On Choice Rules in Dependent Type Theory

Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10185)

Abstract

In a dependent type theory satisfying the propositions as types correspondence together with the proofs-as-programs paradigm, the validity of the unique choice rule or even more of the choice rule says that the extraction of a computable witness from an existential statement under hypothesis can be performed within the same theory.

Here we show that the unique choice rule, and hence the choice rule, are not valid both in Coquand’s Calculus of Constructions with indexed sum types, list types and binary disjoint sums and in its predicative version implemented in the intensional level of the Minimalist Foundation. This means that in these theories the extraction of computational witnesses from existential statements must be performed in a more expressive proofs-as-programs theory.

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Notes

Acknowledgements

We acknowledge very fruitful discussions on this topic with Claudio Sacerdoti Coen, Ferruccio Guidi, Giuseppe Rosolini, Giovanni Sambin, Thomas Streicher. We also thank Tatsuji Kawai and Fabio Pasquali very much for their comments on this paper.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PadovaPaduaItaly

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