Abstract
Representation determines how we can reason about a specific problem. Sometimes one representation helps us find a proof more easily than others. Most current automated reasoning tools focus on reasoning within one representation. There is, therefore, a need for the development of better tools to mechanise and automate formal and logically sound changes of representation.
In this paper we look at examples of representational transformations in discrete mathematics, and show how we have used Isabelle’s Transfer tool to automate the use of these transformations in proofs. We give a brief overview of a general theory of transformations that we consider appropriate for thinking about the matter, and we explain how it relates to the Transfer package. We show our progress towards developing a general tactic that incorporates the automatic search for representation within the proving process.
D. Raggi—This work has been supported by a scholarship from the Mexican Council of Science and Technology (CONACYT).
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- 1.
These can be found in http://homepages.inf.ed.ac.uk/s1052074/AutoTransfer/. They are updated regularly.
- 2.
\(\mathbb {B}\) stands for type of booleans.
- 3.
This one is actually by construction using typedef and the Lifting package, which automatically declares transfer rules from definitions lifted by the user from an old type to the newly declared type.
- 4.
The mechanisation of these transformations have been submitted to the Archive of Formal Proofs, along with some examples of their use.
- 5.
using Isabelle tactics like auto.
- 6.
The examples of this second (more interesting) class have been selected from either maths textbooks for undergraduate students, or from training material for contests such as the Mathematical Olympiads.
- 7.
We thank the anonymous referees of this paper for suggested these possibilities. They remain as future work.
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Raggi, D., Bundy, A., Grov, G., Pease, A. (2015). Automating Change of Representation for Proofs in Discrete Mathematics. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds) Intelligent Computer Mathematics. CICM 2015. Lecture Notes in Computer Science(), vol 9150. Springer, Cham. https://doi.org/10.1007/978-3-319-20615-8_15
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