A direct proof of the confluence of combinatory strong reduction

I give a proof of the confluence of combinatory strong reduction that does not use the one of lambda-calculus. I also give simple and direct proofs of a standardization theorem for this reduction and the strong normalization of simply typed terms.Comment: To appear in TC

A short proof that adding some permutation rules to beta preserves SN

I show that, if a term is $SN$ for $\beta$, it remains $SN$ when some permutation rules are added

The Brauer-Manin obstruction on Kummer varieties and ranks of twists of abelian varieties

Let r > 0 be an integer. We present a sufficient condition for an abelian variety A over a number field k to have infinitely many quadratic twists of rank at least r, in terms of density properties of rational points on the Kummer variety Km(A^r) of the r-fold product of A with itself. One consequence of our results is the following. Fix an abelian variety A over k, and suppose that for some r > 0 the Brauer-Manin obstruction to weak approximation on the Kummer variety Km(A^r) is the only one. Then A has a quadratic twist of rank at least r. Hence if the Brauer-Manin obstruction is the only one to weak approximation on all Kummer varieties, then ranks of twists of any positive-dimensional abelian variety are unbounded. This relates two significant open questions.Comment: 12 pages; final versio

Une preuve simple de résultats classiques en lambda-calcul

International audienceNous donnons une preuve simple de 3 théorèmes " de base " du lambda calcul pur : le théorème de Church - Rosser, le théorème de standardisation et le théorème des développements finis. Nous donnons également une généralisation - dans le lambda calcul pur - du théorème des développements finis qui a comme corollaire immédiat le théorème de normalisation forte dans le système de types simples avec intersection

Un algorithme primitif récursif pour la fonction Inf

International audienceWe give a primitive recursive algorithm (for the structure of integers - in unary - and lists of integers ) that computes the minimum of two integers in linear time of this minimum. This algorithm refutes a conjecture of L.Colso