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Representing Nested Inductive Types Using W-Types
[chapter]
2004
Lecture Notes in Computer Science
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Here we present a more general result which shows that nested inductive types can be constructed using only W-types and we analyse the categorical framework in more detail. ...
However, Dybjer (1997) does not consider any nesting of inductive types, e.g. the example FTree is not covered by his definition. ...
doi:10.1007/978-3-540-27836-8_8
fatcat:s4qrxvtlofaqfkhdeim4j3xhwq
Program Extraction from Nested Definitions
[chapter]
2013
Lecture Notes in Computer Science
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We report on extending Minlog to deal with predicates defined using a particular combination of induction and coinduction, via so-called nested definitions. ...
Related work Nested definitions are used by Ghani, Hancock and Pattinson [11, 12] to define uniformly continuous functions. ...
Danielsson and Altenkirch [10] study so-called mixed induction and coinduction, using Agda. ...
doi:10.1007/978-3-642-39634-2_27
fatcat:pd3cksivondwjfryiw36c4byiq
Nested General Recursion and Partiality in Type Theory
[chapter]
2001
Lecture Notes in Computer Science
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The technique applies also to the formalisation of partial functions as proper type-theoretic functions, rather than relations representing their graphs. ...
We extend Bove's technique for formalising simple general recursive algorithms in constructive type theory to nested recursive algorithms. ...
The basic type comprises sets and propositions and we call it Set. Both sets and propositions are inductively defined. A proposition is interpreted as a set whose elements represent its proofs. ...
doi:10.1007/3-540-44755-5_10
fatcat:pt6uozvufnhvnml6ghkzu7a2ji
Functional pearl: getting a quick fix on comonads
2015
SIGPLAN notices
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As a result, the well known loeb fixed point "cheats" by using functorial strength to implement its recursion. ...
Interpreting modal necessity as an arbitrary Functor (in Haskell), the "type" of Löb's theorem is inhabited by a fixed point function allowing each part of a structure to refer to the whole. ...
Introducing Inductive Instances The Nested type enables us to encapsulate inductive patterns in typeclass definitions for composed types. ...
doi:10.1145/2887747.2804310
fatcat:iuxkqhvxnvdgnp4q4j5dmeznd4
Proving termination of normalization functions for conditional expressions
1986
Journal of automated reasoning
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Three termination proofs are compared: (1) using a measure function, (2) in domain theory using LCF, (3) showing that its recursion relation, defined by the pattern of recursive calls, is well-founded. ...
A related function makes nested recursive calls. The three termination proofs become more complex: termination and correctness must be proved simultaneously. ...
Coquand builds a proof object resembling my Constructive Type theory one, using a similar Lemma. ...
doi:10.1007/bf00246023
fatcat:lqdumyub3bhsxingu5in4oaygy
Dependently Typed Folds for Nested Data Types
[article]
2018
arXiv
pre-print
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. (2) Dependently typed folds do not depend on maps, map functions and many terminating functions can be defined using dependently typed folds. (3) The induction principles for nested data types follow ...
We present an approach to develop folds for nested data types using dependent types. ...
CASE STUDY II: DE BRUIJN NOTATION AS THE NESTED DATA TYPE TERME In this section we consider representing the de Bruijn lambda terms using the following nested data type. ...
arXiv:1806.05230v1
fatcat:jua4dtgmz5g6ddnbi2tv2pfrqi
Nested Abstract Syntax in Coq
2010
Journal of automated reasoning
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We illustrate Nested Abstract Syntax as a high-level alternative representation of langages with binding constructs, based on nested datatypes. ...
Note that the fact that Coq handles such nested inductive types is pretty recent (version 8.1). ...
Furthermore Coq provides us with tactics dedicated to inductive types, notably − injection derives new equalities from old ones by applying the injectivity of the constructions of our inductive type. − ...
doi:10.1007/s10817-010-9207-9
fatcat:3qcjbs4tuzcvzcafo3fgvkmpq4
Combinatorial properties of Thompson's group F
[article]
2002
arXiv
pre-print
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We explore connections between the tree pair diagram representing an element w of F, its normal form in the infinite presentation, its word length, and minimal length representatives of it. ...
We estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates. ...
w = x −1 10 x −1 7 x −1 6 x −1 4 x −2 2 x −2 0 via the nested traversal method. It is easily checked that the final tree represents w. We begin with the base case of the induction, and w = x −1 k . ...
arXiv:math/0208117v1
fatcat:kdec4iog35hh5omkjanitywyya
Datalog extension for nested relations
1995
Computers and Mathematics with Applications
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Finally, we relate the expressive power of nonrecursive nested Datalog to the power of nested relational algebra and safe nested tuple relational calculus. ...
The notion of soundness, safety and consistency is defined to avoid undesirable nested Datalog programs. The evaluation of nested Datalog is given in terms of the nested relational algebra. ...
Let us suppose that the arguments are X1,..., Xn and the corresponding types are A1,..., An. Then let A be the following scheme: A = {(Xl,Ax),..., (Xn, An)}. (Ai denotes a representative from Ai.) ...
doi:10.1016/0898-1221(95)00174-w
fatcat:itb5yeimajbipkry23oscw5efa
Inductive Invariants for Nested Recursion
[chapter]
2003
Lecture Notes in Computer Science
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We demonstrate the usefulness of inductive invariants on a large example of the BDD algorithm Apply. ...
We show that certain input-output relations, termed inductive invariants are of central importance for termination proofs of algorithms defined by nested recursion. ...
We would like to thank the anonymous referees for their detailed and useful comments. ...
doi:10.1007/10930755_17
fatcat:q3ky7aeddrhlllfhxcmollf2yi
Nested Session Types
2022
ACM Transactions on Programming Languages and Systems
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In this article, we present the metatheory of session types extended with prenex polymorphism and, as a result, nested recursive datatypes. ...
We have implemented our ideas by extending the Rast programming language with nested session types. We conclude with several examples illustrating the expressivity of our enhanced type system. ...
Nested session types do not restrict communication to balanced represented words. ...
doi:10.1145/3539656
fatcat:52xplh2a7rd3vjxrs4oxlwok5y
Induce-statements and induce-expressions: Constructs for inductive programming
[chapter]
1993
Lecture Notes in Computer Science
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The example in this paper use the following (polymorphic) types. tree 2( ) = empty j node( tree 2( ) tree 2( )) list( ) = nil j :list( ) tree n( ) = empty j node( list(tree n( ))) natural = 0 j natural ...
A for-loop is somewhat similar to an inductive argument. ...
A n y function expressible in this language using induce-expressions nested at most k deep is said to be k F O inductive. ...
doi:10.1007/3-540-57529-4_62
fatcat:3gbejdm46bbsbigfvnervy27pe
Lattice-structured Domains, Imperfect Data and Inductive Queries
[chapter]
2000
Lecture Notes in Computer Science
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in attributes de ned over such domains, and demonstrates how lattices allow the accommodation of some forms of inductive queries. ...
While our model is applied to at relations, many of the results given are applicable also to nested relations. ...
Incomplete and imprecise information of the type that can be represented in concept and interval lattices can arise in many w ays, such a s b y summarising or sampling data, from space restrictions imposed ...
doi:10.1007/3-540-44469-6_62
fatcat:5etrdagghvb55hy5objwrdgske
Structural Recursion as a Query Language on Lists and Ordered Trees
2008
Theory of Computing Systems
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Note that every value v is naturally represented as a string str(v) over : the atom a is represented by itself; a tree tree(a, v) is represented by the string (str(a)str(v)); a pair (v, w) is represented ...
The Nested Tree Language Let us first introduce the nested tree language N T L, our ambient query language throughout the article. ...
An easy induction on linear N T A expressions then shows that linearity implies linear boundedness: Proposition 23 Every query definable by a linear N T L expression is linearly bounded. ...
doi:10.1007/s00224-008-9110-5
fatcat:3vc32pqpf5hkhdpsxj3fopd4jy
Associated types and constraint propagation for mainstream object-oriented generics
2005
SIGPLAN notices
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Constraint propagation allows certain constraints on type parameters to be inferred from other constraints on those parameters and their use in base class type expressions. ...
One particular design option, that of using subtyping to constrain the type parameters of generic functions, has been chosen for the generics extensions to Java and C#. ...
represent associated types match in their uses and definitions. ...
doi:10.1145/1103845.1094813
fatcat:vp6bj7tahbdgjegrk7ttx3rsey
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