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Weak Normalization for the SimplyTyped LambdaCalculus in Twelf (Extended Abstract)
 In Logical Frameworks and Metalanguages (LFM 04), IJCAR
, 2004
"... Andreas Abel Department of Computer Science, Chalmers University of Technology Rannvagen 6, SWE41296 Goteborg, Sweden Abstract. Weak normalization for the simplytyped calculus is proven in Twelf, an implementation of the Edinburgh Logical Framework. Since due to prooftheoretical restrict ..."
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Cited by 11 (3 self)
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Andreas Abel Department of Computer Science, Chalmers University of Technology Rannvagen 6, SWE41296 Goteborg, Sweden Abstract. Weak normalization for the simplytyped calculus is proven in Twelf, an implementation of the Edinburgh Logical Framework. Since due to prooftheoretical restrictions Twelf Tait's computability method does not seem to be directly usable, a combinatorical proof is adapted and formalized instead.
Fixed points of type constructors and primitive recursion
 Computer Science Logic, 18th International Workshop, CSL 2004, 13th Annual Conference of the EACSL, Karpacz, Poland, September 2024, 2004, Proceedings, volume 3210 of Lecture Notes in Computer Science
, 2004
"... Our contribution to CSL 04 [AM04] contains a little error, which is easily corrected by 2 elementary editing steps (replacing one character and deleting another). Definition of wellformed contexts (fifth page). Typing contexts should, in contrast to kinding contexts, only contain type variable decla ..."
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Cited by 7 (3 self)
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Our contribution to CSL 04 [AM04] contains a little error, which is easily corrected by 2 elementary editing steps (replacing one character and deleting another). Definition of wellformed contexts (fifth page). Typing contexts should, in contrast to kinding contexts, only contain type variable declarations without variance information. Hence, the second rule is too liberal; we must insist on p = ◦. The corrected set of rules is then: ⋄ cxt ∆ cxt ∆, X ◦κ cxt ∆ cxt ∆ ⊢ A: ∗ ∆, x:A cxt Definition of welltyped terms (immediately following). Since wellformed typing contexts ∆ contain no variance information, hence ◦ ∆ = ∆, we might drop the “◦ ” in the instantiation rule (fifth rule). The new set of rules is consequently, (x:A) ∈ ∆ ∆ cxt ∆ ⊢ x: A ∆, X ◦κ ⊢ t: A ∆ ⊢ t: ∀X κ. A ∆, x:A ⊢ t: B ∆ ⊢ λx.t: A → B ∆ ⊢ t: ∀X κ. A ∆ ⊢ F: κ
Strong normalization and equi(co)inductive types
 Proc. of the 8th Int. Conf. on Typed Lambda Calculi and Applications, TLCA 2007, volume 4583 of Lect. Notes in Comput. Sci. SpringerVerlag (2007), 8–22
"... Abstract. A type system for the lambdacalculus enriched with recursive and corecursive functions over equiinductive andcoinductive types is presented in which all welltyped programs are strongly normalizing. The choice of equiinductive types, instead of the more common isoinductive types, in ue ..."
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Cited by 3 (1 self)
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Abstract. A type system for the lambdacalculus enriched with recursive and corecursive functions over equiinductive andcoinductive types is presented in which all welltyped programs are strongly normalizing. The choice of equiinductive types, instead of the more common isoinductive types, in uences both reduction rules and the strong normalization proof. By embedding iso into equitypes, the latter ones are recognized as more fundamental. A model based on orthogonality is constructed where a semantical type corresponds to a set of observations, and soundness of the type system is proven. 1
On Zucker's isomorphism for LJ and its extension to Pure Type Systems
, 2003
"... It is shown how the sequent calculus LJ can be embedded into a simple extension of the calculus by generalized applications, called J. The reduction rules of cut elimination and normalization can be precisely correlated, if explicit substitutions are added to J. The resulting system J2 is prove ..."
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Cited by 2 (0 self)
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It is shown how the sequent calculus LJ can be embedded into a simple extension of the calculus by generalized applications, called J. The reduction rules of cut elimination and normalization can be precisely correlated, if explicit substitutions are added to J. The resulting system J2 is proved strongly normalizing, thus showing strong normalization for Gentzen's cut elimination steps. This re nes previous results by Zucker, Pottinger and Herbelin on the isomorphism between natural deduction and sequent calculus.
Acknowledgments
"... Leaning back and surveying this work, I am glad to admit that it is quite different from what I have anticipated when I started on it years ago, because this divergence reflects the influence of the people I have worked with. I owe much more than just my interest in λcalculi and natural deduction t ..."
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Leaning back and surveying this work, I am glad to admit that it is quite different from what I have anticipated when I started on it years ago, because this divergence reflects the influence of the people I have worked with. I owe much more than just my interest in λcalculi and natural deduction to my Doktorvater Prof. Dr. Helmut Schwichtenberg. It is his scientific impetus that inspires his working group and so many fruitful discussions; the openminded atmosphere that he creates is so valuable to all of us. In particular, I thank him for his confidence in my mathematical fidelity, and his stamina to endure the neverending evolution of my confluence and normalization proofs. I hope that some of his constructive understanding of logic is mirrored in my work. Ralph Matthes is probably the person to blame most for this thesis: Without him as kind of mathematical alter ego, all my ideas would have collapsed in a woeful heap of counterexamples and errors. The basic techniques that served as a starting point for this thesis have been conceived jointly in an intellectual symbiosis, that I count among the most exciting experiences of my life.
Polarized Subtyping for Sized Types
, 2006
"... We present an algorithm for deciding polarized higherorder subtyping without bounded quantification. Constructors are identified not only modulo β, but also η. We give a direct proof of completeness, without constructing a model or establishing a strong normalization theorem. Inductive and coinduct ..."
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We present an algorithm for deciding polarized higherorder subtyping without bounded quantification. Constructors are identified not only modulo β, but also η. We give a direct proof of completeness, without constructing a model or establishing a strong normalization theorem. Inductive and coinductive types are enriched with a notion of size and the subtyping calculus is extended to account for the arising inclusions between the sized types. 1.
Fixed Points of Type Operators and Primitive Recursion
, 2004
"... Abstract. For nested or heterogeneous datatypes, terminating recursion schemes considered so far have been instances of iteration, excluding efficient definitions of fixedpoint unfolding. Two solutions of this problem are proposed: The first one is a system with equirecursive nonstrictly positive ..."
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Abstract. For nested or heterogeneous datatypes, terminating recursion schemes considered so far have been instances of iteration, excluding efficient definitions of fixedpoint unfolding. Two solutions of this problem are proposed: The first one is a system with equirecursive nonstrictly positive type operators of arbitrary finite kinds, where fixedpoint unfolding is computationally invisible due to its treatment on the level of type equality. Positivity is ensured by a polarized kinding system, and strong normalization is proven by a model construction based on saturated sets. The second solution is a formulation of primitive recursion for arbitrary type constructors of any rank. Although without positivity restriction, the second system embeds—even operationally—into the first one. 1