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A Sequent Calculus with Implicit Term Representation
[chapter]

2010
*
Lecture Notes in Computer Science
*

We define

doi:10.1007/978-3-642-15205-4_28
fatcat:o5ctxuk5rrb6lizbbcau7kdrgq
*a*cutelimination procedure for this*calculus*and show that it produces the same cut-free proofs as the standard*calculus*, but, due to the*implicit**representation*of*terms*, it provides exponentially ... We investigate*a*modification of the*sequent**calculus*which separates*a*first-order proof into its abstract deductive structure and*a*unifier which renders this structure*a*valid proof. ... On the other hand the*implicit**representation*of*terms*is used to give*a*considerably simplified proof of*a*characterisation of the form of witness*terms*obtainable by cut-elimination in*terms*of*a*regular ...##
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Intuitionistic Sequent-Style Calculus with Explicit Structural Rules
[chapter]

2011
*
Lecture Notes in Computer Science
*

*Sequent*

*calculus*-LJ

*Implicit*structural rules (Ax) Γ,

*A*

*A*(→ L ) Γ

*A*Γ, B C Γ,

*A*→ B C (→ R ) Γ,

*A*B Γ

*A*→ B (Cut) Γ

*A*Γ,

*A*B Γ B

*Sequent*

*term*calculi Pottinger, Zucker 1970s comparing cut-elimination ... Natural deduction λ-

*calculus*; Hilbert's axiomatic system combinators;

*Sequent*

*calculus*λ Gtz -

*calculus*

*Sequent*

*calculus*

*with*explicit structural rules ??? ...

##
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Compressing Polarized Boxes

2013
*
2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
*

The sequential nature of

doi:10.1109/lics.2013.49
dblp:conf/lics/Accattoli13
fatcat:vg26sbwapne7fchwqpkd56rh4y
*sequent**calculus*provides*a*simple definition of cut-elimination rules that duplicate or erase sub-proofs. ... Moreover,*implicit*boxes are more parallel than explicit boxes, as they realize*a*larger quotient. ... We deal*with*MELLP*sequent**calculus*only indirectly: we will relate our syntax to Laurent's proof nets, which are themselves related to*sequent**calculus*. Nets. ...##
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Importing SMT and Connection proofs as expansion trees

2015
*
Electronic Proceedings in Theoretical Computer Science
*

By representing the proofs in the same framework, all the algorithms and tools available for expansion trees (compression, visualization,

doi:10.4204/eptcs.186.3
fatcat:frjp276fq5amfcljaic4z44oii
*sequent**calculus*proof construction, proof checking, etc.) can ... The expansion proofs can also be used as*a*validation tool for the proof objects produced. ... Expansion proofs are*a*compact*representation*for first and higher order*sequent**calculus*proofs. ...##
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The λ-Calculus and the Unity of Structural Proof Theory

2009
*
Theory of Computing Systems
*

In the unified

doi:10.1007/s00224-009-9183-9
fatcat:ihe5exp4ovbrjozo3qgq7fgsla
*calculus*,*a**sequent**term*behaves like in the*sequent**calculus*, whereas the reduction steps of*a*natural deduction*term*are interleaved*with*explicit steps for bringing heads to focus. ...*A*variant of the*calculus*has the symmetric role of improving*sequent**calculus*in dealing*with*tail-active permutative conversions. ... Kleene [19] showed how to define*sequent**calculus**with**implicit*structural rules, built in the logical rules. ...##
###
Completing Herbelin's Programme
[chapter]

2007
*
Lecture Notes in Computer Science
*

Herbelin worked

doi:10.1007/978-3-540-73228-0_10
fatcat:pdfhumqqf5cslfeysqp6lbvmd4
*with**a*fragment of*sequent**calculus**with*constraints on left introduction. ... This requires the introduction of*a*lambda-like*calculus*for full*sequent**calculus*and an extension of natural deduction that gives meaning to "applicative contexts" and "applicative*terms*". ... Conclusions: Herbelin's seminal suggestion in [11] is that the (computational) difference between*sequent**calculus*and natural deduction may be reduced to*a*mere question of*representation*of λ-*terms*...##
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Structural Cut Elimination

2000
*
Information and Computation
*

The classical

doi:10.1006/inco.1999.2832
fatcat:olw6f7svunaalkzziyiullzcom
*sequent**calculus**with*proof*terms*satisfies the following properties. Proof. By simple structural inductions. ... Appendix B gives*a*formulation of cut elimination as*a*translation from*a**sequent**calculus**with*cut to*a**sequent**calculus*without cut. ... Let D :: (1 wÄ d + 2) be*a*classical*sequent*derivation possibly containing cut. Thus there exists*a*cut-free derivation D$ :: (1 wÄ d $ 2) Proof. ...##
###
CONSTRUCTIVE CLASSICAL LOGIC AS CPS-CALCULUS

2000
*
International Journal of Foundations of Computer Science
*

We establish the Curry-Howard isomorphism between constructive classical logic and CPS-

doi:10.1142/s0129054100000065
fatcat:qt4xrvsvsrc5xf62vheeqvavkq
*calculus*. CPS-*calculus*exactly means the target language of Continuation Passing Style(CPS) transforms. ... i.e. t :*A*x ) )*with*t-cut*with**implicit*contraction inFigureB.1. ... Our*term*assignment faithfully reects the structure of*sequent**calculus*. ...##
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Proving isomorphism of first-order logic proof systems in HOL
[chapter]

1998
*
Lecture Notes in Computer Science
*

Besides, by p r o ving

doi:10.1007/bfb0055143
fatcat:l3zojkhjgbckdbjkhiktlwmpoa
*a*theorem which states that provability in attened*sequent**calculus*implies provability in standard*sequent**calculus*, we show how some meta-logical results about Hilbertian axiomatization ... and natural deduction can be translated to*sequent**calculus*. ...*Sequent**calculus*(SC) represents*a*special case, being*a*proof system for*sequents**with*several (more than one) formulae in the succedent. ...##
###
Refocusing Generalised Normalisation
[chapter]

2007
*
Lecture Notes in Computer Science
*

In the unified

doi:10.1007/978-3-540-73001-9_27
fatcat:yike52mbgneqxdfmbnikmo44n4
*calculus*,*a**sequent**term*behaves like in the*sequent**calculus*, whereas the reduction steps of*a*natural deduction*term*are interleaved*with*explicit steps for bringing heads to focus. ...*A*variant of the*calculus*has the symmetric role of improving*sequent**calculus*in dealing*with*tail-active permutative conversions. ... This rules needs*a*mix of head and tail focus. Maybe*a*good system for dealing*with*such rules is*a*variant of the unified*calculus**with*reduction modulo the equation θ(F N, L) = θ(F, N :: L). 1 ...##
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Cut Elimination for Classical Proofs as Continuation Passing Style Computation
[chapter]

1998
*
Lecture Notes in Computer Science
*

These works are all based on logic in Gentzen-style

doi:10.1007/3-540-49366-2_5
fatcat:hkdngckiqjhrffbrhdnujunnhy
*sequent**calculus*[7] . We unify the proof theoretical approach (i.e. SN and CR cut-elimination procedure) and the reduction system approach (CPS). ... In Grin's result, Plotkin's call-by-name (CBN) CPS translation on simply-typed -*calculus*induces*a*G odel's double-negation translation on their types. ... Our*term*assignment faithfully reects the structure of*sequent**calculus*. Thus inductive denition of substitution on*terms*agrees*with*induction on the length of derivation for*sequent**calculus*. ...##
###
A Sequent Calculus for Type Theory
[chapter]

2006
*
Lecture Notes in Computer Science
*

In order to express proof-search in such theories, we introduce the Pure Type

doi:10.1007/11874683_29
fatcat:252c4vsmevfrzoe2x66m75gdfe
*Sequent*Calculi (PTSC) by enriching*a**sequent**calculus*due to Herbelin, adapted to proof-search and strongly related to natural ... PTSC are equipped*with**a*normalisation procedure, adapted from Herbelin's and defined by local rewrite rules as in Cut-elimination, using explicit substitutions. ... Although encodings from natural deduction to*sequent**calculus*and viceversa have been widely studied [Gen35, Pra65, Zuc74] , the*representation*in*sequent**calculus*of type theories is relatively undeveloped ...##
###
A tutorial on computational classical logic and the sequent calculus

2018
*
Journal of functional programming
*

We begin by reviewing Gentzen's LK

doi:10.1017/s0956796818000023
fatcat:ff6fprv6vfhrjgqhwnll3vqcwe
*sequent**calculus*and show how the Curry–Howard isomorphism still applies to give us*a*different basis for expressing computation. ... We present*a*model of computation that heavily emphasizes the concept of duality and the interaction between opposites–production interacts*with*consumption. ... In*terms*of provability-the question of which*sequents*can conclude*a*valid proof tree-the versions of LK*with*explicit and*implicit*structural rules are the same. ...##
###
On the intuitionistic force of classical search

2000
*
Theoretical Computer Science
*

We develop

doi:10.1016/s0304-3975(99)00178-4
fatcat:bc26d3pcs5fojbmexgaed6ddbu
*a*system of realizers (proof-objects) for*sequents*in classical propositional logic (the types) by extending Parigot's -*calculus*,*a*system of realizers for classical free deduction (cf. natural ... As an application, we develop*a*proof procedure based on the natural extension of the notion of uniform proof to the multiple-conclusioned classical*sequent**calculus*Harrop fragment of intuitionistic logic ... Acknowledgements We gratefully acknowledge the UK EPSRC for supporting part of this work via Research Grants GR=J46616 and GR=K41687 under the common title "Search Modules I:*Representation*and Combination ...##
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Deduction versus Computation: The Case of Induction
[chapter]

2002
*
Lecture Notes in Computer Science
*

When mechanizing proof construction, explicit induction is used in proof assistants and

doi:10.1007/3-540-45470-5_3
fatcat:4bjblblahbhkrhbmrrmxx2ivsu
*implicit*induction is used in rewrite based automated theorem provers. ... We show how this applies to*a*uniform understanding of the so called induction by rewriting method and how this relates directly to the general use of an induction principle. ... This presentation of first-order logic relies on the*sequent**calculus*modulo*a*congruence defined on*terms*and propositions. ...
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