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Iteration and Coiteration Schemes for HigherOrder and Nested Datatypes
, 2004
"... This article studies the implementation of inductive and coinductive constructors of higher kinds (higherorder nested datatypes) in typed term rewriting, with emphasis on the choice of the iteration and coiteration constructions to support as primitive. We propose and compare several wellbehaved e ..."
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Cited by 16 (5 self)
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This article studies the implementation of inductive and coinductive constructors of higher kinds (higherorder nested datatypes) in typed term rewriting, with emphasis on the choice of the iteration and coiteration constructions to support as primitive. We propose and compare several wellbehaved extensions of System with some form of iteration and coiteration uniform in all kinds. In what we call Mendlerstyle systems, the iterator and coiterator have a computational behavior similar to the general recursor, but their types guarantee termination. In conventionalstyle systems, monotonicity witnesses are used for a notion of monotonicity de ned uniformly for all kinds. Our most expressive systems GMIt of generalized Mendler resp. conventional (co)iteration encompass Martin, Gibbons and Bailey's ecient folds for rank2 inductive types. Strong normalization of all systems considered is proved by providing an embedding of the basic Mendlerstyle system MIt into System F .
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
Verification of the Redecoration Algorithm for Triangular Matrices
, 2007
"... Abstract. Triangular matrices with a dedicated type for the diagonal elements can be profitably represented by a nested datatype, i. e., a heterogeneous family of inductive datatypes. These families are fully supported since the version 8.1 of the Coq theorem proving environment, released in 2007. R ..."
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Cited by 2 (1 self)
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Abstract. Triangular matrices with a dedicated type for the diagonal elements can be profitably represented by a nested datatype, i. e., a heterogeneous family of inductive datatypes. These families are fully supported since the version 8.1 of the Coq theorem proving environment, released in 2007. Redecoration of triangular matrices has a succinct implementation in this representation, thus giving the challenge of proving it correct. This has been achieved within Coq, using also induction with measures. An axiomatic approach allowed a verification in the Isabelle theorem prover, giving insights about the differences of both systems. 1
Substitution in nonwellfounded . . .
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 82 NO. 1 (2003)
, 2003
"... Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable bin ..."
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Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable binding operators as well as operators of explicit substitution. This is done in an extensional mathematical setting of initial algebras and final coalgebras of endofunctors on a functor category. In the nonwellfounded case, the fundamental operation of substitution is more beneficially defined in terms of primitive corecursion than coiteration.
Substitution in Nonwellfounded Syntax with Variable Binding ⋆ Abstract
"... Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable bin ..."
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Inspired from the recent developments in theories of nonwellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and nonwellfounded terms is studied for a very general notion of signature permitting both simple variable binding operators as well as operators of explicit substitution. This is done in an extensional mathematical setting of initial algebras and final coalgebras of endofunctors on a functor category. The main technical tool is a novel concept of heterogeneous substitution systems.
Iteration and Coiteration Schemes for HigherOrder and Nested Datatypes
"... This article studies the implementation of inductive and coinductive constructors of higher kinds (higherorder nested datatypes) in typed term rewriting, with emphasis on the choice of the iteration and coiteration constructions to support as primitive. We propose and compare several wellbehaved e ..."
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This article studies the implementation of inductive and coinductive constructors of higher kinds (higherorder nested datatypes) in typed term rewriting, with emphasis on the choice of the iteration and coiteration constructions to support as primitive. We propose and compare several wellbehaved extensions of System F ω with some form of iteration and coiteration uniform in all kinds. In what we call Mendlerstyle systems, the iterator and coiterator have a computational behavior similar to the general recursor, but their types guarantee termination. In conventionalstyle systems, monotonicity witnesses are used for a notion of monotonicity defined uniformly for all kinds. Our most expressive systems GMIt ω and GIt ω of generalized Mendler resp. conventional (co)iteration encompass Martin, Gibbons and Bailey’s efficient folds for rank2 inductive types. Strong normalization of all systems considered is proved by providing an embedding of the basic Mendlerstyle system MIt ω into System F ω.
A Datastructure for Iterated Powers
, 2006
"... Abstract. Bushes are considered as the first example of a truly nested datatype, i. e., a family of datatypes indexed over all types where a constructor argument not only calls this family with a changing index but even with an index that involves the family itself. For the time being, no induction ..."
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Abstract. Bushes are considered as the first example of a truly nested datatype, i. e., a family of datatypes indexed over all types where a constructor argument not only calls this family with a changing index but even with an index that involves the family itself. For the time being, no induction principles for these datatypes are known. However, the author has introduced with Abel and Uustalu (TCS 333(12), pp. 366, 2005) iteration schemes that guarantee to define only terminating functions on those datatypes. The article uses a generalization of Bushes to nfold selfapplication and shows how to define elements of these types that have a number of data entries that is obtained by iterated raising to the power of n. Moreover, the data entries are just all the nbranching trees up to a certain height. The real question is how to extract this list of trees from that complicated data structure and to prove this extraction correct. Here, we use the “refined conventional iteration ” from the cited article for the extraction and describe a verification that has been formally verified inside Coq with its predicative notion of set. 1