Sequence Types and Infinitary Semantics

We introduce a new representation of non-idempotent intersection types, using \textbf{sequences} (families indexed with natural numbers) instead of lists or multisets. This allows scaling up \textbf{intersection type} theory to the infinitary $\lambda$-calculus. We thus characterize hereditary head normalization, which gives a positive answer to a question known as \textbf{Klop's Problem}. On our way, we use \textbf{non-idempotent intersection} to retrieve some well-known results on infinitary terms.