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A Linear Logical Framework
, 1996
"... We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF c ..."
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Cited by 216 (44 self)
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We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of MiniML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cutelimination. 1 Introduction A logical framework is a formal system desig...
Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherord ..."
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Cited by 45 (17 self)
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,
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 42 (20 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 set theory: Cardinal arithmetic and the axiom of choice
 Journal of Automated Reasoning
, 1996
"... Abstract. Fairly deep results of ZermeloFrænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ ⊗ κ = κ, where κ is any infinite cardinal. Proving this resu ..."
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Cited by 16 (9 self)
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Abstract. Fairly deep results of ZermeloFrænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplication is κ ⊗ κ = κ, where κ is any infinite cardinal. Proving this result required developing theories of orders, orderisomorphisms, order types, ordinal arithmetic, cardinals, etc.; this covers most of Kunen, Set Theory, Chapter I. Furthermore, we have proved the equivalence of 7 formulations of the Wellordering Theorem and 20 formulations of AC; this covers the first two chapters of Rubin and Rubin, Equivalents of the Axiom of Choice, and involves highly technical material. The definitions used in the proofs are
An Equational ReEngineering of Set Theories
 Automated Deduction in Classical and NonClassical Logics, LNCS 1761 (LNAI
, 1998
"... New successes in dealing with set theories by means of stateoftheart theoremprovers may ensue from terse and concise axiomatizations, such as can be moulded in the framework of the (fully equational) TarskiGivant map calculus. In this paper we carry out this task in detail, setting the ground fo ..."
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Cited by 6 (6 self)
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New successes in dealing with set theories by means of stateoftheart theoremprovers may ensue from terse and concise axiomatizations, such as can be moulded in the framework of the (fully equational) TarskiGivant map calculus. In this paper we carry out this task in detail, setting the ground for a number of experiments. Key words: Set theory, relation algebras, firstorder theoremproving, algebraic logic. 1 Introduction Like other mature fields of mathematics, Set Theory deserves sustained efforts that bring to light richer and richer decidable fragments of it [5], general inference rules for reasoning in it [23, 2], effective proof strategies based on its domainknowledge, and so forth. Advances in this specialized area of automated reasoning tend, in spite of their steadiness, to be slow compared to the overall progress in the field. Many experiments with set theories have hence been carried out with standard theoremproving systems. Still today such experiments pose consider...
Map calculus: Initial application scenarios and experiments based on Otter
, 1998
"... Properties of a few familiar structures (natural numbers, nested lists, lattices) are formally specified in TarskiGivant's map calculus, with the aim of bringing to light new translation techniques that may bridge the gap between firstorder predicate calculus and the map calculus. It is also highl ..."
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Cited by 6 (6 self)
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Properties of a few familiar structures (natural numbers, nested lists, lattices) are formally specified in TarskiGivant's map calculus, with the aim of bringing to light new translation techniques that may bridge the gap between firstorder predicate calculus and the map calculus. It is also highlighted to what extent a stateoftheart theoremprover for firstorder logic, namely Otter, can be exploited not only to emulate, but also to reason about, map calculus. 3 1 Introduction Everybody remembers that Boole's Laws of thought (1854), Frege's Begriffsschrift (1879), and the WhiteheadRussell's Principia Mathematica (1910) have been three major milestones in the development of contemporary logic (cf. [3, 8, 15, 4]). Only a few people are aware that very important prePrincipia milestones were laid down by C.S. Peirce and E. Schroder and culminated in the monumental work [11, 12] on the Algebra der Logik . The "rather capricious line of historical development" of the algebraic for...
A Ruby Proof System
, 1996
"... floorplan SYNTHESIS SYSTEM LIBRARY CELL REWRITE SYSTEM Proof obligations ... ENTITY conv264 IS END ruby; ... ARCHITECTURE ruby OF... BEGIN END conv264; ( ... ); PORT b= ? 1 3 0... Dynamic behaviour a= 1 3 0 7... SIMULATOR Data flow CAUSALITY ANALYSIS PROVED EQUIVALENCES PARSER/PRINTER TYPE CHEC ..."
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Cited by 4 (2 self)
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floorplan SYNTHESIS SYSTEM LIBRARY CELL REWRITE SYSTEM Proof obligations ... ENTITY conv264 IS END ruby; ... ARCHITECTURE ruby OF... BEGIN END conv264; ( ... ); PORT b= ? 1 3 0... Dynamic behaviour a= 1 3 0 7... SIMULATOR Data flow CAUSALITY ANALYSIS PROVED EQUIVALENCES PARSER/PRINTER TYPE CHECKER (internal representation) RUBY terms PROOF SYSTEM RubyZF TRuby System RUBYTOVHDL TRANSLATOR RUBY expressions RUBY expressions Figure 1.1: The TRuby design system (after [SR95a]) is regarded as clumsy and not very well suited for doing automated proofs but with the extensive work of e.g. Larry Paulson, it has become possible to use ZF in connection with more practical reasoning. This means that from a Ruby point of view ZF is natural and from a ZF point of view Ruby is feasible. We show that even though we make a shallow embedding of Ruby within ZF, as opposed to a deep embedding 1 , we are able to prove an induction theorem enabling us to perform proofs by structural induction o...
Computer Proofs about Transitive Closure
 in International Joint Conference on Automated Reasoning, IJCAR2001 Short Papers
, 2001
"... As a part of ongoing research on automated reasoning in set theory, we focus here on an example of a computer proof that involves a recursive definition. ..."
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Cited by 2 (1 self)
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As a part of ongoing research on automated reasoning in set theory, we focus here on an example of a computer proof that involves a recursive definition.