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Complete Bidirectional Typing for the Calculus of Inductive Constructions [article]

Meven Lennon-Bertrand
2021 arXiv   pre-print
This article presents a bidirectional type system for the Calculus of Inductive Constructions (CIC).  ...  Although it plays an important role in an ongoing completeness proof for a realistic typing algorithm, the interest of bidirectionality is wider, as it gives insights and structure when trying to prove  ...  The central point is to distinguish in a judgment between the subject, 23:4 Complete Bidirectional Typing for the Calculus of Inductive Constructions ⊢ Γ ⊢ · Empty ⊢ Γ Γ ⊢ A : □ i ⊢ Γ, x : A Ext Γ ⊢  ... 
arXiv:2102.06513v2 fatcat:uel5dizklzgjnghu67rkjjfocm

Complete Bidirectional Typing for the Calculus of Inductive Constructions

Meven Lennon-Bertrand, Liron Cohen, Cezary Kaliszyk
2021
This article presents a bidirectional type system for the Calculus of Inductive Constructions (CIC).  ...  Although it plays an important role in an ongoing completeness proof for a realistic typing algorithm, the interest of bidirectionality is wider, as it gives insights and structure when trying to prove  ...  I T P 2 0 2 1 24:14 Complete Bidirectional Typing for the Calculus of Inductive Constructions But let us first give a bit of context about this extension.  ... 
doi:10.4230/lipics.itp.2021.24 fatcat:p4y3sktb5bhpdnsi6sgtam5l44

Touring the MetaCoq Project (Invited Paper)

Matthieu Sozeau
2021 Electronic Proceedings in Theoretical Computer Science  
the following artefacts: a specification of Coq's syntax and type theory, the Polymorphic Cumulative Calculus of (Co)-Inductive Constructions (PCUIC); a monad for the manipulation of raw syntax and interaction  ...  for PCUIC; a sound type and proof erasure procedure from PCUIC to untyped lambda-calculus, i.e., the core of the extraction mechanism of Coq.  ...  Typing, Reduction and Conversion The calculus at the basis of COQ is the Polymorphic Cumulative Calculus of (Co-)Inductive Constructions (PCUIC).  ... 
doi:10.4204/eptcs.337.2 fatcat:vkzxls6psredfkeo7xwazecbpa

Page 5464 of Mathematical Reviews Vol. , Issue 88j [page]

1988 Mathematical Reviews  
Thecalculus of constructions”, developed by the authors, is based on a typed lambda calculus.  ...  A minor difference in the defi- nitions is that types in the calculus of constructions are required to be in normal form.  ... 

Programming Language Foundations in Agda [chapter]

Philip Wadler
2018 Lecture Notes in Computer Science  
After five years using SF in the classroom, I have come to the conclusion that Coq is not the best vehicle for this purpose, as too much of the course needs to focus on learning tactics for proof derivation  ...  Second, that constructive proofs of preservation and progress give immediate rise to a prototype evaluator.  ...  Bidirectional type inference is shown to be both sound and complete.  ... 
doi:10.1007/978-3-030-03044-5_5 fatcat:5hotvlnxujat5nqvs6o6udfq2y

Programming language foundations in Agda

Wen Kokke, Jeremy G. Siek, Philip Wadler
2020 Science of Computer Programming  
For comments on a draft of this paper, an extra thank you to James McKinna, Ulf Norell, Andreas Abel, and Benjamin Pierce. This research was supported by EPSRC Programme Grant EP/K034413/1.  ...  For inventing ideas on which PLFA is based, and for handholding, many thanks to Conor McBride, James McKinna, Ulf Norell, and Andreas Abel.  ...  Bidirectional type inference is shown to be both sound and complete. Untyped: Untyped calculus with full normalisation.  ... 
doi:10.1016/j.scico.2020.102440 fatcat:czwjxx35krfvtclrrmor4gwziy

Strict bidirectional type checking

Adam Chlipala, Leaf Petersen, Robert Harper
2005 Proceedings of the 2005 ACM SIGPLAN international workshop on Types in languages design and implementation - TLDI '05  
An additional optimization that is used in some proof and type systems is to take advantage of the context of occurrence of terms to further elide type information using bidirectional type checking rules  ...  Completely annotated lambda terms (such as are arrived at via the straightforward encodings of various types from System F) contain much redundant type information.  ...  In this case, the context of occurrence defines the type of the function completely, and hence the type decorations on the functions are completely redundant.  ... 
doi:10.1145/1040294.1040301 dblp:conf/tldi/ChlipalaPH05 fatcat:iktpd3rtqrh77civ4linuxcq5q

Towards Normalization by Evaluation for the βη-Calculus of Constructions [chapter]

Andreas Abel
2010 Lecture Notes in Computer Science  
We consider the Calculus of Constructions with typed beta-eta equality and an algorithm which computes long normal forms.  ...  This models also yields consistency of the beta-eta-Calculus of Constructions. The model construction can be carried out directly in impredicative type theory, enabling a formalization in Coq.  ...  Thanks also to the anonymous referees for their comments which helped to improve the quality of this paper.  ... 
doi:10.1007/978-3-642-12251-4_17 fatcat:gjroecnajzgl3oskh2x5kqyr4y

Characterization of strong normalizability for a sequent lambda calculus with co-control

José Espírito Santo, Silvia Ghilezan
2017 Proceedings of the 19th International Symposium on Principles and Practice of Declarative Programming - PPDP '17  
In this sequent lambda calculus, the management of formulas on the left hand side of typing judgements is "dual" to the management of formulas on the right hand side of the typing judgements in Parigot's  ...  We study strong normalization in a lambda calculus of proof-terms with co-control for the intuitionistic sequent calculus.  ...  This work was partly supported by FCT-Fundação para a Ciência e a Tecnologia, within the project UID-MAT-00013/2013; by COST Action CA15123 -The European research network on types for programming and verification  ... 
doi:10.1145/3131851.3131867 dblp:conf/ppdp/SantoG17 fatcat:oidfojtapjfhpnbfhbxspaydoe

Bidirectional Type Checking for Relational Properties [article]

Ezgi Çiçek, Weihao Qu, Gilles Barthe, Marco Gaboardi and Deepak Garg
2018 arXiv   pre-print
Our approach achieves the benefits of bidirectional type checking, in a relational setting.  ...  In particular, it significantly reduces the need for typing annotations through the combination of type checking and type inference.  ...  ensure completeness of bidirectional typechecking (this will cease to be the case for later type systems).  ... 
arXiv:1812.05067v1 fatcat:vpebbvnycbbvdahboho5ntzgrq

π with leftovers: a mechanisation in Agda [article]

Uma Zalakain, Ornela Dardha
2021 arXiv   pre-print
Linear types are important in the π-calculus: they ensure privacy and safety of communication and avoid race conditions, while graded and shared types allow for more flexible programming.  ...  We use leftover typing to define a type system for a resource-aware π-calculus, a process algebra used to model concurrent systems.  ...  Acknowledgments We want to thank Erika, Wen Kokke, James Wood, Guillaume Allais, Bob Atkey, and Conor McBride for their valuable suggestions.  ... 
arXiv:2005.05902v4 fatcat:bq5rz72qancbbowukn55tsg3vq

Second-Order Equational Logic (Extended Abstract) [chapter]

Marcelo Fiore, Chung-Kil Hur
2010 Lecture Notes in Computer Science  
Two completeness results are established: the semantic completeness of equational derivability, and the derivability completeness of (bidirectional) Second-Order Term Rewriting.  ...  We develop an algebraic model theory for second-order equational presentations, generalising the semantics of (first-order) algebraic theories and of (untyped and simply-typed) lambda calculi. 3.  ...  We are most grateful to Pierre-Louis Curien for detailed comments on a preliminary version of the paper.  ... 
doi:10.1007/978-3-642-15205-4_26 fatcat:edxg56j7ifdbvcvosjelqgzppa

Copatterns

Andreas Abel, Brigitte Pientka, David Thibodeau, Anton Setzer
2013 Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '13  
We prove type soundness for our language and sketch how copatterns open new directions for solving problems in the interaction of coinductive and dependent types.  ...  Dual to pattern matching, a tool for analyzing finite data, we develop the concept of copattern matching, which allows us to synthesize infinite data.  ...  Acknowledgments The first author thanks Thierry Coquand and Nils Anders Danielsson for email exchange about copatterns and the regular participants of the Agda Implementor's Meetings from 2008 on for lively  ... 
doi:10.1145/2429069.2429075 dblp:conf/popl/AbelPTS13 fatcat:7wevbhmshfewdncsc3tqo34cwm

Copatterns

Andreas Abel, Brigitte Pientka, David Thibodeau, Anton Setzer
2013 SIGPLAN notices  
We prove type soundness for our language and sketch how copatterns open new directions for solving problems in the interaction of coinductive and dependent types.  ...  Dual to pattern matching, a tool for analyzing finite data, we develop the concept of copattern matching, which allows us to synthesize infinite data.  ...  Acknowledgments The first author thanks Thierry Coquand and Nils Anders Danielsson for email exchange about copatterns and the regular participants of the Agda Implementor's Meetings from 2008 on for lively  ... 
doi:10.1145/2480359.2429075 fatcat:zrbooidzpjfhro4qinjzcotaxa

Elaborating intersection and union types

JANA DUNFIELD
2014 Journal of functional programming  
We describe a foundation for compiling unrestricted intersection and union types: an elaboration type system that generates ordinary lambda-calculus terms.  ...  With this construct, not all reductions of the source program preserve types; however, we prove that ordinary call-by-value evaluation of the elaborated program corresponds to a type-preserving evaluation  ...  Acknowledgments In 2008, Adam Megacz suggested (after I explained the idea of compiling intersection to product) that one could use an existing ML compiler "as a backend".  ... 
doi:10.1017/s0956796813000270 fatcat:ctnnaravtfgxlf3pcog5pkbzae
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