# Type-theory in color

@article{Bernardy2013TypetheoryIC, title={Type-theory in color}, author={Jean-Philippe Bernardy and Guilhem Moulin}, journal={Proceedings of the 18th ACM SIGPLAN international conference on Functional programming}, year={2013} }

Dependent type-theory aims to become the standard way to formalize mathematics at the same time as displacing traditional platforms for high-assurance programming. However, current implementations of type theory are still lacking, in the sense that some obvious truths require explicit proofs, making type-theory awkward to use for many applications, both in formalization and programming. In particular, notions of erasure are poorly supported. In this paper we propose an extension of type-theory…

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## References

SHOWING 1-10 OF 22 REFERENCES

Towards a practical programming language based on dependent type theory

- Computer Science
- 2007

This thesis is concerned with bridging the gap between the theoretical presentations of type theory and the requirements on a practical programming language.

On Irrelevance and Algorithmic Equality in Predicative Type Theory

- PhilosophyLog. Methods Comput. Sci.
- 2012

Pfenning's type theory with irrelevant quantification is considered which is compatible with a type-based notion of equality that respects eta-laws and its meta-theory is extended to universes and large eliminations and developed.

Canonicity for 2-dimensional type theory

- MathematicsPOPL '12
- 2012

This paper develops a novel judgemental formulation of a two-dimensional type theory, which enjoys a canonicity property: a closed term of boolean type is definitionally equal to true or false.

Lambda Calculus with Types

- Computer SciencePerspectives in logic
- 2013

This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype…

Proofs for free

- MathematicsJournal of Functional Programming
- 2012

It is shown how a typing judgement in System F can be translated into a relational statement (in second-order predicate logic) about inhabitants of the type, and it is obtained that for pure type systems (PTSs) there is a PTS that can be used as a logic for parametricity.

A Computational Interpretation of Parametricity

- Computer Science, Mathematics2012 27th Annual IEEE Symposium on Logic in Computer Science
- 2012

This paper describes an extension of the Pure Type Systems with a special parametricity rule (with computational content), and proves fundamental properties such as Church-Rosser's and strong normalization.

Erasure and Polymorphism in Pure Type Systems

- Computer ScienceFoSSaCS
- 2008

The execution model of EPTS generalizes the familiar notions of type erasure and parametric polymorphism, and it is believed functional programmers will find it quite natural to program in such a setting.

Realizability and Parametricity in Pure Type Systems

- Computer ScienceFoSSaCS
- 2011

A systematic method to build a logic from any programming language described as a Pure Type System (PTS) that defines a parametricity theory about programs and a realizability theory for the logic, which is expressive enough to internalize both theories.

Intensionality, extensionality, and proof irrelevance in modal type theory

- PhilosophyProceedings 16th Annual IEEE Symposium on Logic in Computer Science
- 2001

A uniform type theory that integrates intensionality, extensionality and proof irrelevance as judgmental concepts is developed that contrasts with previous approaches that, a priori, distinguished propositions from specifications.

Parametricity and dependent types

- PhilosophyICFP '10
- 2010

Reynolds' abstraction theorem shows how a typing judgement in System F can be translated into a relational statement about inhabitants of the type, and a similar result is obtained for a single lambda calculus, in which terms, types and their relations are expressed.