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Explicit substitutions calculi with one step Eta-reduction decided explicitly

D. Ventura, M. Ayala-Rincon, F. Kamareddine
2009 Logic Journal of the IGPL  
In particular, we show that the explicit definitions of the eta rules coincide with the Eta rule for pure λ-terms and that moreover, their application is decidable in the sense that Eta redices are effectively  ...  In this paper constructive Eta rules are proposed for both the λσ-and the λse-calculi of explicit substitutions.  ...  One has b[s] = b + s = 0. Then, by Lemma 4.7, (s • η i i ) ES Calculi with One Step Eta-reduction Decided Explicitly  ... 
doi:10.1093/jigpal/jzp027 fatcat:3wsbhxuoqjen7f3msj455ljx2q

Eta-Equivalence in Core Dependent Haskell

Anastasiya Kravchuk-Kirilyuk, Antoine Voizard, Stephanie Weirich, Assia Mahboubi, Marc Bezem
2020 Types for Proofs and Programs  
We extend the core semantics for Dependent Haskell with rules for η-equivalence. This semantics is defined by two related calculi, Systems D and DC.  ...  Our work builds on and extends the existing metatheory for these systems developed using the Coq proof assistant.  ...  One difficulty with η-reduction in this setting is the problem with confluence for Churchstyle calculi.  ... 
doi:10.4230/lipics.types.2019.7 dblp:conf/types/Kravchuk-Kirilyuk19 fatcat:kjlierzyn5at3ls35xsdzomumm

More Church-Rosser proofs (in Isabelle/HOL) [chapter]

Tobias Nipkow
1996 Lecture Notes in Computer Science  
All proofs are based on a general theory of commutating relations which supports an almost geometric style of con uence proofs.  ...  For -reduction, both the standard proof and the variation by Takahashi are given and compared.  ...  H o wever, after I had nished my o wn development I found out about the following two formalizations of in -calculi with explicit substitutions.  ... 
doi:10.1007/3-540-61511-3_125 fatcat:k6rjkyixgzhpvosc5dpqd742li

Second-Order Matching via Explicit Substitutions [chapter]

Flávio L. C. de Moura, Fairouz Kamareddine, Mauricio Ayala-Rincón
2005 Lecture Notes in Computer Science  
In this paper, we will make use of explicit substitutions to facilitate matching: we develop a second-order matching algorithm via the λσ-style of explicit substitutions.  ...  The past few years have established the benefit of using explicit substitutions for theorem proving and higher-order unification.  ...  We write M → r M to denote one step reduction by one application of rule r.  ... 
doi:10.1007/978-3-540-32275-7_29 fatcat:bfzqutroerhbtcdgys4mro6yei

Higher-Order Rewriting: Framework, Confluence and Termination [chapter]

Jean-Pierre Jouannaud
2005 Lecture Notes in Computer Science  
Acknowledgments: This paper would not exist without the pionneering work of Klop on higher-order rewriting and the work I have done myself with my coauthors Femke van Raamsdonk and Albert Rubio, my colleague  ...  Mitsuhiro Okada with who I started getting into that subject, and my students Maribel Fernandez, Frederic Blanqui and Daria Walukiewicz who investigated some of these questions within the framework of  ...  In both cases, the conventional expression is transformed by the parser into the one with explicit application.  ... 
doi:10.1007/11601548_14 fatcat:iwdbwyqmbre2bkbza545hxkjfu

Type Destructors

Martin Hofmann, Benjamin C. Pierce
2002 Information and Computation  
We study a variant of System F that integrates and generalizes several existing proposals for calculi with structural typing rules. To the usual type constructors !  ...  We illustrate this by giving elementary presentations of two w ell-known encodings of objects, one based on recursive record types and the other based on existential packages.  ...  Moreover, most of the de nition in Section 3.2 is mirrored directly here: Eta-A-Refl is an explicit symmetry rule, while Eta-A-Any through Eta-A-Rec give explicit congruence rules for all the type constructors  ... 
doi:10.1006/inco.2001.2926 fatcat:alca7rajqrgrhl5g66ygpefpzu

Full Reduction in the Face of Absurdity [chapter]

Gabriel Scherer, Didier Rémy
2015 Lecture Notes in Computer Science  
Core calculi that model the essence of computations use full reduction semantics to be built on solid grounds.  ...  To preserve type soundness, reduction must then be delayed until logical hypotheses on which the computation depends have been proved consistent.  ...  We know how to design calculi with explicit uses of logical assumptions and full reduction, or calculi with implicit uses of assumptions and weak reduction.  ... 
doi:10.1007/978-3-662-46669-8_28 fatcat:yy7nhn76eraapc7w5wzufrgptq

Polymorphism and separation in hoare type theory

Aleksandar Nanevski, Greg Morrisett, Lars Birkedal
2006 Proceedings of the eleventh ACM SIGPLAN international conference on Functional programming - ICFP '06  
Each judgment relates an expression with its one-step reduct. The inference rules of the evaluation judgments are straightforward, so we omit them here.  ...  Monadic and hereditary substitutions. The equational theory of HTT is based on the usual beta and eta reductions for the various type constructors.  ... 
doi:10.1145/1159803.1159812 dblp:conf/icfp/NanevskiMB06 fatcat:2rfoxzsbxba4fncioxrqbxa3hy

Polymorphism and separation in hoare type theory

Aleksandar Nanevski, Greg Morrisett, Lars Birkedal
2006 SIGPLAN notices  
Each judgment relates an expression with its one-step reduct. The inference rules of the evaluation judgments are straightforward, so we omit them here.  ...  Monadic and hereditary substitutions. The equational theory of HTT is based on the usual beta and eta reductions for the various type constructors.  ... 
doi:10.1145/1160074.1159812 fatcat:2xtpzu7l2zckzhketfmhjftdba

Resource operators for λ-calculus

Delia Kesner, Stéphane Lengrand
2007 Information and Computation  
We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic's proof-nets  ...  simulation of -reduction and full composition.  ...  on the calculus with explicit substitutions to mimic exactly the calculus without explicit substitutions [25] .  ... 
doi:10.1016/j.ic.2006.08.008 fatcat:3rmvqhcrpjbd3gs24yqzmvgvtm

Separability in the Ambient Logic

Etienne Lozes, Daniel Hirschkoff, Davide Sangiorgi, Andrew Gordon
2008 Logical Methods in Computer Science  
We also present some (un)decidability results that are related to the above separation properties for AL: the undecidability of _=L on MA and its decidability on the subcalculus.  ...  In these calculi, we provide two operational characterisations of _=L: a coinductive one (as a form of bisimilarity) and an inductive one (based on structual properties of processes).  ...  Unless explicitely stated, we use P, Q, . . . to range over closed processes in our definitions and results. Substitutions, ranged over with σ, are partial functions from variables to names.  ... 
doi:10.2168/lmcs-4(3:4)2008 fatcat:5cgoodeasjenbjefxkft2ntbua

Mechanizing the Metatheory of LF [article]

Christian Urban, James Cheney, Stefan Berghofer
2010 arXiv   pre-print
In the process, we identified and resolved a gap in one of the proofs and a small number of minor lacunae in others.  ...  However, correct use of LF relies on nontrivial metatheoretic developments such as proofs of correctness of decision procedures for LF's judgments.  ...  Benjamin Pierce and Stephanie Weirich have also made helpful comments on drafts of this paper.  ... 
arXiv:0804.1667v3 fatcat:a372v4ivi5cxhl555b5kamvnzi

Logical foundations for hybrid type-logical grammars [article]

Richard Moot, Symon Stevens-Guille
2020 arXiv   pre-print
This paper explores proof-theoretic aspects of hybrid type-logical grammars , a logic combining Lambek grammars with lambda grammars.  ...  Term labels have been added to the vertices of the graph to make the correspondence with beta-reduction explicit.  ...  Finally, the rules in δ ′ 3 are therefore those in δ 3 with possibly a single β reduction removed and with P either identical to P ′ or β-reducible to it in one step. 4 In what follows, we assume that  ... 
arXiv:2009.10387v1 fatcat:6toldo6dozccnkjjltcjvvibaa

Mechanizing the metatheory of LF

Christian Urban, James Cheney, Stefan Berghofer
2011 ACM Transactions on Computational Logic  
substitution) by (nominal) primitive recursion, and (3) strong induction principles for datatypes and inductive definitions that build in Barendregt-style renaming conventions.  ...  In the process, we identified and resolved a gap in one of the proofs and a small number of minor lacunae in others.  ...  Benjamin Pierce and Stephanie Weirich have also made helpful comments on drafts of this paper.  ... 
doi:10.1145/1877714.1877721 fatcat:asxhcrfavbhrnnm6cvtqpi7nhq

Mechanizing the Metatheory of LF

Christian Urban, James Cheney, Stefan Berghofer
2008 Logic in Computer Science  
substitution) by (nominal) primitive recursion, and (3) strong induction principles for datatypes and inductive definitions that build in Barendregt-style renaming conventions.  ...  In the process, we identified and resolved a gap in one of the proofs and a small number of minor lacunae in others.  ...  Benjamin Pierce and Stephanie Weirich have also made helpful comments on drafts of this paper.  ... 
doi:10.1109/lics.2008.29 dblp:conf/lics/UrbanCB08 fatcat:dtiowrtba5ey7i2cwryfeocdnq
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