We present a formalization of a constructive proof of weak normalization for the simply-typed λ-calculus in the theorem prover Isabelle/HOL, and show how a program can be extracted from it. Unlike many other proofs of weak normalization based on Tait’s strong computability predicates, which require a logic supporting strong eliminations and can give rise to dependent types in the extracted program, our formalization requires only relatively simple proof principles. Thus, the program obtained from this proof is typable in simply-typed higher-order logic as implemented in Isabelle/HOL, and a proof of its correctness can automatically be derived within the system.

It is shown how the sequent calculus LJ can be embedded into a simple extension of the -calculus by generalized applications, called J. The reduction rules of cut elimination and normalization can be precisely correlated, if explicit substitutions are added to J. The resulting system J2 is proved strongly normalizing, thus showing strong normalization for Gentzen's cut elimination steps. This re nes previous results by Zucker, Pottinger and Herbelin on the isomorphism between natural deduction and sequent calculus.

Arithmetical proofs of strong normalization results for symmetric λ-calculi