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21
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 217 (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...
Some lambda calculus and type theory formalized
 Journal of Automated Reasoning
, 1999
"... Abstract. We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified, proof of standardization for beta reduction, that does not mention ..."
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Cited by 53 (7 self)
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Abstract. We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified, proof of standardization for beta reduction, that does not mention redex positions or residuals. Then we outline the meta theory of Pure Type Systems, leading to the strengthening lemma. One novelty is our use of named variables for the formalization. Along the way we point out what we feel has been learned about general issues of formalizing mathematics, emphasizing the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts.
More ChurchRosser Proofs (in Isabelle/HOL)
 Journal of Automated Reasoning
, 1996
"... The proofs of the ChurchRosser theorems for fi, j and fi [ j reduction in untyped calculus are formalized in Isabelle/HOL, an implementation of Higher Order Logic in the generic theorem prover Isabelle. ..."
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Cited by 39 (4 self)
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The proofs of the ChurchRosser theorems for fi, j and fi [ j reduction in untyped calculus are formalized in Isabelle/HOL, an implementation of Higher Order Logic in the generic theorem prover Isabelle.
Closure Under AlphaConversion
 In The Informal Proceeding of the 1993 Workshop on Types for Proofs and Programs
, 1993
"... this paper appears in Types for Proofs and Programs: International Workshop TYPES'93, Nijmegen, May 1993, Selected Papers, LNCS 806. abstraction, compute a type for its body in an extended context; to compute a type for an application, compute types for its left and right components, and check that ..."
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Cited by 24 (3 self)
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this paper appears in Types for Proofs and Programs: International Workshop TYPES'93, Nijmegen, May 1993, Selected Papers, LNCS 806. abstraction, compute a type for its body in an extended context; to compute a type for an application, compute types for its left and right components, and check that they match appropriately. Lets use the algorithm to compute a type for a = [x:ø ][x:oe]x. FAILURE: no rule applies because x 2 Dom (x:ø )
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
The ChurchRosser Theorem in Isabelle: A Proof Porting Experiment
, 1995
"... This paper describes a proof of the ChurchRosser theorem for the pure calculus formalised in the Isabelle theorem prover. The initial version of the proof is ported from a similar proof done in the Coq proof assistant by Gérard Huet, but a number of optimisations have been performed. The developme ..."
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Cited by 10 (0 self)
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This paper describes a proof of the ChurchRosser theorem for the pure calculus formalised in the Isabelle theorem prover. The initial version of the proof is ported from a similar proof done in the Coq proof assistant by Gérard Huet, but a number of optimisations have been performed. The development involves the introduction of several inductive and recursive definitions and thus gives a good presentation of the inductive package of Isabelle.
A Full Formalisation of πCalculus Theory in the Calculus of Constructions
, 1997
"... A formalisation of picalculus in the Coq system is presented. Based on a de Bruijn notation for names, our... ..."
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Cited by 8 (0 self)
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A formalisation of picalculus in the Coq system is presented. Based on a de Bruijn notation for names, our...
Upper Bounds for Standardizations and an Application
 The Journal of Symbolic Logic
, 1996
"... We first present a new proof for the standardization theorem, a fundamental theorem in calculus. Since our proof is largely built upon structural induction on lambda terms, we can extract some bounds for the number of fireduction steps in the standard fireduction sequences obtained from transfor ..."
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Cited by 7 (1 self)
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We first present a new proof for the standardization theorem, a fundamental theorem in calculus. Since our proof is largely built upon structural induction on lambda terms, we can extract some bounds for the number of fireduction steps in the standard fireduction sequences obtained from transforming any given fireduction sequences. This result sharpens the standardization theorem and establishes a link between lazy and eager evaluation orders in the context of computational complexity. As an application, we establish a superexponential bound for the number of fireduction steps in fireduction sequences from any given simply typed terms. 1 Introduction The standardization theorem of Curry and Feys [CF58] is a very useful result, stating that if u reduces to v for terms u and v, then there is a standard fireduction from u to v. Using this theorem, we can readily prove the normalization theorem, i.e., a term has a normal form if and only if the leftmost fireduction sequence f...
Polishing Up the TaitMartinLöf Proof of the ChurchRosser Theorem
, 1995
"... Introduction The TaitMartinLof proof is the best known and simplest proof of confluence (the ChurchRosser theorem) for various lambda calculi. It is explained in detail, for example, in [Bar84, HS86, Rev88]. The desire to clarify this proof has inspired work on concrete representation of bindi ..."
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Cited by 6 (0 self)
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Introduction The TaitMartinLof proof is the best known and simplest proof of confluence (the ChurchRosser theorem) for various lambda calculi. It is explained in detail, for example, in [Bar84, HS86, Rev88]. The desire to clarify this proof has inspired work on concrete representation of binding [dB72, Coq91]. Perhaps the best modern version is given in [Tak95]. Formal proofs are reported in [Hue94, MP93, Pfe92, Sha88] 1 . In this note I outline the innovation given in [Tak95] (and formalized by McKinna [MP93]), and present a further improvement which I believe has not appeared in the literature before. 1.1 Preliminary Definitions Let Rel2 be the class of binary relations, and R; T 2 Rel2 ; we write aRb for (a; b) 2 R . For R 2 Rel2 the transitive reflexive closure of R , wri
Formalizing Rewriting in the ACL2 Theorem Prover
"... We present an application of the ACL2 theorem prover to formalize and reason about rewrite systems theory. This can be seen as a first approach to apply formal methods, using ACL2, to the design of symbolic computation systems, since the notion of rewriting or simplification is ubiquitous in such sy ..."
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Cited by 6 (6 self)
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We present an application of the ACL2 theorem prover to formalize and reason about rewrite systems theory. This can be seen as a first approach to apply formal methods, using ACL2, to the design of symbolic computation systems, since the notion of rewriting or simplification is ubiquitous in such systems. We concentrate here in formalization and representation aspects of abstract reduction and term rewriting systems, using the firstorder, quantifierfree ACL2 logic based on Common Lisp.