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Types in lambda calculi and programming languages [chapter]

Henk Barendregt, Kees Hemerik
1990 Lecture Notes in Computer Science  
TYPE-FREE LAMBDA CALCULUS The aspects of lambda calculus important for functional programming consist of the syntax of terms and the reduction (rewrite) relation on these. 1.1.1. Definition.  ...  In this way the { ~, V } fragment of (manysorted constructive) predicate logic can be interpreted too.  ... 
doi:10.1007/3-540-52592-0_53 fatcat:qgywhxbgjvhyvdgi4lmqabsbxm

Context-Free Language Theory Formalization [article]

Marcus Vinícius Midena Ramos, Ruy J. G. B. de Queiroz
2015 arXiv   pre-print
This proposal aims at formalizing parts of context-free language theory in the Coq proof assistant.  ...  The results obtained so far include the formalization of closure properties for context-free grammars (under union, concatenation and closure) and the formalization of grammar simplification.  ...  The so-called lambda language is a notation for the representation of lambda terms, which are then manipulated by the rules of the calculus.  ... 
arXiv:1505.00061v1 fatcat:kqugpa34jvcjlpdomfabtdl2aq

A Formalisation of Weak Normalisation (with Respect to Permutations) of Sequent Calculus Proofs

A.A. Adams
2000 LMS Journal of Computation and Mathematics  
The formalisation is restricted to the implicational fragment of intuitionistic logic.  ...  AbstractDyckhoff and Pinto present a weakly normalising system of reductions on derivations are characterised as the fixed points of the composition of the Prawitz translations into natural deduction and  ...  The formal development was performed with Version 6.1 of the Coq proof system, and may not work with more recent releases.  ... 
doi:10.1112/s1461157000000188 fatcat:a7z76x7w4jg3pnmzr44j4dadsi

Foundations [article]

Jeremy Avigad
2021 arXiv   pre-print
This is a draft of a chapter on mathematical logic and foundations for an upcoming handbook of computational proof assistants.  ...  If we read the term : as saying that is a proof of , we can interpret the simply typed lambda calculus as a calculus of proof terms for the corresponding fragment of intuitionistic logic.  ...  Such a type theory was the basis of N. G. de Bruijn's seminal Automath proof assistant from the late 1960s.  ... 
arXiv:2009.09541v4 fatcat:yatuz32vondo3eburgtj6jmp6e

Typed Nominal Rewriting

Elliot Fairweather, Maribel Fernández
2018 ACM Transactions on Computational Logic  
First studied as a means to overcome the paradoxes of set theory [64], they were later incorporated into the lambda calculus by Church [1940].  ...  For example, the nominal terms lam [a] X and app (lam [a] X , X ), where the permutation suspended over X is the identity and has been omitted, represent the lambda term 'contexts' λx .M (or λx .[−]) and  ...  It also highlights a conflict between the types for the lambda calculus being modelled and the type system for nominal terms explored here that specifies the syntactic representations of the lambda terms  ... 
doi:10.1145/3161558 fatcat:ghyr5ru2sjhi3ib4wdtsny23uq

Innovations in computational type theory using Nuprl

S.F. Allen, M. Bickford, R.L. Constable, R. Eaton, C. Kreitz, L. Lorigo, E. Moran
2006 Journal of Applied Logic  
fragment.  ...  The Gödel dialectica interpretation [107] also applied to the constructive fragment. Serious and difficult issues are lurking one small step away from elementary logic.  ...  Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.  ... 
doi:10.1016/j.jal.2005.10.005 fatcat:eq3eqvtacvbmnd3vywzioi7kbm

Used Formal Methods

Holger Blasum, Oto Havle, Sergey Tverdyshev, Bruno Langenstein, Werner Stephan, Abderrahmane Feliachi, Yakoub Nemouchi, Burkhart Wolff, Cyril Proch, Freek Verbeek, Julien Schmaltz
2015 Zenodo  
We state how, in the EURO-MILS project, the developednbsp;theories are compliant with (1) and (2)./li> /ul>  ...  This document consists of three chapters: Chapter 1 describes how Isabelle/HOL works and how to use it in a certification processnbsp;in a sound way./li> li>Chapter 2: Style Guide.  ...  One can include the fragment: which defines a new tactic that applies just the existential-introduction rule of HOL.  ... 
doi:10.5281/zenodo.47297 fatcat:oopbiyjrhvh3hizcbokwimobsi

Proofs and Programs about Open Terms [article]

Francisco Ferreira Ruiz
2018 arXiv   pre-print
framework LF with the expressivity of dependent types.  ...  We discuss three aspects of this: the reconstruction of implicit parameters that makes writing proofs and programs with dependent types easier, the addition of contextual objects to an existing programming  ...  RECONSTRUCTION OF IMPLICIT PARAMETERS 35 Figure 3 . 2 : 32 Well-formed Kinds and Types highlight that free index variables are only allowed in declarations when Figure 3 . 3 : 33 Example: A Simply-typed  ... 
arXiv:1805.07176v1 fatcat:pxvtcretendzxgpq7bgnsgdhiu

ASPfun : A typed functional active object calculus

Ludovic Henrio, Florian Kammüller, Bianca Lutz
2012 Science of Computer Programming  
Most importantly, ASP fun has been formalised; its properties have been formalised and proved using the Isabelle theorem prover and we consider this as an important step in the formalization of distributed  ...  This work was also an opportunity to study different binder representations and experiment with two of them in the Isabelle/HOL theorem prover.  ...  lambda calculus with futures.  ... 
doi:10.1016/j.scico.2010.12.008 fatcat:5ry772fd2bhpzjztbxbkb7ufvy

Mixing Computations and Proofs

Michael Beeson
2014 Journal of Formalized Reasoning  
We compare the various approaches to proofs with a significant computational component, including (i) verifying the algorithms, (ii) verifying the results of the unverified algorithms, and (iii) trusting  ...  We examine the relationship between proof and computation in mathematics, especially in formalized mathematics.  ...  What exactly are the "difficulties to be overcome" that are highlighted by this example? Certainly one of them is the matter of the "rich enough library."  ... 
doi:10.6092/issn.1972-5787/4552 dblp:journals/jfrea/Beeson16 fatcat:ieften5wunhnnesihejategbku

Variations on a theme of Curry and Howard : the Curry-Howard isomorphism and the proofs-as-programs paradigm adapted to imperative and structured program synthesis

Poernomo, Iman Hafiz, 1976-
The idea of separating proofs is essential to the Curry-Howard protocol, identified by Crossley and the author in [PC01], and described in Part II of this thesis.  ...  Parts III and IV of this thesis are an applications of the protocol, and can be seen as a further argument for the separation of proofs from programs, to achieve a practical approaches to constructive  ...  the usual notions of free and bound variables of the lambda terms of Term(E).  ... 
doi:10.4225/03/59c9fe4995db8 fatcat:xtbydkjgrrb2xh6n75nvery7ka

Extensional Paramodulation for Higher-Order Logic and Its Effective Implementation Leo-III

Alexander Steen
2019 Künstliche Intelligenz  
In the practically motivated main part of this thesis, the design and architecture of the new higher-order theorem prover Leo-III is presented.  ...  An evaluation on a heterogeneous set of benchmark problems confirms that Leo-III is one of the most effective and versatile higher-order automated reasoning systems to date. vii  ...  Note that "extensionality principles" usually only refer to the validity of ( f ντ X τ = g ντ X τ ) ⇒ Note that the formulation of "Lambda-free higher-order logic" by Bentkamp et al. does not come  ... 
doi:10.1007/s13218-019-00628-8 fatcat:b6yfenxgxndb3myskqushsm6uy

A formalisation of design methods : a lambda-calculus approach to system design with an application to text editing [article]

LMG Loe Feijs, FEJ Frans Kruseman Aretz, JA Jan Bergstra
cycle-free.  ...  In fact the specification of the editor itself is only a fragment of the total specification.  ...  As usual the renaming is done in the parameterised description and as a result we get a parameterised description whose parameter restrictions are renamed versionsof CLASS SORT Inst FREE END and ITEM respectively  ... 
doi:10.6100/ir324749 fatcat:jgws4v7lz5bpdigaxkezwue2uy

Refinement Types for Logical Frameworks

William Lovas
several case studies that highlight the utility of refinement types for formalized mathematics.  ...  In the end I find that refinement types and LF are a match made in heaven: refinements enable many rich new modes of expression, and the simplicity of LF ensures that they come at a modest cost.  ...  (Compare with the declarations above for the sort lambda.) v/lambda ⊏ tm. v/lambda :: sort. lam : : ⊤ → (v/value → ⊤) → v/lambda.  ... 
doi:10.1184/r1/6721286.v1 fatcat:sfuxh7ibcjf37b7d5embncwhkm

Assertion level proof planning with compiled strategies [article]

Dominik Dietrich, Universität Des Saarlandes, Universität Des Saarlandes
permission of the Publisher.  ...  objective of this thesis is to ease the formalization of proofs by being able to verify as well as to automatically construct abstract human-style proofs.  ...  Note that the fragment of geometry theorem proving is decidable [Har09] .  ... 
doi:10.22028/d291-26394 fatcat:ubhz423ftrcn7ez35xgx3ggczy
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