eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-11-05
10:1
10:27
10.4230/LIPIcs.TYPES.2016.10
article
Permutability in Proof Terms for Intuitionistic Sequent Calculus with Cuts
Espírito Santo, José
Frade, Maria João
Pinto, Luís
This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol097-types2016/LIPIcs.TYPES.2016.10/LIPIcs.TYPES.2016.10.pdf
sequent calculus
permutative conversion
Curry-Howard isomorphism
vector of arguments
generalized application
normal proof
natural proof
cut eli