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We describe a type system for the linear-algebraic lambda-calculus. The typesystem accounts for the part of the language emulating linear operators andvectors, i.e. it is able to statically describe the linear combinations ofterms resulting from the reduction of programs. This gives rise to an originaltype theory where types, in the same way as terms, can be superposed intolinear combinations. We show that the resulting typed lambda-calculus isstrongly normalizing and features a weak subject-reduction.

A Type System for the Vectorial Aspect of the Linear-Algebraic Lambda-Calculus