## Inductive Data Type Systems (1997)

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Venue: | THEORETICAL COMPUTER SCIENCE |

Citations: | 44 - 10 self |

### BibTeX

@ARTICLE{Blanqui97inductivedata,

author = {Frédéric Blanqui and Jean-pierre Jouannaud and Mitsuhiro Okada},

title = {Inductive Data Type Systems},

journal = {THEORETICAL COMPUTER SCIENCE},

year = {1997},

pages = {349--391}

}

### OpenURL

### Abstract

In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λ-calculus enriched by pattern-matching definitions following a certain format, called the “General Schema”, which generalizes the usual recursor definitions for natural numbers and similar “basic inductive types”. This combined language was shown to be strongly normalizing. The purpose of this paper is to reformulate and extend the General Schema in order to make it easily extensible, to capture a more general class of inductive types, called “strictly positive”, and to ease the strong normalization proof of the resulting system. This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive types.

### Citations

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Citation Context ... uniform formalism with a strong rewriting flavor. In the sequel, we assume the reader familiar with the notions of l-calculus and term rewriting, as presented in [4] for the simply-typed l-calculus, =-=[16]-=- for term rewriting and [31,39,49] for the several variants of higher-order rewriting existing in the literature. We first introduce the term language before to move on with the definition of higher-o... |

518 | Lambda calculi with types
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Citation Context ...ar case of the latter, resulting in a uniform formalism with a strong rewriting flavor. In the sequel, we assume the reader familiar with the notions of l-calculus and term rewriting, as presented in =-=[4]-=- for the simply-typed l-calculus, [16] for term rewriting and [31,39,49] for the several variants of higher-order rewriting existing in the literature. We first introduce the term language before to m... |

352 |
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Citation Context ...mitted to Elsevier Preprint 1 February 2008In retrospect, the quest for an expressive language allowing to specify and prove mathematical properties of software started with system F on the one hand =-=[23,24]-=- and the Automath project on the other hand [15]. Much later, Coquand and Huet combined both calculi, resulting in the Calculus of Constructions [13]. Making use of impredicativity, data structures co... |

342 |
Intuitionistic type theory
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Citation Context .... Making use of impredicativity, data structures could be encoded in this calculus, but these encodings were far too complex to be used by nonspecialists. A different approach was taken by Martin-Löf =-=[32,33]-=-, whose theory was based on the notion of inductive definition, originating in Gödel’s system T [25]. Coquand and Paulin-Möhring later incorporated a similar notion to the Calculus of Constructions un... |

287 | A logic programming language with lambda-abstraction, function variables, and simple unification
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Citation Context ...ive types is however open. 4.6 Matching modulo βη In this section, we address the case of higher-order rewrite rules à la Nipkow [39], based on higher-order pattern-matching with patterns à la Miller =-=[37]-=-. We give here several examples taken from [39], [48] or [42], and recall why plain pattern-matching does not really make sense for them. On the other hand, we will see that all these examples follow ... |

285 |
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Citation Context ...mitted to Elsevier Preprint 1 February 2008In retrospect, the quest for an expressive language allowing to specify and prove mathematical properties of software started with system F on the one hand =-=[23,24]-=- and the Automath project on the other hand [15]. Much later, Coquand and Huet combined both calculi, resulting in the Calculus of Constructions [13]. Making use of impredicativity, data structures co... |

245 |
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Citation Context ...n θ. Matching here is syntactic, that is, u is α-convertible to the instance of l. In contrast, the more sophisticated notions of higher-order rewriting defined by Klop (Combinatory Reduction Systems =-=[30,31]-=-), Nipkow (Higher-order Rewrite Systems [39,34]) and van Raamsdonk and van Oostrom (Higher-Order Rewriting Systems [49,50], generalizing both) are based on higher-order pattern-matching, that is, u mu... |

236 |
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Citation Context ...orm. In this case, the initial algebra of the inductive type is equivalent to its normal form algebra and the latter can be represented by the accepting states of a finite tree automaton of some form =-=[7,11]-=-. The important property of this automaton is that the set of terms recognized at every accepting state is recursive and the predicate of this state is actually easy to define. We show the constructio... |

229 |
Une extension de l’interprétation de Gödel à l’analyse et son application à l’élimination de coupures dans l’analyse et la théorie des types
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Citation Context .... Here, computability refers to Tait’s computability predicate method for proving the termination of the simply-typed l-calculus [46], which was later extended by Girard to the polymorphic l-calculus =-=[22,24]-=-. To explain our construction, we need to recall the basics of Tait’s method. The starting observation is that it is not possible to prove the termination of β-reduction directly by induction on the s... |

213 |
Intensional Interpretations of Functionals of Finite Type
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- 1967
(Show Context)
Citation Context ...her generalizations of the General Schema allowing for dependent and polymorphic inductive types. The strong normalization proof of our new calculus is based on Tait’s computability predicates method =-=[46,24]-=-. In contrast with [28], the whole structure of the proof is made quite modular thanks to a novel formulation of our new version of the General Schema. Here, given a left-hand side f( l), we define ... |

157 |
An intuitionistic theory of types: Predicative part
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Citation Context .... Making use of impredicativity, data structures could be encoded in this calculus, but these encodings were far too complex to be used by nonspecialists. A different approach was taken by Martin-Löf =-=[32,33]-=-, whose theory was based on the notion of inductive definition, originating in Gödel’s system T [25]. Coquand and Paulin-Möhring later incorporated a similar notion to the Calculus of Constructions un... |

124 |
Higher-order critical pairs
- Nipkow
- 1991
(Show Context)
Citation Context ...rong rewriting flavor. In the sequel, we assume the reader familiar with the notions of l-calculus and term rewriting, as presented in [4] for the simply-typed l-calculus, [16] for term rewriting and =-=[31,39,49]-=- for the several variants of higher-order rewriting existing in the literature. We first introduce the term language before to move on with the definition of higher-order rewrite rules and of the new ... |

109 | Irlductively defined types - Coquand, Paulin - 1990 |

104 | Specification and proof in membership equational logic
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Citation Context ...orm. In this case, the initial algebra of the inductive type is equivalent to its normal form algebra and the latter can be represented by the accepting states of a finite tree automaton of some form =-=[7,11]-=-. The important property of this automaton is that the set of terms recognized at every accepting state is recursive and the predicate of this state is actually easy to define. We show the constructio... |

91 |
The mathematical language AUTOMATH, its usage, and some of its extensions
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Citation Context ...ospect, the quest for an expressive language allowing to specify and prove mathematical properties of software started with system F on the one hand [23,24] and the Automath project on the other hand =-=[15]-=-. Much later, Coquand and Huet combined both calculi, resulting in the Calculus of Constructions [13]. Making use of impredicativity, data structures could be encoded in this calculus, but these encod... |

85 | Pattern matching with dependent types
- Coquand
- 1992
(Show Context)
Citation Context ...oquand, for a calculus with dependent types, in which functions can be defined by pattern-matching, provided all righthand side recursive calls are “structurally smaller” than the left-hand side call =-=[12]-=-. His notion is very abstract, though, and relies on a well-foundedness assumption which is satisfied in practice. Concurrently, following the pioneering works of Tannen [8], Tannen and Gallier [9,10]... |

85 |
Counterexamples to termination for the direct sum of term rewriting systems
- Toyama
- 1987
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Citation Context ...d are non-duplicating (ie. no free variable occurs more often in the right-hand side than in the left-hand side), a condition needed to avoid Toyama’s counter-example to the modularity of termination =-=[47]-=-. It is of course possible to do the same here, using Lemma 24 of [28], an analog of Lemma 18 for first-order functions symbols. Below, we give an example which cannot be proved to terminate by our me... |

84 | Raamsdonk: Combinatory reduction systems: Introduction and survey
- Klop, Oostrom, et al.
- 1993
(Show Context)
Citation Context ...rong rewriting flavor. In the sequel, we assume the reader familiar with the notions of l-calculus and term rewriting, as presented in [4] for the simply-typed l-calculus, [16] for term rewriting and =-=[31,39,49]-=- for the several variants of higher-order rewriting existing in the literature. We first introduce the term language before to move on with the definition of higher-order rewrite rules and of the new ... |

76 | Theorem proving modulo - Dowek, Hardin, et al. |

74 |
Confluence for Abstract and Higher-Order Rewriting
- Oostrom
- 1994
(Show Context)
Citation Context ...rong rewriting flavor. In the sequel, we assume the reader familiar with the notions of l-calculus and term rewriting, as presented in [4] for the simply-typed l-calculus, [16] for term rewriting and =-=[31,39,49]-=- for the several variants of higher-order rewriting existing in the literature. We first introduce the term language before to move on with the definition of higher-order rewrite rules and of the new ... |

71 |
Codifying guarded definitions with recursive schemes
- Giménez
- 1994
(Show Context)
Citation Context ...actical needs, making it possible to have nested recursive calls, an important facility that Coquand’s ordering cannot provide with. Finally, it is important to note that, in contrast with other work =-=[35,20]-=-, our definitions allow nonlinear and overlapping left-hand sides, to the price of checking confluence via the computation of critical pairs. The fact that the General Schema covers only a limited por... |

65 |
Recursive Definition in Type Theory
- Mendler
- 1987
(Show Context)
Citation Context ...ve types is the largest class that one can consider within the framework of the simply-typed l-calculus, since any non-positive type is inhabited by non-terminating well-typed terms in this framework =-=[36]-=-. In this paper, we restrict ourselves to strictly positive inductive types, as in the Calculus of Inductive Constructions [51], and prove the strong normalization property of our calculus under this ... |

62 | Higher-order rewrite systems and their confluence
- Mayr, Nipkow
- 1998
(Show Context)
Citation Context ...α-convertible to the instance of l. In contrast, the more sophisticated notions of higher-order rewriting defined by Klop (Combinatory Reduction Systems [30,31]), Nipkow (Higher-order Rewrite Systems =-=[39,34]-=-) and van Raamsdonk and van Oostrom (Higher-Order Rewriting Systems [49,50], generalizing both) are based on higher-order pattern-matching, that is, u must be βηα-convertible to the instance of l. Def... |

58 |
Une Théorie des Constructions Inductives
- Werner
- 1994
(Show Context)
Citation Context ... type is inhabited by non-terminating well-typed terms in this framework [36]. In this paper, we restrict ourselves to strictly positive inductive types, as in the Calculus of Inductive Constructions =-=[51]-=-, and prove the strong normalization property of our calculus under this assumption. However, we conjecture that strong normalization holds in the non-strictly positive case too. Definition 4 (Functio... |

57 |
Combining Algebra and Higher-order Types
- Breazu-Tannen
- 1988
(Show Context)
Citation Context ...n the left-hand side call [12]. His notion is very abstract, though, and relies on a well-foundedness assumption which is satisfied in practice. Concurrently, following the pioneering works of Tannen =-=[8]-=-, Tannen and Gallier [9,10] and Okada [40], the last two authors of the present paper proposed another solution, for a polymorphically typed l-calculus, based on pattern-matching functional definition... |

52 |
Polymorphic Rewriting Conserves Algebraic Strong Normalization and Confluence
- Gallier, Breazu-Tannen
- 1989
(Show Context)
Citation Context ...l [12]. His notion is very abstract, though, and relies on a well-foundedness assumption which is satisfied in practice. Concurrently, following the pioneering works of Tannen [8], Tannen and Gallier =-=[9,10]-=- and Okada [40], the last two authors of the present paper proposed another solution, for a polymorphically typed l-calculus, based on pattern-matching functional definitions following the so-called “... |

45 | Proof Normalization Modulo - Dowek, Werner - 2003 |

44 | The higher-order recursive path ordering
- Jouannaud, Rubio
- 1999
(Show Context)
Citation Context ...accessibility”, or new computability preserving operations in the “computable closure”. (3) The notion of “computable closure” is an important concept which has already be used in a different context =-=[29]-=-. (4) Several precise conjectures have been stated. The most important two, in our view, are the use of the General Schema to prove the strong normalization of higher-order rewriting à la Nipkow on th... |

34 | Solving higher-order equations: From logic to programming. Birkhäuser PCTS Series
- Prehofer
- 1997
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Citation Context ... + the choice operator and Σ the dependent choice), ord for well-founded trees, i.e. Brouwer’s ordinals [45], form for formulas of the predicate calculus and R for expressions built upon real numbers =-=[42]-=-: • bool = true : bool | false : bool • nat = 0 : nat | s : nat → nat • listnat = nil : listnat | cons : nat → listnat → listnat • tree = node : listtree → tree • listtree = nil : listtree | cons : tr... |

33 | On Girard’s “Candidats de Réductibilité
- Gallier
- 1990
(Show Context)
Citation Context ...Schema. Due to the formulation of the schema, our proof here is much simpler than the one in [28], although the schema is more general. It is again based on Tait’s computability predicate method. See =-=[19]-=- for a comprehensive survey of the method. We first define the interpretation of types and prove important properties about it. In a second part, we prove a computability property for the function sym... |

33 |
Executable higher-order algebraic specification languages
- Jouannaud, Okada
- 1991
(Show Context)
Citation Context ... the last two authors of the present paper proposed another solution, for a polymorphically typed l-calculus, based on pattern-matching functional definitions following the so-called “General Schema” =-=[27,28]-=-. This work was then generalized so as to cover the full Calculus of Constructions [1,2,3]. As in Coquand [12], the idea of the General Schema is to control the arguments of the right-hand side recurs... |

32 | Structural recursive definitions in type theory
- Giménez
- 1998
(Show Context)
Citation Context ...ws the General Schema while / does not and that the last rule is duplicating the variable y: 0 − y → 0 s(x) − 0 → s(x) s(x) − s(y) → x − y x / 0 → x 0 / s(y) → 0 s(x) / s(y) → s((x − y) / s(y)) 23In =-=[21]-=-, Giménez proposes a terminating schema using a notion of subtyping which allows to prove the strong normalization property of this example. However, we do not think this is a real issue. Non-terminat... |

27 | The calculus of algebraic constructions
- Blanqui, Jouannaud, et al.
- 1999
(Show Context)
Citation Context ...the theoretical one. The first conjecture has been recently solved by the first author in [5]. Another kind of extension should now be considered, by considering a richer type system, which we did in =-=[6]-=-, keeping the same definition for the rules and the General Schema. But a richer type system allows us to have richer forms of rewrite rules: the General Schema should therefore be adapted so as to al... |

25 | H.: Modularity of strong normalization and confluence in the algebraic-λ-cube
- Barbanera, Fernández, et al.
- 1994
(Show Context)
Citation Context ...y typed l-calculus, based on pattern-matching functional definitions following the so-called “General Schema” [27,28]. This work was then generalized so as to cover the full Calculus of Constructions =-=[1,2,3]-=-. As in Coquand [12], the idea of the General Schema is to control the arguments of the right-hand side recursive calls of a rule-based definition by checking that they are smaller than the left-hand ... |

24 |
Strong Normalizability for the Combined System of the Typed λ Calculus and an Arbitrary Convergent Term Rewrite System
- Okada
- 1989
(Show Context)
Citation Context ...n is very abstract, though, and relies on a well-foundedness assumption which is satisfied in practice. Concurrently, following the pioneering works of Tannen [8], Tannen and Gallier [9,10] and Okada =-=[40]-=-, the last two authors of the present paper proposed another solution, for a polymorphically typed l-calculus, based on pattern-matching functional definitions following the so-called “General Schema”... |

22 |
Combining first and higher order rewrite systems with type assignment systems
- Barbanera, Fernández
- 1993
(Show Context)
Citation Context ...y typed l-calculus, based on pattern-matching functional definitions following the so-called “General Schema” [27,28]. This work was then generalized so as to cover the full Calculus of Constructions =-=[1,2,3]-=-. As in Coquand [12], the idea of the General Schema is to control the arguments of the right-hand side recursive calls of a rule-based definition by checking that they are smaller than the left-hand ... |

15 |
Modularity of Termination and Confluence in Combinations of Rewrite Systems with λω
- Barbanera, Fernández
- 1993
(Show Context)
Citation Context ...y typed l-calculus, based on pattern-matching functional definitions following the so-called “General Schema” [27,28]. This work was then generalized so as to cover the full Calculus of Constructions =-=[1,2,3]-=-. As in Coquand [12], the idea of the General Schema is to control the arguments of the right-hand side recursive calls of a rule-based definition by checking that they are smaller than the left-hand ... |

15 |
On intuitionistic arithmetic and number theory
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(Show Context)
Citation Context ... were far too complex to be used by nonspecialists. A different approach was taken by Martin-Löf [32,33], whose theory was based on the notion of inductive definition, originating in Gödel’s system T =-=[25]-=-. Coquand and Paulin-Möhring later incorporated a similar notion to the Calculus of Constructions under the name of inductive type [14]. But despite their legitimate success, inductive types are not y... |

14 | Termination and confluence of higher-order rewrite systems
- Blanqui
- 2000
(Show Context)
Citation Context ... patterns in the left-hand sides. To prove our conjecture, we essentially need to show that higher-order pattern-matching preserves computability. This has been recently proved by the first author in =-=[5]-=-, where the framework described here is extended into a typed version of Klop’s higher-order rewriting framework [31], and where Nipkow’s higherorder Critical Pair Lemma is shown to apply to this exte... |

13 | Termination Proofs for Higher-Order Rewrite Systems
- Pol
- 1994
(Show Context)
Citation Context ...this section, we address the case of higher-order rewrite rules à la Nipkow [39], based on higher-order pattern-matching with patterns à la Miller [37]. We give here several examples taken from [39], =-=[48]-=- or [42], and recall why plain pattern-matching does not really make sense for them. On the other hand, we will see that all these examples follow the General Schema: we explain the first example in d... |

9 |
Confluence and Normalization for Higher-Order Rewriting.PhDthesis,VrijeUniversiteit,Amsterdam,1996
- Raamsdonk
(Show Context)
Citation Context ...ions of higher-order rewriting defined by Klop (Combinatory Reduction Systems [30,31]), Nipkow (Higher-order Rewrite Systems [39,34]) and van Raamsdonk and van Oostrom (Higher-Order Rewriting Systems =-=[49,50]-=-, generalizing both) are based on higher-order pattern-matching, that is, u must be βηα-convertible to the instance of l. Definition 6 (Rewrite rules and rewriting) A rewrite rule is a pair l → r of t... |

5 |
A note on rewriting theory for uniqueness of iteration, Theory and Applications of Categories 6
- Okada, Scott
- 1999
(Show Context)
Citation Context ...of arbitrary strictly positive inductive types. The general case is no more difficult apart for the more complex notations. The uniqueness rules for recursors of basic inductive types were studied in =-=[41]-=- and extended to the strictly positive case in [26]. In both cases, the termination proof did not use the General Schema since the uniqueness rules do not seem to fit the General Schema. It is open wh... |

3 |
second-order lambda calculi with recursive types
- First-
- 1986
(Show Context)
Citation Context ...actical needs, making it possible to have nested recursive calls, an important facility that Coquand’s ordering cannot provide with. Finally, it is important to note that, in contrast with other work =-=[35,20]-=-, our definitions allow nonlinear and overlapping left-hand sides, to the price of checking confluence via the computation of critical pairs. The fact that the General Schema covers only a limited por... |

3 |
Automates de formes normales et réductibilité inductive
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- 1997
(Show Context)
Citation Context ...term rewriting system {s(p(x)) → x, p(s(x)) → x} whose normal forms are recognized by the automaton given at Figure 1. This automaton can be easily constructed by solving disequations over terms (see =-=[11,38]-=-). s,p Int ε p s ε s Pos ε Neg p s p Zero 0 Fig. 1. Automaton Then, the recursor on integers may be defined by the following set of constraint rules: intrect(X, Y, Z, 0) → X intrect(X, Y, Z, s(x)) → (... |

3 |
unification of higher-order patterns
- Linear
- 1993
(Show Context)
Citation Context ...o cover all cases. The local confluence of these rules can be checked on higher-order critical 27pairs, as shown by Nipkow [39,34]. The computation of these critical pairs can be done in linear time =-=[43]-=-, thanks to the hypothesis that the left-hand sides are patterns. We now show that this example follows the General Schema, by showing first that the free variables of the right-hand sides are accessi... |

2 |
On extensions of Gödel’s System T
- Hasebe
- 2000
(Show Context)
Citation Context ... general case is no more difficult apart for the more complex notations. The uniqueness rules for recursors of basic inductive types were studied in [41] and extended to the strictly positive case in =-=[26]-=-. In both cases, the termination proof did not use the General Schema since the uniqueness rules do not seem to fit the General Schema. It is open whether one could modify the schema to cover this kin... |