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Set Theory for Verification: II  Induction and Recursion
 Journal of Automated Reasoning
, 2000
"... A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning. ..."
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Cited by 43 (21 self)
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A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning.
Mechanizing Coinduction and Corecursion in Higherorder Logic
 Journal of Logic and Computation
, 1997
"... A theory of recursive and corecursive definitions has been developed in higherorder logic (HOL) and mechanized using Isabelle. Least fixedpoints express inductive data types such as strict lists; greatest fixedpoints express coinductive data types, such as lazy lists. Wellfounded recursion expresse ..."
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Cited by 41 (5 self)
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A theory of recursive and corecursive definitions has been developed in higherorder logic (HOL) and mechanized using Isabelle. Least fixedpoints express inductive data types such as strict lists; greatest fixedpoints express coinductive data types, such as lazy lists. Wellfounded recursion expresses recursive functions over inductive data types; corecursion expresses functions that yield elements of coinductive data types. The theory rests on a traditional formalization of infinite trees. The theory is intended for use in specification and verification. It supports reasoning about a wide range of computable functions, but it does not formalize their operational semantics and can express noncomputable functions also. The theory is illustrated using finite and infinite lists. Corecursion expresses functions over infinite lists; coinduction reasons about such functions. Key words. Isabelle, higherorder logic, coinduction, corecursion Copyright c fl 1996 by Lawrence C. Paulson Content...
XML, Stylesheets and the Remathematization of Formal Content
, 2001
"... An important part of the descriptive power of mathematics derives from its ability to represent formal concepts in a highly evolved, twodimensional system of symbolic notations. Tools for the mechanisation of mathematics and the automation of formal reasoning must eventually face the problem of re ..."
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Cited by 2 (2 self)
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An important part of the descriptive power of mathematics derives from its ability to represent formal concepts in a highly evolved, twodimensional system of symbolic notations. Tools for the mechanisation of mathematics and the automation of formal reasoning must eventually face the problem of remathematization of the logical, symbolic content of the information, especially in view of their integration with the World Wide Web. In a different work [APSS00c], we already discussed the pivotal role that XML [eXtensible Markup Language] technology [XML] is likely to play in such an integration. In this paper, we focus on the problem of (Web) publishing, advocating the use of XSL [eXtensible Stylesheet Language] Transformations, in conjunction with MathML [Mathematical Markup Language], as a standard, application independent and modular way for associating notation to formal mathematical content.
A Classical Realizability Model arising from a Stable Model of Untyped Lambda Calculus
, 2013
"... In [SR98] it has been shown that λcalculus with control can be interpreted in any domain D which is isomorphic to the domain of functions from D ω to the 2element (Sierpiński) lattice Σ. By a theorem of A. Pitts there exists a unique subset P of D such that f ∈ P iff f ( d) = ⊥ for all d ∈ P ω ..."
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In [SR98] it has been shown that λcalculus with control can be interpreted in any domain D which is isomorphic to the domain of functions from D ω to the 2element (Sierpiński) lattice Σ. By a theorem of A. Pitts there exists a unique subset P of D such that f ∈ P iff f ( d) = ⊥ for all d ∈ P ω. The domain D gives rise to a realizability structure in the sense of [Kri11] where the set of prooflike terms is given by P. When working in Scott domains the ensuing realizability model coincides with the ground model Set but when taking D within coherence spaces we obtain a classical realizability model of set theory different from any forcing model. We will show that this model validates countable and dependent choice since an appropriate form of bar recursion is available in stable domains. 1