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

Abstract

Abstract. 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.

Learning-Assisted Automated Reasoning with Flyspeck

“…Also, thanks to the Flyspeck project [32], HOL Light is becoming less of a "single, very knowledgable formalizer" tool, and is getting increasingly used as a "tool for interested mathematicians" (such as the Flyspeck team in Hanoi) 3 who may know the large libraries much less and have less experience with crafting their own proof tactics. For such ITP users it is good to provide a small number of strong methods that allow fast progress, which can perhaps also complement the declarative modes [89] pioneered by HOL Light [36] in the LCF world.…”

Progress in the Independent Certification of Mizar Mathematical Library in Isabelle

“…The result of this work was the environment Mizar Mode for HOL enabling writing proofs in a Mizar declarative way [24]. The similar solutions were implemented in other procedural proof assistants, e.g., Declare [25], Isar language for Isabelle [18], Mizar-light for HOL Light [26], miz3 for HOL Light [27], [28]. However, the similarity between these environments and Mizar system generally is limited to a few rules that are similar to the rules of the S. Jaśkowski natural deduction style [29], responsible for the universal quantifier introduction, the thesis indication, the implication elimination, the introduction of the reasoning by cases.…”

Four Decades of Mizar

“…The contacts gain momentum with the discussions and research collaboration stemming from the QED initiative. To list the most important cases of Mizar's influence on other systems we can mention the Declare system developed by Syme [24], the Mizar mode for HOL by Harrison [10], the Isar language for Isabelle by Wenzel [37], Mizar-light for HOL Light by Wiedijk [38], the declarative proof language (DPL) for Coq by Corbineau [4] and finally the Wiedijk's miz3 proof interface for HOL Light [40] that combines both the procedural and declarative style of writing proofs. The Mizar way of writing proofs was also the model for the notion of 'formal proof sketches' developed by Wiedijk [39].…”