## Realisability and adequacy for (co)induction

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@MISC{Berger_realisabilityand,

author = {Ulrich Berger},

title = {Realisability and adequacy for (co)induction},

year = {}

}

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### Abstract

Abstract. We prove the correctness of a formalised realisability interpretation of extensions of first-order theories by inductive and coinductive definitions in an untyped λ-calculus with fixed-points. We illustrate the use of this interpretation for program extraction by some simple examples in the area of exact real number computation, and hint at further non-trivial applications in computable analysis. 1

### Citations

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Citation Context ...∅) =⇒ d. The proof of the Adequacy Theorem is uses a technique that has been used for a similar purpose in [Win93] and [CS06]. It can be viewed as transformation of Plotkin’s Adequacy Theorem for PCF =-=[Plo77]-=- to the untyped setting. To carry out the proof, we first exploit the algebraicity of the domain D. Every element of D is the directed supremum of compact elements, which are generated at some finite ... |

227 | Une extension de l’interprétation de Gödel à l’analyse, et son application à l’elimination des coupures dans l’analyse et la théorie des types - Girard - 1971 |

95 |
Inductive types and type constraints in the second-order lambda calculus
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Citation Context ... Theorem relating the operational with a (domain-theoretic) denotational semantics. On the other hand, Miranda extracts typed terms and uses the more general “Mendler-style” (co)inductive definitions =-=[Men91]-=- which extract strongly normalising terms in extensions of the second-order polymorphic λ-calculus or stronger systems [Mat01,AMU05]. Furthermore, Tatsuta studies realisability with truth while we omi... |

67 |
W.: Iterated Inductive Definitions and Subsystems of Analysis
- Buchholz, Feferman, et al.
- 1981
(Show Context)
Citation Context ...inductive definitions which lead to this enormous strength. If one forbids these interleavings, one obtains the proof-theoretically much weaker system ID <ω of finitely iterated inductive definitions =-=[BFPS81]-=-. The realisability interpretation we are going to study is related to interpretations given by Tatsuta [Tat98] and Miranda-Perea [MP05]. We try to point out the main similarities and differences. Lik... |

47 | A realizability interpretation of the theory of species - Tait - 1975 |

28 | Modal mu-calculi
- Bradfield, Stirling
- 2007
(Show Context)
Citation Context ... maximised in the outer coinductive definition. In the context of classical propositional modal logic a system allowing similar “interleaved” least and largest fixed points is known as the µ-calculus =-=[BS07b]-=-. Möllerfeld [M¨03] studied the first-order version of the µ-calculus (which is essentially the classical version of our system) and showed that it has the same prooftheoretic strength as Π 1 2 -compr... |

16 | R.: Computing with real numbers: I. The LFT approach to real number computation; II. A domain framework for computational geometry - Edalat, Heckmann - 2002 |

13 | A proof of strong normalisation using domain theory
- Coquand, Spiwack
- 2006
(Show Context)
Citation Context ...rring Plotkin’s Adequacy Theorem for PCF [Plo77] to the untyped setting by using a a domaintheoretic variant of the reducibility or candidate method [Gir71,Tai75] introduced by Coquand and Spiwack in =-=[CS06]-=-. In loc. cit. and in [Ber08] the technique was applied to prove strong normalisation of λ-calculi with constants and rewrite rules. To carry out the proof, we first exploit the algebraicity of the do... |

12 | Iteration and coiteration schemes for higher-order and nested datatypes - Abel, Matthes, et al. - 2005 |

12 | Affine functions and series with co-inductive real numbers - Bertot - 2007 |

12 | Gianantonio, P.: A certified, corecursive implementation of exact real numbers - Ciaffaglione, Di - 2006 |

10 | Monotone inductive and coinductive constructors of rank 2 - Matthes - 2001 |

9 | D.: Continuous functions on final coalgebras
- Ghani, Hancock, et al.
- 2006
(Show Context)
Citation Context ...structive proof of C1(f) contains a program that implements f as a non-wellfounded tree acting as a (signed digit) stream transformer similar to the structures studied by Ghani, Hancock and Pattinson =-=[GHP06]-=-. More precisely, this interpretation is the computational content of a constructive proof the formula ∀f (C1(f) → ∀x (C0(x) → C0(f(x)))), which is a special case of a constructive composition theorem... |

9 | F.: Constructive analysis, types and exact real numbers - Geuvers, Niqui, et al. - 2007 |

7 | From coinductive proofs to exact real arithmetic
- Berger
- 2009
(Show Context)
Citation Context ...hich is a special case of a constructive composition theorem for analogous predicates Cn of n-ary functions. Details as well as concrete applications with extracted Haskell programs are worked out in =-=[Ber]-=-. Note that in the definition of C1, the inner inductive definition depends on the set parameter F which is then maximised in the outer coinductive definition. In the context of classical propositiona... |

6 | RZ: a tool for bringing constructive and computable mathematics closer to programming practice - Bauer, Stone - 2007 |

6 | Realizability interpretation of proofs in constructive analysis
- Schwichtenberg
- 2008
(Show Context)
Citation Context ...iomatically) without assuming them to be constructed in a particular way. The ignorance w.r.t. the first-order part can also be seen as a special case of the interpretations studied by Schwichtenberg =-=[Sch08]-=-, Hernest and Oliva [HO08] and Ratiu and Trifonov [RT09], which allow for a fine control of the amount of computational information extracted from proofs. 2 Induction and coinduction We fix a first-or... |

4 | P.: Hybrid functional interpretations
- Hernest
- 2008
(Show Context)
Citation Context ...ng them to be constructed in a particular way. The ignorance w.r.t. the first-order part can also be seen as a special case of the interpretations studied by Schwichtenberg [Sch08], Hernest and Oliva =-=[HO08]-=- and Ratiu and Trifonov [RT09], which allow for a fine control of the amount of computational information extracted from proofs. 2 Induction and coinduction We fix a first-order language L. Terms, r, ... |

4 |
Realizability for monotone clausular (co)inductive definitions
- Miranda-Perea
- 2005
(Show Context)
Citation Context ... particular way. The ignorance w.r.t. the first-order part can also be seen as a special case of the interpretations studied by Schwichtenberg [Sch08], Hernest and Oliva [HO08] and Ratiu and Trifonov =-=[RT09]-=-, which allow for a fine control of the amount of computational information extracted from proofs. 2 Induction and coinduction We fix a first-order language L. Terms, r, s, t . . ., are built from con... |

2 | A domain model characterising strong normalisation
- Berger
(Show Context)
Citation Context ...orem for PCF [Plo77] to the untyped setting by using a a domaintheoretic variant of the reducibility or candidate method [Gir71,Tai75] introduced by Coquand and Spiwack in [CS06]. In loc. cit. and in =-=[Ber08]-=- the technique was applied to prove strong normalisation of λ-calculi with constants and rewrite rules. To carry out the proof, we first exploit the algebraicity of the domain D. Every element of D is... |

1 |
Realizability interpretation of proofs in constructive
- Schwichtenberg
(Show Context)
Citation Context ...iomatically) without assuming them to be constructed in a particular way. The ignorance w.r.t. the first-order part can also be seen as a special case of the interpretations studied by Schwichtenberg =-=[Sch09]-=- and Hernest and Oliva [HO08] which allow for a fine control of the amount of computational information extracted from proofs. We state most of the results without proof. Full proofs will be given in ... |

1 |
Realizability of monotone coinductive definitions and its application to program synthesis
- Sys, appear
- 1998
(Show Context)
Citation Context ...provable. (b) If c =⇒ d, then c = d is provable. Theorem 2 (Adequacy). If [M ] = d, then (M, ∅) =⇒ d. The proof of the Adequacy Theorem is uses a technique that has been used for a similar purpose in =-=[Win93]-=- and [CS06]. It can be viewed as transformation of Plotkin’s Adequacy Theorem for PCF [Plo77] to the untyped setting. To carry out the proof, we first exploit the algebraicity of the domain D. Every e... |