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Types in lambda calculi and programming languages
[chapter]
1990
Lecture Notes in Computer Science
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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]
2015
arXiv
pre-print
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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
2000
LMS Journal of Computation and Mathematics
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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]
2021
arXiv
pre-print
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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
2018
ACM Transactions on Computational Logic
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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
2006
Journal of Applied Logic
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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
2015
Zenodo
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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]
2018
arXiv
pre-print
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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
2012
Science of Computer Programming
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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
2014
Journal of Formalized Reasoning
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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
2017
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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
2019
Künstliche Intelligenz
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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]
1990
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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
2018
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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]
2012
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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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