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Inductive-Inductive Definitions
[chapter]
2010
Lecture Notes in Computer Science
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Compared to indexed 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. ...
doi:10.1007/978-3-642-15205-4_35
fatcat:hrog3qrvhfhkdkon3o3yqxvg3q
Partial inductive definitions
1991
Theoretical Computer Science
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We will in a certain sense think of such data types as 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. ...
doi:10.1016/s0304-3975(06)80007-1
fatcat:uusnzfuyifdrzekgl2ohqknz2a
Elaborating Inductive Definitions
[article]
2012
arXiv
pre-print
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about 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. ...
arXiv:1210.6390v2
fatcat:oe7zdyq4zffunbyqtvrpkwvgmu
A Categorical Semantics for Inductive-Inductive Definitions
[chapter]
2011
Lecture Notes in Computer Science
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This covers many forms of 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 ...
doi:10.1007/978-3-642-22944-2_6
fatcat:hn6wrqk65rfrxjj3kgs2ohqx2q
Positive Inductive-Recursive Definitions
2015
Logical Methods in Computer Science
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This theory, which we call positive 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 ...
doi:10.2168/lmcs-11(1:13)2015
fatcat:2ubrqz3hh5fgxecftuxqctvife
Non-deterministic inductive definitions
[article]
2012
arXiv
pre-print
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We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call "non-deterministic 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. ...
arXiv:1104.2744v2
fatcat:ivqrqi6trvcnjg7vh7xhwottgi
Non-deterministic inductive definitions
2012
Archive for Mathematical Logic
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We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call "non-deterministic 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. ...
doi:10.1007/s00153-012-0309-4
fatcat:x23a2m2rerhg3are4ggksstztu
Inductive Definitions and Computability
1971
Transactions of the American Mathematical Society
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Sets 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. ...
doi:10.2307/1995907
fatcat:6zzauhczcfglhhxi2hmfg65pxi
Positive Inductive-Recursive Definitions
[chapter]
2013
Lecture Notes in Computer Science
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This theory, which we call positive 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. ...
doi:10.1007/978-3-642-40206-7_3
fatcat:bmbq64zvbzdzjaj45xl5fyfbje
Functional interpretation and inductive definitions
[article]
2009
arXiv
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Extending Gödel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic 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] . ...
arXiv:0802.1938v4
fatcat:yxwafhc5rze7zcijmmxaidtbx4
Structuring Metatheory on Inductive Definitions
2000
Information and Computation
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We show how metatheories based on 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. ...
doi:10.1006/inco.2000.2858
fatcat:am3mm6zv2vegjf4syvnx2c7ifu
Inductive definitions over finite structures
1990
Information and Computation
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INTRODUCTION 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 ...
doi:10.1016/0890-5401(90)90006-4
fatcat:s4rvi2bbwfd7hin4ku4jldgl7y
Elaborating Inductive Definitions and Course-of-Values Induction in Cedille
[article]
2019
arXiv
pre-print
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This datatype subsystem and elaborator are implemented in Cedille, establishing for the first time a complete translation of 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. ...
arXiv:1903.08233v3
fatcat:rgzfv32azfcwfmhrfi5kv5mwbe
Induction and inductive definitions in fragments of second order arithmetic
2005
Journal of Symbolic Logic (JSL)
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The amount of 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 ). ...
doi:10.2178/jsl/1129642116
fatcat:qs6hrkuxlvc2hamlvy6foqdwfa
Proof-theoretic strengths of weak theories for positive inductive definitions
[article]
2018
arXiv
pre-print
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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. ...
arXiv:1603.01342v7
fatcat:sddiwn2iunenlje3fjcsfeqmzi
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