New Notions of Reduction and Non-Semantic Proofs of β-Strong Normalization in Typed λ-Calculi

Two new notions of reduction for terms of the λ-calculus are introduced and the question of whether a λ-term is beta-strongly normalizing is reduced to the question of whether a λ-term is merely normalizing under one of the new notions of reduction. This leads to a new way to prove beta-strong normalization for typed λ-calculi. Instead of the usual semantic proof style based on Girard's "candidats de réductibilité'', termination can be proved using a decreasing metric over a well-founded ordering in a style more common in the field of term rewriting. This new proof method is applied to the simply-typed λ-calculus and the system of intersection types.

Addendum to "New Notions of Reduction and Non-Semantic Proofs of β-Strong Normalization in Typed λ-Calculi"

This is an addendum to our technical report BUCS TR-94-014 of December 19, 1994. It clarifies some statements, adds information on some related research, includes a comparison with research be de Groote, and fixes two minor mistakes in a proof.

Didactical reflections about some proofs of the pythagorean proposition

This theoretical paper presents some dimensions considered in the literature to analyze proof in the teaching and learning of mathematics. In order to show how different types of proofs can be used with students of different levels, we use these dimensions to analyze four proofs of the Pythagorean proposition.

Analytic Methods for the Logic of Proofs

The logic of proofs (lp) was proposed as Gdels missed link between Intuitionistic and S4-proofs, but so far the tableau-based methods proposed for lp have not explored this closeness with S4 and contain rules whose analycity is not immediately evident. We study possible formulations of analytic tableau proof methods for lp that preserve the subformula property. Two sound and complete tableau decision methods of increasing degree of analycity are proposed, KELP and preKELP. The latter is particularly inspired on S4-proofs. The crucial role of proof constants in the structure of lp-proofs methods is analysed. In particular, a method for the abduction of proof constant specifications in strongly analytic preKELP proofs is presented; abduction heuristics and the complexity of the method are discussed.