Calculational Reasoning Revisited An Isabelle/Isar Experience

Bauer, Gertrud; Wenzel, Markus

doi:10.1007/3-540-44755-5_7

Cited by 17 publications

(13 citation statements)

References 10 publications

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“…Part (iv) is most interesting from the mathematical viewpoint. Technically, we merely conduct a few steps of calculational reasoning [3] in Isar (with "glue statements" also and finally), while results are composed in a mostly trivial manner; we rarely invoke automated reasoning support here. In line 14, syntactic case analysis converts between m + 1 and Suc m, which is the preferred representation in the Isabelle/HOL library.…”

Section: Case-study: Complete Induction With Local Definitionsmentioning

confidence: 99%

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Structured Induction Proofs in Isabelle/Isar

Wenzel

2006

Lecture Notes in Computer Science

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Abstract. Isabelle/Isar is a generic framework for human-readable formal proof documents, based on higher-order natural deduction. The Isar proof language provides general principles that may be instantiated to particular object-logics and applications. We discuss specific Isar language elements that support complex induction patterns of practical importance. Despite the additional bookkeeping required for induction with local facts and parameters, definitions, simultaneous goals and multiple rules, the resulting Isar proof texts turn out well-structured and readable. Our techniques can be applied to non-standard variants of induction as well, such as co-induction and nominal induction. This demonstrates that Isar provides a viable platform for building domain-specific tools that support fully-formal mathematical proof composition. Motivation The Isar philosophyIsabelle/Isar [15,16,7,17] is intended as a generic framework for developing formal mathematical documents with full proof checking. The Isabelle/Isar system is well integrated with existing theorem prover interface technology [1] and document preparation based on PDF-L A T E X. 1 The main objective is the design of a human-readable structured proof language, which is called the "primary proof format" in Isar terminology. Such a primary proof language is somewhere in the middle between the extremes of primitive proof objects and actual natural language. In this respect, Isar is a bit more formalistic than Mizar [12,10], using explicit logical connectives for certain reasoning schemes where Mizar would prefer English words; see [19,18] for further comparisons of these systems. We argue that any effort of building a library of formalized mathematics heavily depends on a reasonable notion of structured proofs -the Mizar Mathematical Library [6] provides some empirical evidence for this. So Isar challenges the traditional way of recording informal proofs in mathematical prose, as well as the common tendency to see fully formal proofs directly as objects of some logical calculus (e.g. λ-terms in a version of type theory). In fact, Isar is better understood as an interpreter of a simple block-structured language for describing data flow of local facts and goals, interspersed with occasional invocations of proof methods.1 In fact, the present paper has been prepared as an Isabelle/Isar theory document, which means that the proofs and proof outlines encountered here have been checked by the system.

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Section: Case-study: Complete Induction With Local Definitionsmentioning

confidence: 99%

“…We merely need to support an additional "avoiding" argument, similar to the present "fixing". The resulting nominal-induct method has been used by Urban for some key induction proofs of the POPLmark challenge 3 . The complex pattern of §4.4 reflects the structure of a key induction proof in this application.…”

Section: Conclusion and Related Workmentioning

confidence: 99%

Structured Induction Proofs in Isabelle/Isar

Wenzel

2006

Lecture Notes in Computer Science

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“…Isabelle has this, but proving inequalities on complex terms remains tedious as often only very small proof steps are possible. However, the calculational proof style [5] (inspired by Mizar) is very helpful here.…”

Section: Discussionmentioning

confidence: 99%

Proof Pearl: A Probabilistic Proof for the Girth-Chromatic Number Theorem

Noschinski

2012

Interactive Theorem Proving

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“…Reasoning by chains of (in)equations is described elsewhere [1]. Reasoning about axiomatically defined structures by means of so called "locales" was first described in [3] but has evolved much since then.…”

Section: Overview Of the Papermentioning

confidence: 99%

“…The latter happens only with datatypes and inductively defined sets, but not with recursive functions. The third case is only shown in gory detail (see [1] for how to reason with chains of equations) to demonstrate that the 'case ( constructor vars) ' notation also works for arbitrary induction theorems with numbered cases. The order of the vars corresponds to the order of the -quantified variables in each case of the induction theorem.…”

Section: More Inductionmentioning

confidence: 99%