Foundational Extensible Corecursion

This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under well-behaved operations, including constructors. Corecursive functions that are well behaved can be registered as such, thereby increasing the corecursor's expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic

Foundational extensible corecursion: a proof assistant perspective

This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under “friendly” operations, including constructors. Friendly corecursive functions can be registered as such, thereby increasing the corecursor’s expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic

Foundational extensible corecursion: a proof assistant perspective

This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under “friendly” operations, including constructors. Friendly corecursive functions can be registered as such, thereby increasing the corecursor’s expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic

Friends with benefits: implementing corecursion in foundational proof assistants

We introduce AmiCo, a tool that extends a proof assistant, Isabelle/HOL, with flexible function definitions well beyond primitive corecursion. All definitions are certified by the assistant’s inference kernel to guard against inconsistencies. A central notion is that of friends: functions that preserve the productivity of their arguments and that are allowed in corecursive call contexts. As new friends are registered, corecursion benefits by becoming more expressive. We describe this process and its implementation, from the user’s specification to the synthesis of a higher-order definition to the registration of a friend. We show some substantial case studies where our approach makes a difference

Friends with benefits: implementing corecursion in foundational proof assistants

We introduce AmiCo, a tool that extends a proof assistant, Isabelle/HOL, with flexible function definitions well beyond primitive corecursion. All definitions are certified by the assistant’s inference kernel to guard against inconsistencies. A central notion is that of friends: functions that preserve the productivity of their arguments and that are allowed in corecursive call contexts. As new friends are registered, corecursion benefits by becoming more expressive. We describe this process and its implementation, from the user’s specification to the synthesis of a higher-order definition to the registration of a friend. We show some substantial case studies where our approach makes a difference

Foundational (co)datatypes and (co)recursion for higher-order logic

We describe a line of work that started in 2011 towards enriching Isabelle/HOL's language with coinductive datatypes, which allow infinite values, and with a more expressive notion of inductive datatype than previously supported by any system based on higher-order logic. These (co)datatypes are complemented by definitional principles for (co)recursive functions and reasoning principles for (co)induction. In contrast with other systems offering codatatypes, no additional axioms or logic extensions are necessary with our approach

Foundational extensible corecursion: a proof assistant perspective

This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under "friendly" operations, including constructors. Friendly corecursive functions can be registered as such, thereby increasing the corecursor's expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic

Coiterative Morphisms: Interactive Equational Reasoning for Bisimulation, using Coalgebras

ter: SEN 3 Abstract: We study several techniques for interactive equational reasoning with the bisimulation equivalence. Our work is based on a modular library, formalised in Coq, that axiomatises weakly final coalgebras and bisimulation. As a theory we derive some coalgebraic schemes and an associated coinduction principle. This will help in interactive proofs by coinduction, modular derivation of congruence and co-fixed point equations and enables an extensional treatment of bisimulation. Finally we present a version of the lambda-coinduction proof principle in our framework

Nominal Recursors as Epi-Recursors: Extended Technical Report

We study nominal recursors from the literature on syntax with bindings and compare them with respect to expressiveness. The term "nominal" refers to the fact that these recursors operate on a syntax representation where the names of bound variables appear explicitly, as in nominal logic. We argue that nominal recursors can be viewed as epi-recursors, a concept that captures abstractly the distinction between the constructors on which one actually recurses, and other operators and properties that further underpin recursion.We develop an abstract framework for comparing epi-recursors and instantiate it to the existing nominal recursors, and also to several recursors obtained from them by cross-pollination. The resulted expressiveness hierarchies depend on how strictly we perform this comparison, and bring insight into the relative merits of different axiomatizations of syntax. We also apply our methodology to produce an expressiveness hierarchy of nominal corecursors, which are principles for defining functions targeting infinitary non-well-founded terms (which underlie lambda-calculus semantics concepts such as B\"ohm trees). Our results are validated with the Isabelle/HOL theorem prover

Program extraction from coinductive proofs and its application to exact real arithmetic

Program extraction has been initiated in the field of constructive mathematics, and it attracts interest not only from mathematicians but also from computer scientists nowadays. From a mathematical viewpoint its aim is to figure out computational meaning of proofs, while from a computer-scientific viewpoint its aim is the study of a method to obtain correct programs. Therefore, it is natural to have both theoretical results and a practical computer system to develop executable programs via program extraction. In this Thesis we study the computational interpretation of constructive proofs involving inductive and coinductive reasoning. We interpret proofs by translating the computational content of proofs into executable program code. This translation is the procedure we call program extraction and it is given through Kreisel's modified realizability. Here we study a proof-theoretic foundation for program extraction, enriching the proof assistant system Minlog based on this theoretical improvement. Once a proof of a formula is written in Minlog, a program can be extracted from the proof by the system itself, and the extracted program can be executed in Minlog. Moreover, extracted programs are provably correct with respect to the proven formula due to a soundness theorem which we prove. We practice program extraction by elaborating some case studies from exact real arithmetic within our formal theory. Although these case studies have been studied elsewhere, here we offer a formalization of them in Minlog, and also machine-extraction of the corresponding programs.Die Methode der Programmextraktion hat ihren Ursprung im Bereich der konstruktiven Mathematik, und stößt in letzter Zeit auf viel Interesse nicht nur bei Mathematikern sondern auch bei Informatikern. Vom Standpunkt der Mathematik ist ihr Ziel, aus Beweisen ihre rechnerische Bedeutung abzulesen, während vom Standpunkt der Informatik ihr Ziel die Untersuchung einer Methode ist, beweisbar korrekte Programme zu erhalten. Es ist deshalb naheliegend, neben theoretischen Ergebnissen auch ein praktisches Computersystem zur Verfügung zu haben, mit dessen Hilfe durch Programmextraktion lauffähige Programme entwickelt werden können. In dieser Doktorarbeit wird eine rechnerische Interpretation konstruktiver Beweise mit induktiven und koinduktiven Definitionen angegeben und untersucht. Die Interpretation geschieht dadurch, daß der rechnerische Gehalt von Beweisen in eine Programmiersprache übersetzt wird. Diese übersetzung wird Programmextraktion genannt; sie basiert auf Kreisels modifizierter Realisierbarkeit. Wir untersuchen die beweistheoretischen Grundlagen der Programmextraktion und erweitern den Beweisassistenten Minlog auf der Basis der erhaltenen theoretischen Resultate. Wenn eine Formel in Minlog formal bewiesen ist, läßt sich ein Programm aus dem Beweis extrahieren, und dieses extrahierte Programm kann in Minlog ausgeführt werden. Ferner sind extrahierte Programme beweisbar korrekt bezüglich der entsprechenden Formel aufgrund eines Korrektheitsatzes, den wir beweisen werden. Innerhalb unserer formalen Theorie bearbeiten wir einige aus der Literatur bekannte Fallstudien im Bereich der exakten reellen Arithmetik. Wir entwickeln eine vollständige Formalisierung der entsprechenden Beweise und diskutieren die in Minlog automatisch extrahierten Programme