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The Theory of LEGO  A Proof Checker for the Extended Calculus of Constructions
, 1994
"... LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO ..."
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Cited by 68 (10 self)
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LEGO is a computer program for interactive typechecking in the Extended Calculus of Constructions and two of its subsystems. LEGO also supports the extension of these three systems with inductive types. These type systems can be viewed as logics, and as meta languages for expressing logics, and LEGO is intended to be used for interactively constructing proofs in mathematical theories presented in these logics. I have developed LEGO over six years, starting from an implementation of the Calculus of Constructions by G erard Huet. LEGO has been used for problems at the limits of our abilities to do formal mathematics. In this thesis I explain some aspects of the metatheory of LEGO's type systems leading to a machinechecked proof that typechecking is decidable for all three type theories supported by LEGO, and to a verified algorithm for deciding their typing judgements, assuming only that they are normalizing. In order to do this, the theory of Pure Type Systems (PTS) is extended and f...
A proof of strong normalisation using domain theory
 In LICS’06
, 2006
"... U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show tha ..."
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Cited by 13 (1 self)
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U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show that, using ideas from the theory of intersection types [2, 6, 7, 21] and MartinLöf’s domain interpretation of type theory [18], we can in turn simplify U. Berger’s argument in the construction of such a domain model. We think that our domain model can be used to give modular proofs of strong normalization for various type theory. As an example, we show in some details how it can be used to prove strong normalization for MartinLöf dependent type theory extended with bar recursion, and with some form of proofirrelevance. 1
Proving Strong Normalization of CC by Modifying Realizability Semantics
 IN TYPES, VOLUME 806 OF LNCS
, 1994
"... ..."
A predicative strong normalisation proof for a λcalculus with interleaving inductive types
 TYPES FOR PROOF AND PROGRAMS, INTER40 A. ABEL AND T. ALTENKIRCH NATIONAL WORKSHOP, TYPES '99, SELECTED PAPERS. LECTURE NOTES IN COMPUTER SCIENCE
, 1999
"... We present a new strong normalisation proof for a λcalculus with interleaving strictly positive inductive types λ^µ which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metaleve ..."
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Cited by 8 (5 self)
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We present a new strong normalisation proof for a λcalculus with interleaving strictly positive inductive types λ^µ which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel. To achieve this we show that every strictly positive operator on types gives rise to an operator on saturated sets which is not only monotone but also (deterministically) set based  a concept introduced by Peter Aczel in the context of intuitionistic set theory. We also extend this to coinductive types using greatest fixpoints of strictly monotone
A Predicative Strong Normalisation Proof for a lambdaCalculus with Interleaving Inductive Types
, 2000
"... We present a new strong normalisation proof for a calculus with interleaving strictly positive inductive types which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel ..."
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Cited by 4 (0 self)
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We present a new strong normalisation proof for a calculus with interleaving strictly positive inductive types which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel. To achieve this we show that every strictly positive operator on types gives rise to an operator on saturated sets which is not only monotone but also (deterministically) set based  a concept introduced by Peter Aczel in the context of intuitionistic set theory. We also extend this to coinductive types using greatest fixpoints of strictly monotone operators on the metalevel. 1
Coding binding and substitution explicitly in isabelle
 University of Cambridge Computer Laboratory
, 1995
"... Logical frameworks provide powerful methods of encoding objectlogical binding and substitution using metalogical λabstraction and application. However, there are some cases in which these methods are not general enough: in such cases objectlogical binding and substitution must be explicitly code ..."
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Cited by 3 (0 self)
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Logical frameworks provide powerful methods of encoding objectlogical binding and substitution using metalogical λabstraction and application. However, there are some cases in which these methods are not general enough: in such cases objectlogical binding and substitution must be explicitly coded. McKinna and Pollack [MP93] give a novel formalization of binding, where they use it principally to prove metatheorems of Type Theory. We analyse the practical use of McKinnaPollack binding in Isabelle objectlogics, and illustrate its use with a simple example logic. 1
Lego and Related Work
, 1999
"... ence card):  \Pitypes, abstraction and applications: fx:AgB, A?B, [x:A]b, (f a).  Inductive types: macro Inductive with options such as Theorems, Relation, Inversion, Double, etc. For example (also see examples like the lessthan relation in exercises): Inductive [List : Type] Theorems P ..."
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ence card):  \Pitypes, abstraction and applications: fx:AgB, A?B, [x:A]b, (f a).  Inductive types: macro Inductive with options such as Theorems, Relation, Inversion, Double, etc. For example (also see examples like the lessthan relation in exercises): Inductive [List : Type] Theorems Parameters [A : Type] Constructors [nil : List] [cons : A?List?List]; Lecture notes for Types Summer School'99: Theory and Practice of Formal Proofs, Giens, France, 1999. 1  Predicative universes (with `typical ambiguity'): Type(i), Type.  Logical universe (impredicative, giving HOL): Prop.  Local definitions: [x=a]b.  Argument synthesis: fxA
A Predicative Strong Normalisation Proof for a
"... We present a new strong normalisation proof for a calculus with interleaving strictly positive inductive types which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel ..."
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We present a new strong normalisation proof for a calculus with interleaving strictly positive inductive types which avoids the use of impredicative reasoning, i.e., the theorem of KnasterTarski. Instead it only uses predicative, i.e., strictly positive inductive definitions on the metalevel. To achieve this we show that every strictly positive operator on types gives rise to an operator on saturated sets which is not only monotone but also (deterministically) set based  a concept introduced by Peter Aczel in the context of intuitionistic set theory. We also extend this to coinductive types using greatest fixpoints of strictly monotone operators on the metalevel.
Proving Correctness of Modular Functional Programs
"... and for Mum. I whacked the back of the driver’s seat with my fist. “This is important, goddamnit! This is a true story! ” The car swerved sickeningly, thenstraightenedout....Thekidinthebacklookedlikehewasready to jump right out of the car and take his chances. Our vibrations were getting nasty—but w ..."
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and for Mum. I whacked the back of the driver’s seat with my fist. “This is important, goddamnit! This is a true story! ” The car swerved sickeningly, thenstraightenedout....Thekidinthebacklookedlikehewasready to jump right out of the car and take his chances. Our vibrations were getting nasty—but why? I was puzzled, frustrated. Was there no communication in this car? Had we deteriorated to the level of dumb beasts? Because my story was true. I was certain of that. And it was extremely important, I felt, for the meaning of our journey to be made absolutelyclear....Andwhenthecallcame, I wasready. One reason for studying and programming in functional programming languages is that they are easy to reason about, yet there is surprisingly little work on proving the correctness of large functional programs. In this dissertation I show