Une réponse négative à la conjecture de E. Tronci pour les systémes numériques typés

A numeral system is a sequence of an infinite different closed normal λ-terms which has closed λ-terms for successor and zero test. A numeral system is said adequate iff it has a closed λ-term for predecessor. A storage operator for a numeral system is a closed λ-term which simulate “call-by-value” in the context of a “call-by-name” strategy. E. Tronci conjectured the following result : a numeral system is adequate if it has a storage operator. This paper gives a negative answer to this conjecture for the numeral systems typable in the J.-Y. Girard type system F . The E. Tronci’s conjecture remains open in pure λ-calculus.