The basic motivation of this work is to make formal theory developments with machine-checked proofs accessible to a broader audience. Our particular approach is centered around the Isar formal proof language that is intended to support adequate composition of proof documents that are suitable for human consumption. Such primary proofs written in Isar may be both checked by the machine and read by human-beings; final presentation merely involves trivial pretty printing of the sources. Sound logical foundations of Isar are achieved by interpretation within the generic Natural Deduction framework of Isabelle, reducing all high-level reasoning steps to primitive inferences. The resulting Isabelle/Isar system...

We discuss the general concept of calculational reasoning within Isabelle/Isar, which provides a framework for high-level natural deduction proofs that may be written in a human-readable fashion. Setting out from a few basic logical concepts of the underlying meta-logical framework of Isabelle, such as higher-order unification and resolution, calculational commands are added to the basic Isar proof language in a flexible and non-intrusive manner. Thus calculational proof style may be combined with the remaining natural deduction proof language in a liberal manner, resulting in many useful proof patterns. A case-study on formalizing Computational Tree Logic (CTL) in simply-typed set-theory demonstrates common calculational idioms in practice.

The mathematical proof checker Mizar by Andrzej Trybulec uses a proof input language that is much more readable than the input languages of most other proof assistants. This system also di#ers in many other respects from most current systems. John Harrison has shown that one can have a Mizar mode on top of a tactical prover, allowing one to combine a mathematical proof language with other styles of proof checking. Currently the only fully developed Mizar mode in this style is the Isar proof language for the Isabelle theorem prover. In fact the Isar language has become the o#cial input language to the Isabelle system, even though many users still use its low-level tactical part only.

There are two dierent approaches to formalizing proofs in a computer: the procedural approach (which is the one of the HOL system) and the declarative approach (which is the one of the Mizar system).

Abstract. The mathematical proof checker Mizar by Andrzej Trybulec uses a proof input language that is much more readable than the input languages of most other proof assistants. This system also differs in many other respects from most current systems. John Harrison has shown that one can have a Mizar mode on top of a tactical prover, allowing one to combine a mathematical proof language with other styles of proof checking. Currently the only fully developed Mizar mode in this style is the Isar proof language for the Isabelle theorem prover. In fact the Isar language has become the official input language to the Isabelle system, even though many users still use its low-level tactical part only. In this paper we compare Mizar and Isar. A small example, Euclid’s proof of the existence of infinitely many primes, is shown in both systems. We also include slightly higher-level views of formal proof sketches. Moreover a list of differences between Mizar and Isar is presented, highlighting the strengths of both systems from the perspective of end-users. Finally, we point out some key differences of the