Abstract
Traditionally a rigorous mathematical document consists of a sequence of definition – statement – proof. Taking this basic outline as starting point we investigate how these three categories of text can be represented adequately in the formal language of Isabelle/Isar.
Proofs represented in human-readable form have been the initial motivation of Isar language design 10 years ago. The principles developed here allow to turn deductions of the Isabelle logical framework into a format that transcends the raw logical calculus, with more direct description of reasoning using pseudo-natural language elements.
Statements describe the main result of a theorem in an open format as a reasoning scheme, saying that in the context of certain parameters and assumptions certain conclusions can be derived. This idea of turning Isar context elements into rule statements has been recently refined to support the dual form of elimination rules as well.
Definitions in their primitive form merely name existing elements of the logical environment, by stating a suitable equation or logical equivalence. Inductive definitions provide a convenient derived principle to describe a new predicate as the closure of given natural deduction rules. Again there is a direct connection to Isar principles, rules stemming from an inductive characterization are immediately available in structured reasoning.
All three categories benefit from replacing raw logical encodings by native Isar language elements. The overall formality in the presented mathematical text is reduced. Instead of manipulating auxiliary logical connectives and quantifiers, the mathematical concepts are emphasized.
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References
Ballarin, C.: Interpretation of locales in Isabelle: Theories and proof contexts. In: Borwein, J.M., Farmer, W.M. (eds.) MKM 2006. LNCS (LNAI), vol. 4108, pp. 31–43. Springer, Heidelberg (2006)
Barras, B., et al.: The Coq Proof Assistant Reference Manual. INRIA (2006)
Gentzen, G.: Untersuchungen über das logische Schließen (I + II). Mathematische Zeitschrift 39 (1935)
Harrison, J.: Inductive definitions: automation and application. In: Schubert, E.T., Alves-Foss, J., Windley, P. (eds.) HUG 1995. LNCS, vol. 971. Springer, Heidelberg (1995)
Jaskowski, S.: On the rules of suppositions. Studia Logica 1 (1934)
Melham, T.F.: A package for inductive relation definitions in HOL. In: Archer, M., et al. (eds.) Higher Order Logic Theorem Proving and its Applications. IEEE Computer Society Press, Los Alamitos (1992)
Paulin-Mohring, C.: Inductive Definitions in the System Coq – Rules and Properties. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664. Springer, Heidelberg (1993)
Paulson, L.C.: The foundation of a generic theorem prover. Journal of Automated Reasoning 5(3) (1989)
Paulson, L.C.: Isabelle: the next 700 theorem provers. In: Odifreddi, P. (ed.) Logic and Computer Science. Academic Press, London (1990)
Paulson, L.C.: A fixedpoint approach to (co)inductive and (co)datatype definitions. In: Plotkin, G., Stirling, C., Tofte, M. (eds.) Proof, Language, and Interaction: Essays in Honour of Robin Milner. MIT Press, Cambridge (2000)
Rudnicki, P.: An overview of the MIZAR project. In: 1992 Workshop on Types for Proofs and Programs, Chalmers University of Technology, Bastad (1992)
Trybulec, A.: Some features of the Mizar language (1993)presented at Turin
Wenzel, M.: Isar — a generic interpretative approach to readable formal proof documents. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690. Springer, Heidelberg (1999)
Wenzel, M.: Isabelle/Isar — a versatile environment for human-readable formal proof documents. Ph.D. thesis, Institut für Informatik, TU München (2002)
Wenzel, M.: Isabelle/Isar — a generic framework for human-readable proof documents. In: Matuszewski, R., Zalewska, A. (eds.) From Insight to Proof — Festschrift in Honour of Andrzej Trybulec, University of Białystok. Studies in Logic, Grammar, and Rhetoric, vol. 10(23) (2007), http://www.in.tum.de/~wenzelm/papers/isar-framework.pdf
Wiedijk, F.(ed.): The Seventeen Provers of the World. LNCS (LNAI), vol. 3600. Springer, Heidelberg (2006)
Wiedijk, F., Wenzel, M.: A comparison of the mathematical proof languages Mizar and Isar. Journal of Automated Reasoning 29(3-4) (2002)
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Berghofer, S., Wenzel, M. (2008). Logic-Free Reasoning in Isabelle/Isar. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds) Intelligent Computer Mathematics. CICM 2008. Lecture Notes in Computer Science(), vol 5144. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85110-3_31
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