choice. We will also give a short list of open problems

Realizability for programming languages.

We present a toy functional programming language inspired by our work on the PML language [22] together with a criterion ensuring safety and the fact that non termination can only occur via recursive programs. To prove this theorem, we use realizability techniques and a semantical notion of types. Important features of PML like polymorphism, proof-checking, termination criterion for recursive function,... will be covered by forthcoming articles reusing the formalism introduced here. The paper contains the source of the algorithm (some boring parts like the parser are omitted) and the complete source are available from the author webpage

PML -- a new proof assistant

International audienceWe will present our ongoing work on a new proof assistant and deduction system named PML. The basic idea is to start from an ML-like programming language and add specification and proof facilities. On the programming language side, the language unifies certain concepts: PML uses only one notion of sum types (polymorphic variants) and one notion of products (extensible records). These can then be used to encode modules and objects. PML's typing algorithm is based on a new constraint consistency check (as opposed to constraint solving). We transform the programming language into a deduction system by adding specification and proofs into modules. Surprisingly, extending such a powerful programming language into a deduction systems requires very little work. For instance, the syntax of programs can be reused for proofs

GlSurf version 3

This is a software actively maintened and still improved by the author.GlSurf is a program (similar to Surf) to draw surfaces and curves from their implicit equations (that is drawing the set of points (x,y,z) such that f(x,y,z) = 0). It offers an intuitive and simple syntax to construct your functions, it can draw multiple surfaces simultaneously and it can use all the power of OpenGl to animate the surface, use transparency, etc ... web page : http://www.lama.univ-savoie.fr/~raffalli/glsurf.htm