This paper has benefited from the financial support of the Fonds FCAR (Quebec)- programme de soutien aux équipes de recherche. We are also grateful to Raymond Medza of the Groupement des assureurs automobiles du Quebec (GAA) who gave us the permission to use the data and Nathalie Lizotte and Jacques Morazin, also from the GAA, for their technical support. Of course, they do not share the potential mistakes! Also, comments by Philippe

deductibles

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. ILL has a contraction rule and an introduction rule !I for the exponential; in ILL + , instead of a contraction rule, multiple occurrences of labels for assumptions are permitted under certain conditions; moreover, there is a different introduction rule for the exponential, !I + , which is closer

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