Abstract
Families of inductive types defined by recursion arise in the formalization of mathematical theories. An example is the family of term algebras on the type of signatures. Type theory does not allow the direct definition of such families. We state the problem abstractly by defining a notion, strong positivity, that characterizes these families. Then we investigate its solutions. First, we construct a model using wellorderings. Second, we use an extension of type theory, implemented in the proof tool Coq, to construct another model that does not have extensionality problems. Finally, we apply the two level approach: We internalize inductive definitions, so that we can manipulate them and reason about them inside type theory.
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Capretta, V. (2000). Recursive Families of Inductive Types. In: Aagaard, M., Harrison, J. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2000. Lecture Notes in Computer Science, vol 1869. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44659-1_5
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DOI: https://doi.org/10.1007/3-540-44659-1_5
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