A Synthesis of the Procedural and Declarative Styles of Interactive Theorem Proving

We propose a synthesis of the two proof styles of interactive theorem proving: the procedural style (where proofs are scripts of commands, like in Coq) and the declarative style (where proofs are texts in a controlled natural language, like in Isabelle/Isar). Our approach combines the advantages of the declarative style - the possibility to write formal proofs like normal mathematical text - and the procedural style - strong automation and help with shaping the proofs, including determining the statements of intermediate steps. Our approach is new, and differs significantly from the ways in which the procedural and declarative proof styles have been combined before in the Isabelle, Ssreflect and Matita systems. Our approach is generic and can be implemented on top of any procedural interactive theorem prover, regardless of its architecture and logical foundations. To show the viability of our proposed approach, we fully implemented it as a proof interface called miz3, on top of the HOL Light interactive theorem prover. The declarative language that this interface uses is a slight variant of the language of the Mizar system, and can be used for any interactive theorem prover regardless of its logical foundations. The miz3 interface allows easy access to the full set of tactics and formal libraries of HOL Light, and as such has "industrial strength". Our approach gives a way to automatically convert any procedural proof to a declarative counterpart, where the converted proof is similar in size to the original. As all declarative systems have essentially the same proof language, this gives a straightforward way to port proofs between interactive theorem provers

定理証明支援系Coqにおける手続き的証明から宣言的証明への変換

　定理証明支援系Coqにおける証明は、一般に手続き的証明と呼ばれる形式で記述される。これは対話的証明を前提としており、自然言語による証明記述と大きく異なるため、可読性が高いものではない。この問題を解決するためにCoq用宣言的証明言語C-zarが開発された。宣言的証明は可読性が高く、また外部ツールを導入し易い。しかし、C-zar は手続き的証明に対して記述量が多い上に柔軟性が低く、Coq ユーザに受け入れられなかった。本研究では、Coq の手続き的証明からC-zarの証明を生成することで、両者間の橋渡しを行う。一般に手続き的証明から宣言的証明への変換手法としては、証明項や証明木のような中間表現を経由する方法が考えられ、既に定理証明支援系Matitaでは証明項を経由する手続き的証明から宣言的証明への変換が存在する。しかし、中間表現は元の証明と比べて詳細かつ巨大になり、元の手続き的証明1ステップに対して数百ステップの宣言的証明が生成されてしまう場合もある。一方で、C-zar は手続き的証明で用いられるタクティックと呼ばれるコマンドを利用することができ、これによって手続き的証明の1ステップは、多くの場合C-zarの数ステップと対応させることができる。本研究では、元の手続き的証明と証明項の両方を用いて変換を行うことで、元の証明に近い粒度の宣言的証明の生成を実現する。電気通信大学201

Refactoring proofs

Refactoring is an important Software Engineering technique for improving the structure of a program after it has been written. Refactorings improve the maintainability, readability, and design of a program without affecting its external behaviour. In analogy, this thesis introduces proof refactoring to make structured, semantics preserving changes to the proof documents constructed by interactive theorem provers as part of a formal proof development. In order to formally study proof refactoring, the first part of this thesis constructs a proof language framework, Hiscript. The Hiscript framework consists of a procedural tactic language, a declarative proof language, and a modular theory language. Each level of this framework is equipped with a formal semantics based on a hierarchical notion of proof trees. Furthermore, this framework is generic as it does not prescribe an underlying logical kernel. This part contributes an investigation of semantics for formal proof documents, which is proved to construct valid proofs. Moreover, in analogy with type-checking, static well-formedness checks of proof documents are separated from evaluation of the proof. Furthermore, a subset of the SSReflect language for Coq, called eSSence, is also encoded using hierarchical proofs. Both Hiscript and eSSence are shown to have language elements with a natural hierarchical representation. In the second part, proof refactoring is put on a formal footing with a definition using the Hiscript framework. Over thirty refactorings are formally specified and proved to preserve the semantics in a precise way for the Hiscript language, including traditional structural refactorings, such as rename item, and proof specific refactorings such as backwards proof to forwards proof and declarative to procedural. Finally, a concrete, generic refactoring framework, called Polar, is introduced. Polar is based on graph rewriting and has been implemented with over ten refactorings and for two proof languages, including Hiscript. Finally, the third part concludes with some wishes for the future

Assertion level proof planning with compiled strategies

This book presents new techniques that allow the automatic verification and generation of abstract human-style proofs. The core of this approach builds an efficient calculus that works directly by applying definitions, theorems, and axioms, which reduces the size of the underlying proof object by a factor of ten. The calculus is extended by the deep inference paradigm which allows the application of inference rules at arbitrary depth inside logical expressions and provides new proofs that are exponentially shorter and not available in the sequent calculus without cut. In addition, a strategy language for abstract underspecified declarative proof patterns is developed. Together, the complementary methods provide a framework to automate declarative proofs. The benefits of the techniques are illustrated by practical applications.Die vorliegende Arbeit beschäftigt sich damit, das Formalisieren von Beweisen zu vereinfachen, indem Methoden entwickelt werden, um informale Beweise formal zu verifizieren und erzeugen zu können. Dazu wird ein abstrakter Kalkül entwickelt, der direkt auf der Faktenebene arbeitet, welche von Menschen geführten Beweisen relativ nahe kommt. Anhand einer Fallstudie wird gezeigt, dass die abstrakte Beweisführung auf der Fakteneben vorteilhaft für automatische Suchverfahren ist. Zusätzlich wird eine Strategiesprache entwickelt, die es erlaubt, unterspezifizierte Beweismuster innerhalb des Beweisdokumentes zu spezifizieren und Beweisskizzen automatisch zu verfeinern. Fallstudien zeigen, dass komplexe Beweismuster kompakt in der entwickelten Strategiesprache spezifiziert werden können. Zusammen bilden die einander ergänzenden Methoden den Rahmen zur Automatisierung von deklarativen Beweisen auf der Faktenebene, die bisher überwiegend manuell entwickelt werden mussten

Assertion level proof planning with compiled strategies

This book presents new techniques that allow the automatic verification and generation of abstract human-style proofs. The core of this approach builds an efficient calculus that works directly by applying definitions, theorems, and axioms, which reduces the size of the underlying proof object by a factor of ten. The calculus is extended by the deep inference paradigm which allows the application of inference rules at arbitrary depth inside logical expressions and provides new proofs that are exponentially shorter and not available in the sequent calculus without cut. In addition, a strategy language for abstract underspecified declarative proof patterns is developed. Together, the complementary methods provide a framework to automate declarative proofs. The benefits of the techniques are illustrated by practical applications.Die vorliegende Arbeit beschäftigt sich damit, das Formalisieren von Beweisen zu vereinfachen, indem Methoden entwickelt werden, um informale Beweise formal zu verifizieren und erzeugen zu können. Dazu wird ein abstrakter Kalkül entwickelt, der direkt auf der Faktenebene arbeitet, welche von Menschen geführten Beweisen relativ nahe kommt. Anhand einer Fallstudie wird gezeigt, dass die abstrakte Beweisführung auf der Fakteneben vorteilhaft für automatische Suchverfahren ist. Zusätzlich wird eine Strategiesprache entwickelt, die es erlaubt, unterspezifizierte Beweismuster innerhalb des Beweisdokumentes zu spezifizieren und Beweisskizzen automatisch zu verfeinern. Fallstudien zeigen, dass komplexe Beweismuster kompakt in der entwickelten Strategiesprache spezifiziert werden können. Zusammen bilden die einander ergänzenden Methoden den Rahmen zur Automatisierung von deklarativen Beweisen auf der Faktenebene, die bisher überwiegend manuell entwickelt werden mussten

Declarative Representation of Proof Terms

We present a declarative language inspired by the pseudo-natural language used in Matita for the explanation of proof terms. We show how to compile the language to proof terms and how to automatically generate declarative scripts from proof terms. Then we investigate the relationship between the two translations, identifying the amount of proof structure preserved by compilation and re-generation of declarative scripts

Declarative Representation of Proof Terms

We present a declarative language inspired by the pseudo-natural language previously used in Matita for the explanation of proof terms. We show how to compile the language to proof terms and how to automatically generate declarative scripts from proof terms. Then we investigate the relationship between the two translations, identifying the amount of proof structure preserved by compilation and re-generation of declarative scripts