Structural Logical Relations with Case Analysis and Equality Reasoning

Abstract

Formalizing proofs by logical relations in the Twelf proof assistant is known to be notoriously difficult. However, as demonstrated by Schürmann and Sarnat [In Proc. of 23rd Symp. on Logic in Computer Science, 2008] such proofs can be represented and verified in Twelf if done so using a Gentzen-style auxiliary assertion logic which is subsequently proved consistent via cut elimination. We demonstrate in this paper an application of the above methodology to proofs of observational equivalence between expressions in a simply typed lambda calculus with a call-by-name operational semantics. Our use case requires the assertion logic to be extended with reasoning principles not present in the original presentation of the formalization method. We address this by generalizing the assertion logic to include dependent sorts, and demonstrate that the original cut elimination proof continues to apply without modification.

@inproceedings{rasmussen2013a,
  author = {Ulrik Rasmussen, Andrzej Filinski},
  title = {Structural Logical Relations with Case Analysis and Equality Reasoning},
  abstract = {Formalizing proofs by logical relations in the Twelf proof
                  assistant is known to be notoriously difficult. However, as
                  demonstrated by Schürmann and Sarnat [In Proc. of 23rd
                  Symp. on Logic in Computer Science, 2008] such proofs can be
                  represented and verified in Twelf if done so using a
                  Gentzen-style auxiliary assertion logic which is subsequently
                  proved consistent via cut elimination. We demonstrate in this
                  paper an application of the above methodology to proofs of
                  observational equivalence between expressions in a simply
                  typed lambda calculus with a call-by-name operational
                  semantics. Our use case requires the assertion logic to be
                  extended with reasoning principles not present in the original
                  presentation of the formalization method. We address this by
                  generalizing the assertion logic to include dependent sorts,
                  and demonstrate that the original cut elimination proof
                  continues to apply without modification.},
  year = {2013},
  month = {September},
  booktitle = {Proceedings International Workshop on Logical Frameworks and
                  Meta-Languages: Theory and Practice (LFMTP)},
  doi = {10.1145/2503887.2503891},
}