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Coq without Type Casts: A Complete Proof of Coq Modulo Theory

16 pagesPublished: May 4, 2017

Abstract

Incorporating extensional equality into a dependent intensional type system such as the Calculus of Constructions provides with stronger type-checking capabilities and makes the proof development closer to intuition. Since strong forms of extensionality lead to undecidable type-checking, a good trade-off is to extend intensional equality with a decidable first-order theory T, as done in CoqMT, which uses matching modulo T for the weak and strong elimination rules, we call these rules T-elimination. So far, type-checking in CoqMT is known to be decidable in presence of a cumulative hierarchy of universes and weak T-elimination. Further, it has been shown by Wang with a formal proof in Coq that consistency is preserved in presence of weak and strong elimination rules, which actually implies consistency in presence of weak and strong T-elimination rules since T is already present in the conversion rule of the calculus.
We justify here CoqMT’s type-checking algorithm by showing strong normalization as well as the Church-Rosser property of β-reductions augmented with CoqMT’s weak and strong T -elimination rules. This therefore concludes successfully the meta-theoretical study of CoqMT.

Keyphrases: Coq, first-order theory, proof assistant, soundness, type theory

In: Thomas Eiter and David Sands (editors). LPAR-21. 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning, vol 46, pages 474--489

Links:
BibTeX entry
@inproceedings{LPAR-21:Coq_without_Type_Casts,
  author    = {Jean-Pierre Jouannaud and Pierre-Yves Strub},
  title     = {Coq without Type Casts: A Complete Proof of Coq Modulo Theory},
  booktitle = {LPAR-21. 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning},
  editor    = {Thomas Eiter and David Sands},
  series    = {EPiC Series in Computing},
  volume    = {46},
  pages     = {474--489},
  year      = {2017},
  publisher = {EasyChair},
  bibsource = {EasyChair, http://www.easychair.org},
  issn      = {2398-7340},
  url       = {https://easychair.org/publications/paper/BKQ},
  doi       = {10.29007/bjpg}}
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