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Simple Types in Type Theory: Deep and Shallow Encodings

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Theorem Proving in Higher Order Logics (TPHOLs 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4732))

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Abstract

We present a formal treatment of normalization by evaluation in type theory. The involved semantics of simply-typed λ-calculus is exactly the simply typed fragment of the type theory. This means we have constructed and proved correct a decompilation function which recovers the syntax of a program, provided it belongs to the simply typed fragment. The development runs and is checked in Coq. Possible applications include the formal treatment of languages with binders.

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References

  1. Altenkirch, T., Dybjer, P., Hofmann, M., Scott, P.J.: Normalization by evaluation for typed lambda calculus with coproducts. In: LICS, pp. 303–310 (2001)

    Google Scholar 

  2. Berger, U.: Program extraction from normalization proofs. In: Bezem, M., Groote, J.F. (eds.) TLCA 1993. LNCS, vol. 664, pp. 91–106. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  3. Berger, U., Berghofer, S., Letouzey, P., Schwichtenberg, H.: Program extraction from normalization proofs. Studia Logica 82(1), 25–49 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berger, U., Eberl, M., Schwichtenberg, H.: Normalisation by evaluation. In: Möller, B., Tucker, J.V. (eds.) Prospects for Hardware Foundations. LNCS, vol. 1546, pp. 117–137. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  5. Berger, U., Eberl, M., Schwichtenberg, H.: Term rewriting for normalization by evaluation. Inf. Comput. 183(1), 19–42 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Berger, U., Schwichtenberg, H.: An inverse of the evaluation functional for typed lambda-calculus. In: LICS, pp. 203–211. IEEE Computer Society Press, Los Alamitos (1991)

    Google Scholar 

  7. Capretta, V., Felty, A.: Combining de bruijn indices and higher-order abstract syntax in coq. In: Altenkirch, T., McBride, C. (eds.) Proceedings of TYPES 2006. LNCS, Springer, Heidelberg (2007)

    Google Scholar 

  8. Coquand, C.: From semantics to rules: A machine assisted analysis. In: Meinke, K., Börger, E., Gurevich, Y. (eds.) CSL 1993. LNCS, vol. 832, pp. 91–105. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  9. Coquand, C.: A formalised proof of the soundness and completeness of a simply typed lambda-calculus with explicit substitutions. Higher-Order and Symbolic Computation 15(1), 57–90 (2002)

    Article  MATH  Google Scholar 

  10. Danvy, O., Rhiger, M., Rose, K.H.: Normalization by evaluation with typed abstract syntax. J. Funct. Program. 11(6), 673–680 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Despeyroux, J., Pfenning, F., Schürmann, C.: Primitive recursion for higher-order abstract syntax. In: de Groote, P., Hindley, J.R. (eds.) TLCA 1997. LNCS, vol. 1210, pp. 147–163. Springer, Heidelberg (1997)

    Google Scholar 

  12. Gonthier, G.: A computer-checked proof of the four colour theorem (2005), available on http://research.microsoft.com/~gonthier/

  13. Gonthier, G.: Notations of the four colour theorem proof (2005), available on http://research.microsoft.com/~gonthier/

  14. Krivine, J.-L.: Une preuve formelle et intuitionniste du théorème de complétude de la logique classique. Bulletin of Symbolic Logic 2(4), 405–421 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pfenning, F., Elliott, C.: Higher-order abstract syntax. In: PLDI, pp. 199–208 (1988)

    Google Scholar 

  16. Schürmann, C., Despeyroux, J., Pfenning, F.: Primitive recursion for higher-order abstract syntax. Theor. Comput. Sci. 266(1-2), 1–57 (2001)

    Article  MATH  Google Scholar 

  17. Stoughton, A.: Substitution revisited. Theor. Comput. Sci. 59, 317–325 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. École Normale Supéieure, 45 rue d’Ulm, Paris, France, and INRIA-Futurs at  , LIX, École Polytechnique, Palaiseau, France

    François Garillot

  2. INRIA-Futurs at, LIX, École Polytechnique, Palaiseau, France

    Benjamin Werner

Authors
  1. François Garillot
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  2. Benjamin Werner
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Klaus Schneider Jens Brandt

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Garillot, F., Werner, B. (2007). Simple Types in Type Theory: Deep and Shallow Encodings. In: Schneider, K., Brandt, J. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2007. Lecture Notes in Computer Science, vol 4732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74591-4_27

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  • DOI: https://doi.org/10.1007/978-3-540-74591-4_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74590-7

  • Online ISBN: 978-3-540-74591-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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