Parigot’s Second Order λμ-Calculus and Inductive Types
A new proof of strong normalization of Parigot’s (second order) λμ-calculus is given by a reduction-preserving embedding into system F (second order polymorphic λ-calculus). The main idea is to use the least stable supertype for any type. These non-strictly positive inductive types and their associated iteration principle are available in system F, and allow to give a translation vaguely related to CPS translations (corresponding to the Kolmogorov embedding of classical logic into intuitionistic logic). However, they simulate Parigot’s μ-reductions whereas CPS translations hide them.
As a major advantage, this embedding does not use the idea of reducing stability (¬¬¬ ¬ ¬) to that for atomic formulae. Therefore, it even extends to non-interleaving positive fixed-point types. As a non-trivial application, strong normalization of λμ-calculus, extended by primitive recursion on monotone inductive types, is established.
KeywordsClassical Logic Intuitionistic Logic Natural Deduction Elimination Rule Inductive Type
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