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Inductive-Inductive Definitions
[chapter]

2010
*
Lecture Notes in Computer Science
*

Compared to indexed

doi:10.1007/978-3-642-15205-4_35
fatcat:hrog3qrvhfhkdkon3o3yqxvg3q
*inductive**definitions*, the novelty is that the index set A is generated*inductively*simultaneously with B. ... In this new principle a set A is defined*inductively*simultaneously with an A-indexed set B, which is also defined*inductively*. ... -*inductive**definitions*. ...##
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Partial inductive definitions

1991
*
Theoretical Computer Science
*

We will in a certain sense think of such data types as

doi:10.1016/s0304-3975(06)80007-1
fatcat:uusnzfuyifdrzekgl2ohqknz2a
*inductively*defined. A class of*inductive**definitions*will be interpreted as partial*definitions*: partial*inductive**definitions*. ... ., Partial*inductive**definitions*, Theoretical Computer Science 87 (1991) 115-142. An attempt to consider partial*definitions*of semantically oriented data types will be described. ... It is easy to see that*definitions*of level <~1 are all total. So ordinary monotone*inductive**definitions*are the canonical examples of total*inductive**definitions*. ...##
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Elaborating Inductive Definitions
[article]

2012
*
arXiv
*
pre-print

about

arXiv:1210.6390v2
fatcat:oe7zdyq4zffunbyqtvrpkwvgmu
*inductive**definitions*could be carried in the type theory, not in the meta-theory as it is usually the case. ... By elaborating an*inductive**definition*-- a syntactic artifact -- to its code -- its semantics -- we obtain an internalized account of*inductives*inside the type theory itself: we claim that reasoning ... Acknowledgements We would like to thank Pierre Boutillier for pointing us to the relevant literature on Coq's treatment of*inductives*. ...##
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A Categorical Semantics for Inductive-Inductive Definitions
[chapter]

2011
*
Lecture Notes in Computer Science
*

This covers many forms of

doi:10.1007/978-3-642-22944-2_6
fatcat:hn6wrqk65rfrxjj3kgs2ohqx2q
*induction*and data type*definitions*such as indexed*inductive**definitions*[5] and*induction*-recursion [7] (Dybjer and Setzer [8] also give a direct proof for*induction*-recursion ... This gives a new and compact formalisation of*inductive*-*inductive**definitions*, which we prove is equivalent to the usual formulation with elimination rules. ... The Generic Eliminator for an*Inductive*-*Inductive**Definition*Let us now generalise the preceding discussion from*inductive**definitions*(i.e. endofunctors on Set) to*inductive*-*inductive**definitions*(i.e ...##
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Positive Inductive-Recursive Definitions

2015
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Logical Methods in Computer Science
*

This theory, which we call positive

doi:10.2168/lmcs-11(1:13)2015
fatcat:2ubrqz3hh5fgxecftuxqctvife
*inductive*-recursive*definitions*, is a generalisation of Dybjer and Setzer's theory of*inductive*-recursive*definitions*within which C had to be discrete -- our work can ... Finally we justify the existence of positive*inductive*-recursive*definitions*by adapting Dybjer and Setzer's set-theoretic model to our setting. ...*Inductive*-recursive*definitions*In increasing complexity and sophistication,*inductive**definitions*, indexed*inductive**definitions*and*inductive*-recursive*definitions*encode more and more information about ...##
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Non-deterministic inductive definitions
[article]

2012
*
arXiv
*
pre-print

We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call "non-deterministic

arXiv:1104.2744v2
fatcat:ivqrqi6trvcnjg7vh7xhwottgi
*inductive**definitions*". ... Suppose R is a non-deterministic*inductive**definition*on a set X. If * is an element not in X, then R can also be considered as a non-deterministic*inductive**definition*on X ∪ { * }. ... Then consider the following deterministic*inductive**definition*on M: {t(b) : b ∈ B a } {sup a t} a ∈ A, t: B a → M By Proposition 3.6, this*inductive**definition*has a least fixed point. ...##
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Non-deterministic inductive definitions

2012
*
Archive for Mathematical Logic
*

We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call "non-deterministic

doi:10.1007/s00153-012-0309-4
fatcat:x23a2m2rerhg3are4ggksstztu
*inductive**definitions*". ... Suppose R is a non-deterministic*inductive**definition*on a set X. If * is an element not in X, then R can also be considered as a non-deterministic*inductive**definition*on X ∪ { * }. ... Then consider the following deterministic*inductive**definition*on M: {t(b) : b ∈ B a } {sup a t} a ∈ A, t: B a → M By Proposition 3.6, this*inductive**definition*has a least fixed point. ...##
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Inductive Definitions and Computability

1971
*
Transactions of the American Mathematical Society
*

Sets

doi:10.2307/1995907
fatcat:6zzauhczcfglhhxi2hmfg65pxi
*inductively*defined with respect to Il0, Sx, (nonmonotonic) II t and S2 predicates are characterized in terms of the four chief notions of abstract recursion. ... Add to the*inductive**definition*of S the following clauses : Theorem 3. ... These problems are handled by adding some auxiliary clauses to the*inductive**definition*ofS. Theorem 2. ...##
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Positive Inductive-Recursive Definitions
[chapter]

2013
*
Lecture Notes in Computer Science
*

This theory, which we call positive

doi:10.1007/978-3-642-40206-7_3
fatcat:bmbq64zvbzdzjaj45xl5fyfbje
*inductive*-recursive*definitions*, is a generalisation of Dybjer and Setzer's theory of*inductive*-recursive*definitions*within which C had to be discrete -our work can ... We then give an application of positive*inductive*-recursive*definitions*to the theory of nested data types. ... This leads us to consider a new variation IR + of*inductive*-recursive*definitions*which we call positive*inductive*-recursive*definitions*. ...##
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Functional interpretation and inductive definitions
[article]

2009
*
arXiv
*
pre-print

Extending Gödel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic

arXiv:0802.1938v4
fatcat:yxwafhc5rze7zcijmmxaidtbx4
*inductive**definitions*, reducing them to theories of finite-type functionals ... Iterating the interpretation In this section, we consider theories ID n of finitely iterated*inductive**definitions*. ... The interpretation can be further extended to theories of transfinitely iterated*inductive**definitions*, as described in [9] . ...##
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Structuring Metatheory on Inductive Definitions

2000
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Information and Computation
*

We show how metatheories based on

doi:10.1006/inco.2000.2858
fatcat:am3mm6zv2vegjf4syvnx2c7ifu
*inductive**definitions*allow theories and general metatheorems to be organized this way and report on a case study using the theory FS 0 . ] ...*INDUCTIVE**DEFINITIONS*AND THE THEORY FS 0 We use Feferman's FS 0 for our work since it is a particularly simple minimal theory of*inductive**definitions*. ... Other Metatheories Supporting*Inductive**Definitions*We have indicated how parameterized*inductive**definitions*in FS 0 support structured metatheoretic development. ...##
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Inductive definitions over finite structures

1990
*
Information and Computation
*

INTRODUCTION

doi:10.1016/0890-5401(90)90006-4
fatcat:s4rvi2bbwfd7hin4ku4jldgl7y
*Inductive**definitions*have played a central role in the foundations of mathematics for over a century. ... The*inductive*closure, F", of an arbitrary operator F can be defined as the union of the increasing chain F<') = Df uj< i F[F("'], If F is*inductive*, then FCil = F<'> (by*induction*on i), so this*definition*...##
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Elaborating Inductive Definitions and Course-of-Values Induction in Cedille
[article]

2019
*
arXiv
*
pre-print

This datatype subsystem and elaborator are implemented in Cedille, establishing for the first time a complete translation of

arXiv:1903.08233v3
fatcat:rgzfv32azfcwfmhrfi5kv5mwbe
*inductive**definitions*to a small pure typed λ-calculus. ... In the Calculus of Dependent Lambda Eliminations (CDLE), a pure Curry-style type theory, it is possible to generically λ-encode*inductive*datatypes which support course-of-values (CoV)*induction*. ...*Inductive*-recursive and*inductive*-*inductive**definitions*are not part of this proposal. ...##
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Induction and inductive definitions in fragments of second order arithmetic

2005
*
Journal of Symbolic Logic (JSL)
*

The amount of

doi:10.2178/jsl/1129642116
fatcat:qs6hrkuxlvc2hamlvy6foqdwfa
*induction*on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding*induction*for arbitrary formulae increases the strength by one*inductive**definition*... A fragment with the same provably recursive functions as n iterated*inductive**definitions*is obtained by restricting second order arithmetic in the following way. ...*Induction*along the*inductive**Definition*5.2 of d n,(K) Γ = ∀X n A(X n ). ...##
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Proof-theoretic strengths of weak theories for positive inductive definitions
[article]

2018
*
arXiv
*
pre-print

*definitions*over natural numbers. ... Conjunctions of negative and positive formulas in the transfinite

*induction*axiom of ID_1 are shown to be weak, and disjunctions are strong. ...

*Inductive*

*definition*of H ⊢ a d Γ. Let Γ be a sequent, a < Ω · ε 0 and d ≤ 3. ...

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