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**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

In logic and computer science one often needs to constructivize a theorem ∀ f ∃ g.P(f,g), stating that for every infinite sequence f there is an infinite sequence g such that P(f,g). Here P is a computable predicate but g is not necessarily computable from f. In this paper we propose the following constructive version of ∀ f ∃ g.P(f,g): for every f there is a "long enough" finite prefix g₀ of g such that P(f,g₀), where "long enough" is expressed by membership to a bar which is a free parameter of the constructive version.
Our approach with bars generalises the approaches to Higman’s lemma undertaken by Coquand-Fridlender, Murthy-Russell and Schwichtenberg-Seisenberger-Wiesnet. As a first test for our bar technique, we sketch a constructive theory of well-quasi orders. This includes yet another constructive version of Higman’s lemma: that every infinite sequence of words has an infinite ascending subsequence. As compared with the previous constructive versions of Higman’s lemma, our constructive proofs are closer to the original classical proofs.

Stefano Berardi, Gabriele Buriola, and Peter Schuster. A General Constructive Form of Higman’s Lemma. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{berardi_et_al:LIPIcs.CSL.2024.16, author = {Berardi, Stefano and Buriola, Gabriele and Schuster, Peter}, title = {{A General Constructive Form of Higman’s Lemma}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {16:1--16:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.16}, URN = {urn:nbn:de:0030-drops-196599}, doi = {10.4230/LIPIcs.CSL.2024.16}, annote = {Keywords: intuitionistic logic, constructive mathematics, formal proof, inductive predicate, bar induction, well quasi-order, Higman’s lemma} }

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Complete Volume

**Published in:** LIPIcs, Volume 188, 26th International Conference on Types for Proofs and Programs (TYPES 2020)

LIPIcs, Volume 188, TYPES 2020, Complete Volume

26th International Conference on Types for Proofs and Programs (TYPES 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 188, pp. 1-204, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Proceedings{deliguoro_et_al:LIPIcs.TYPES.2020, title = {{LIPIcs, Volume 188, TYPES 2020, Complete Volume}}, booktitle = {26th International Conference on Types for Proofs and Programs (TYPES 2020)}, pages = {1--204}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-182-5}, ISSN = {1868-8969}, year = {2021}, volume = {188}, editor = {de'Liguoro, Ugo and Berardi, Stefano and Altenkirch, Thorsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2020}, URN = {urn:nbn:de:0030-drops-138785}, doi = {10.4230/LIPIcs.TYPES.2020}, annote = {Keywords: LIPIcs, Volume 188, TYPES 2020, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 188, 26th International Conference on Types for Proofs and Programs (TYPES 2020)

Front Matter, Table of Contents, Preface, Conference Organization

26th International Conference on Types for Proofs and Programs (TYPES 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 188, pp. 0:i-0:viii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{deliguoro_et_al:LIPIcs.TYPES.2020.0, author = {de'Liguoro, Ugo and Berardi, Stefano and Altenkirch, Thorsten}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {26th International Conference on Types for Proofs and Programs (TYPES 2020)}, pages = {0:i--0:viii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-182-5}, ISSN = {1868-8969}, year = {2021}, volume = {188}, editor = {de'Liguoro, Ugo and Berardi, Stefano and Altenkirch, Thorsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2020.0}, URN = {urn:nbn:de:0030-drops-138792}, doi = {10.4230/LIPIcs.TYPES.2020.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 41, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015)

Assume that we may prove in Classical Functional Analysis that a primitive recursive relation R is well-founded, using the inductive definition of well-founded. In this paper we prove that such a proof of well-foundation may be made intuitionistic. We conclude that if we are able to formulate any mathematical problem as the inductive well-foundation of some primitive recursive relation, then intuitionistic and classical provability coincide, and for such a statement of well-foundation we may always find an intuitionistic proof if we may find a proof at all.
The core of intuitionism are the methods for computing out data with given properties from input data with given properties: these are the results we are looking for when we do constructive mathematics. Proving that a primitive recursive relation R is inductively well-founded is a more abstract kind of result, but it is crucial as well, because once we proved that R is inductively well-founded, then we may write programs by induction over R. This is the way inductive relation are currently used in intuitionism and in proof assistants based on intuitionism, like Coq.
In the paper we introduce the comprehension axiom for Functional Analysis in the form of introduction and elimination rules for predicates of types Prop, Nat->Prop, ..., in order to use Girard's method of candidates for impredicative arithmetic.

Stefano Berardi. Classical and Intuitionistic Arithmetic with Higher Order Comprehension Coincide on Inductive Well-Foundedness. In 24th EACSL Annual Conference on Computer Science Logic (CSL 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 41, pp. 343-358, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{berardi:LIPIcs.CSL.2015.343, author = {Berardi, Stefano}, title = {{Classical and Intuitionistic Arithmetic with Higher Order Comprehension Coincide on Inductive Well-Foundedness}}, booktitle = {24th EACSL Annual Conference on Computer Science Logic (CSL 2015)}, pages = {343--358}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-90-3}, ISSN = {1868-8969}, year = {2015}, volume = {41}, editor = {Kreutzer, Stephan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2015.343}, URN = {urn:nbn:de:0030-drops-54246}, doi = {10.4230/LIPIcs.CSL.2015.343}, annote = {Keywords: Intuitionism, Inductive Definitions, Proof Theory, impredicativity, omega rule} }

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**Published in:** LIPIcs, Volume 26, 19th International Conference on Types for Proofs and Programs (TYPES 2013)

We produce a first order proof of a famous combinatorial result, Ramsey Theorem for pairs and in two colors. Our goal is to find the minimal classical principle that implies the "miniature" version of Ramsey we may express in Heyting Arithmetic HA. We are going to prove that Ramsey Theorem for pairs with recursive assignments of two colors is equivalent in HA to the sub-classical principle Sigma-0-3-LLPO. One possible application of our result could be to use classical realization to give constructive proofs of some combinatorial corollaries of Ramsey; another, a formalization of Ramsey in HA, using a proof assistant. In order to compare Ramsey Theorem with first order classical principles, we express it as a schema in the first order language of arithmetic, instead of using quantification over sets and functions as it is more usual: all sets we deal with are explicitly defined as arithmetical predicates. In particular, we formally define the homogeneous set which is the witness of Ramsey theorem as a Delta-0-3-arithmetical predicate.

Stefano Berardi and Silvia Steila. Ramsey Theorem for Pairs As a Classical Principle in Intuitionistic Arithmetic. In 19th International Conference on Types for Proofs and Programs (TYPES 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 26, pp. 64-83, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{berardi_et_al:LIPIcs.TYPES.2013.64, author = {Berardi, Stefano and Steila, Silvia}, title = {{Ramsey Theorem for Pairs As a Classical Principle in Intuitionistic Arithmetic}}, booktitle = {19th International Conference on Types for Proofs and Programs (TYPES 2013)}, pages = {64--83}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-72-9}, ISSN = {1868-8969}, year = {2014}, volume = {26}, editor = {Matthes, Ralph and Schubert, Aleksy}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2013.64}, URN = {urn:nbn:de:0030-drops-46269}, doi = {10.4230/LIPIcs.TYPES.2013.64}, annote = {Keywords: Formalization, Constructivism, Classical logic, Ramsey Theorem} }

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**Published in:** LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

We present a new Curry-Howard correspondence for HA + EM_1, constructive Heyting Arithmetic with the excluded middle on \Sigma^0_1-formulas. We add to the lambda calculus an operator ||_a which represents, from the viewpoint of programming, an exception operator with a delimited scope, and from the viewpoint of logic, a restricted version of the excluded middle. We motivate the restriction of the excluded middle by its use in proof mining; we introduce new techniques to prove strong normalization for HA + EM_1 and the witness property for simply existential statements. One may consider our results as an application of the ideas of Interactive realizability, which we have adapted to the new setting and used to prove our main theorems.

Federico Aschieri, Stefano Berardi, and Giovanni Birolo. Realizability and Strong Normalization for a Curry-Howard Interpretation of HA + EM1. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 45-60, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{aschieri_et_al:LIPIcs.CSL.2013.45, author = {Aschieri, Federico and Berardi, Stefano and Birolo, Giovanni}, title = {{Realizability and Strong Normalization for a Curry-Howard Interpretation of HA + EM1}}, booktitle = {Computer Science Logic 2013 (CSL 2013)}, pages = {45--60}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-60-6}, ISSN = {1868-8969}, year = {2013}, volume = {23}, editor = {Ronchi Della Rocca, Simona}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.45}, URN = {urn:nbn:de:0030-drops-41894}, doi = {10.4230/LIPIcs.CSL.2013.45}, annote = {Keywords: Interactive realizability, classical Arithmetic, witness extraction, delimited exceptions} }

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**Published in:** LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012)

We propose a theory of learning aimed to formalize some ideas underlying Coquand's game semantics and Krivine's realizability of classical logic. We introduce a notion of knowledge state together with a new topology, capturing finite positive and negative information that guides a learning strategy. We use a leading example to illustrate how non-constructive proofs lead to continuous and effective learning strategies over knowledge spaces, and prove that our learning semantics is sound and complete w.r.t. classical truth, as it is the case for Coquand's and Krivine's approaches.

Stefano Berardi and Ugo de'Liguoro. Knowledge Spaces and the Completeness of Learning Strategies. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 77-91, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{berardi_et_al:LIPIcs.CSL.2012.77, author = {Berardi, Stefano and de'Liguoro, Ugo}, title = {{Knowledge Spaces and the Completeness of Learning Strategies}}, booktitle = {Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL}, pages = {77--91}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-42-2}, ISSN = {1868-8969}, year = {2012}, volume = {16}, editor = {C\'{e}gielski, Patrick and Durand, Arnaud}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.77}, URN = {urn:nbn:de:0030-drops-36656}, doi = {10.4230/LIPIcs.CSL.2012.77}, annote = {Keywords: Classical Logic, Proof Mining, Game Semantics, Learning, Realizability} }

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**Published in:** LIPIcs, Volume 12, Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL (2011)

Does there exist any sequent calculus such that it is a subclassical logic and it becomes classical logic when the exchange rules are added? The first contribution of this paper is answering this question for infinitary Peano arithmetic. This paper defines infinitary Peano arithmetic with non-commutative sequents, called non-commutative infinitary Peano arithmetic, so that the system becomes equivalent to Peano arithmetic with the omega-rule if the the exchange rule is added to this system. This system is unique among other non-commutative systems, since all the logical connectives have standard meaning and specifically the commutativity for conjunction and disjunction is derivable. This paper shows that the provability in non-commutative infinitary Peano arithmetic is equivalent to Heyting arithmetic with the recursive omega rule and the law of excluded middle for Sigma-0-1 formulas. Thus, non-commutative infinitary Peano arithmetic is shown to be a subclassical logic. The cut elimination theorem in this system is also proved. The second contribution of this paper is introducing infinitary Peano arithmetic having antecedent-grouping and no right exchange rules. The first contribution of this paper is achieved through this system. This system is obtained from the positive fragment of infinitary Peano arithmetic without the exchange rules by extending it from a positive fragment to a full system, preserving its 1-backtracking game semantics. This paper shows that this system is equivalent to both non-commutative infinitary Peano arithmetic, and Heyting arithmetic with the recursive omega rule and the Sigma-0-1 excluded middle.

Makoto Tatsuta and Stefano Berardi. Non-Commutative Infinitary Peano Arithmetic. In Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 12, pp. 538-552, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{tatsuta_et_al:LIPIcs.CSL.2011.538, author = {Tatsuta, Makoto and Berardi, Stefano}, title = {{Non-Commutative Infinitary Peano Arithmetic}}, booktitle = {Computer Science Logic (CSL'11) - 25th International Workshop/20th Annual Conference of the EACSL}, pages = {538--552}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-32-3}, ISSN = {1868-8969}, year = {2011}, volume = {12}, editor = {Bezem, Marc}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2011.538}, URN = {urn:nbn:de:0030-drops-32551}, doi = {10.4230/LIPIcs.CSL.2011.538}, annote = {Keywords: proof theory, cut elimination, intuitionistic logic, infinitary logic, recursive omega rules, substructural logic} }

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