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##
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Classical Program Extraction in the Calculus of Constructions
[chapter]

*
Lecture Notes in Computer Science
*

We show how to

doi:10.1007/978-3-540-74915-8_25
fatcat:kuh6lyjvlre7jm563sdyy3wple
*extract**classical**programs*expressed*in*Krivine λc-*calculus*from proof-terms built*in*a proof-irrelevant and*classical*version*of**the**calculus**of**constructions*with universes. ... Finally, we extend*the**extraction*scheme to a primitive data-type*of*natural numbers*in*a way which preserves*the*whole compatibility with*the**classical*realisability interpretation*of*second-order arithmetic ... Introduction*Program**extraction*has been a major concern from*the*early development*of**the**calculus**of**constructions*(CC) [3] to its more recent extensions [12, 16] implemented*in*proof assistants ...##
###
Towards the theory of programming in constructive logic
[chapter]

1990
*
Lecture Notes in Computer Science
*

We develop an approach to

doi:10.1007/3-540-52592-0_78
fatcat:wy7miwvy5jgmxdpzkw5mmhlcly
*the*theory*of**extracting**programs*from proofs based on*constructive*semantics*of**the*first order formulas called*consTrucTive*TruLb.*The*underlying ideas are discussed. ... Using this notion*of*truth we define an appropriate notion*of**constructive**calculus*. ...*the*main ideas and problems*The**extraction**of**programs*from proofs or*programming**in**constructive*logic is based on*the*idea that under some restrictions proofs can be considered as*programs*. ...##
###
Mechanically Verified Calculational Abstract Interpretation
[article]

2015
*
arXiv
*
pre-print

Our proof "by

arXiv:1507.03559v1
fatcat:a2ho4ehqyjaqrprr5nujpaj7ri
*calculus*" closely follows*the*original paper-and-pencil proof and supports*the**extraction**of*a verified static analyzer. ... Calculational abstract interpretation, long advocated by Cousot, is a technique for deriving correct-by-*construction*abstract interpreters from*the*formal semantics*of**programming*languages. ... Acknowledgments We gratefully acknowledge*the*Colony Club*in*Washington, DC for providing a fruitful environment*in*which to do this research. ...##
###
Extending the Calculus of Constructions with Tarski's fix-point theorem
[article]

2006
*
arXiv
*
pre-print

We propose to use Tarski's least fixpoint theorem as a basis to define recursive functions

arXiv:cs/0610055v1
fatcat:bvuamb63wvhknndm4zgvuxk6au
*in**the**calculus**of*inductive*constructions*. ... We claim that*the*extended framework makes it possible to reason about terminating and non-terminating computations and we show that common facilities*of**the**calculus**of*inductive*construction*, like*program*... Acknowledgments Benjamin Werner and Hugo Herbelin played a significant role*in*understanding what form*of**the*axioms*of**classical*logic provide safe extensions*of**the**calculus**of**constructions*. ...##
###
Page 5756 of Mathematical Reviews Vol. , Issue 95j
[page]

1995
*
Mathematical Reviews
*

*In*this respect, it stands

*in*contrast to other frameworks for

*program*

*extraction*, such as

*the*polymorphic lambda-

*calculus*(

*of*J. Girard) or

*the*

*calculus*

*of*

*constructions*(

*of*T. Coquand and G. Huet). ... (For

*the*newcomer to

*the*field

*of*

*program*

*extraction*, we remark that

*the*Curry-Howard correspondence does not work for

*classical*first-order logic.) ...

##
###
Programming and Proving with Classical Types
[chapter]

2017
*
Lecture Notes in Computer Science
*

We present example closed µML

doi:10.1007/978-3-319-71237-6_11
fatcat:2dz6qias4vbmpp3qggznclk3ee
*programs*with*classical*tautologies for types, including some inexpressible as closed*programs**in**the*original λµ-*calculus*, and some example tactic-driven µTP proofs*of**classical*...*In*this work, we use*the*λµ-*calculus*as*the*foundation for a system*of*proof terms for*classical*first-order logic. ... We thank*the*anonymous referees and Peter Sewell for their helpful comments. ...##
###
Proofs-as-Programs in Computable Analysis

2009
*
Electronic Communications of the EASST
*

Since

doi:10.14279/tuj.eceasst.23.332
dblp:journals/eceasst/Berger09
fatcat:nihg2u2vobbmhgxto2k7vzq7wy
*the*work*of*Brouwer, Kolmogorov, Goedel, Kleene and many others we know that*constructive*proofs have computational meaning. ...*In*Computer Science this idea is known as*the*"proofs-as-*programs*paradigm" or "Curry-Howard correspondence". ...*The*interesting aspect*of*this example is that*the**program*is*extracted*from*the**classical*i.e. non-*constructive*proof by Nash-Williams [NW63] using a version*of*Friedman's A-translation [Fri78] that ...##
###
Efficient Translation of Sequent Calculus Proofs Into Natural Deduction Proofs

2018
*
Conference on Automated Deduction
*

We present a simple and efficient translation

dblp:conf/cade/EbnerS18
fatcat:j46vomsdhrgp5j267md73sjcyq
*of**the**classical*multi-succedent sequent*calculus*LK to natural deduction. ... This translation aims to produce few excluded middle inferences, and to keep*the*quantifier complexity*of**the*instances*of*excluded middle low. ... To*extract**programs*from these proofs*in*LK, we want an effective translation to natural deduction*in*order to apply established*program**extraction*methods. ...##
###
Page 1071 of Mathematical Reviews Vol. , Issue 95b
[page]

1995
*
Mathematical Reviews
*

*The*paper treats

*the*

*extraction*

*of*

*programs*out

*of*a proof

*in*some

*constructive*type theory. ...

*The*specification

*of*a

*program*can be represented by a logical formula and

*the*

*program*itself can be

*extracted*from

*the*

*constructive*proof

*of*

*the*specification. ...

##
###
Extracting Herbrand trees from Coq
[article]

2013
*
arXiv
*
pre-print

*Classical*

*program*

*extraction*using Krivine's

*classical*realizability then translates this proof into a certified

*program*that computes Herbrand trees. ... Software certification aims at proving

*the*correctness

*of*

*programs*but

*in*many cases,

*the*use

*of*external libraries allows only a conditional proof: it depends on

*the*assumption that

*the*libraries meet ...

*the*

*calculus*

*of*

*constructions*with universes enriched with

*classical*reasoning at

*the*level

*of*

*the*sort Prop

*of*propositions. ...

##
###
Extracting information from intermediate semiconstructive HA-systems – extended abstract

2001
*
Mathematical Structures in Computer Science
*

Here we will concentrate our attention on semiconstructive calculi, which is a kind

doi:10.1017/s0960129501003358
fatcat:pqquh7r6lra2ngxsdhggbdrvka
*of**calculus*that is*of*interest*in**the*framework*of**program*synthesis and formal verification. ...*In*this abstract we will describe research*in*progress on*the*problem*of**extracting*information from proofs. ... ND HA ++ is a uniformly semiconstructive*calculus**in*ND PA . Hence HA ++ = Theo(ND HA ++ ) is a uniformly semiconstructive HA-system. ...##
###
Proof-search in type-theoretic languages: an introduction

2000
*
Theoretical Computer Science
*

We introduce

doi:10.1016/s0304-3975(99)00169-3
fatcat:gy4nmmsupfeype23pw3jzbrvwu
*the*main concepts and problems*in**the*theory*of*proof-search*in*type-theoretic languages and survey some speciÿc, connected topics. ... We do not claim to cover all*of**the*theoretical and implementation issues*in**the*study*of*proof-search*in*type-theoretic languages; rather, we present some key ideas and problems, starting from well-motivated ... Second-order systems like*the**Calculus**of**Constructions*(CC) [35, 127] or AF 2 [122] , issued from studies*of**the*system F [64] , can be used for*program*synthesis, i.e., to*extract**programs*from proofs ...##
###
Towards the animation of proofs – testing proofs by examples

2002
*
Theoretical Computer Science
*

*In*this paper, we introduce

*the*notion

*of*proof animation, which is a new application

*of*

*the*principle

*of*"Curry-Howard isomorphism" to formal proof development. ...

*The*motivation, aims, problems, and a prototype tool under development are presented

*in*this paper. We also discuss possibility

*of*"proof engineering" guided by

*the*Curry-Howard isomorphism. ... This paper is a sequel to

*the*ÿrst author's previous article [11] , which drew a negative conclusion about

*the*idea

*of*proofs as

*programs*as a methodology for formal methods. ...

##
###
Computational interpretations of classical reasoning: From the epsilon calculus to stateful programs
[article]

2018
*
arXiv
*
pre-print

*The*problem

*of*giving a computational meaning to

*classical*reasoning lies at

*the*heart

*of*logic. ... This article surveys three famous solutions to this problem -

*the*epsilon

*calculus*, modified realizability and

*the*dialectica interpretation - and re-examines them from a modern perspective, with a particular ... This point

*of*view has been explored primarily

*in*

*the*French style

*of*

*program*

*extraction*via e.g. ...

##
###
Page 5731 of Mathematical Reviews Vol. , Issue 2003h
[page]

2003
*
Mathematical Reviews
*

*The*author uses Kreisel’s modified realizability interpretation to

*extract*higher type primitive recur- sive

*programs*(

*in*

*the*sense

*of*Gédel’s

*calculus*7) for

*the*fast growing hierarchy up to €9. ... This paper provides an interesting case study

*in*a

*program*extrac- tion from a

*constructive*proof. ...

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