## Inductively defined types in the calculus of constructions (1990)

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### BibTeX

@INPROCEEDINGS{Pfenning90inductivelydefined,

author = {Frank Pfenning and Christine Paulin-mohring},

title = {Inductively defined types in the calculus of constructions},

booktitle = {},

year = {1990},

pages = {209--228},

publisher = {Springer-Verlag}

}

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### Abstract

We define the notion of an inductively defined type in the Calculus of Constructions and show how inductively defined types can be represented by closed types. We show that all primitive recursive functionals over these inductively defined types are also representable. This generalizes work by Böhm & Berarducci on synthesis of functions on term algebras in the second-order polymorphic λ-calculus (F2). We give several applications of this generalization, including a representation of F2-programs in F3, along with a definition of functions reify, reflect, and eval for F2 in F3. We also show how to define induction over inductively defined types and sketch some results that show that the extension of the Calculus of Construction by induction principles does not alter the set of functions in its computational fragment, Fω. This is because a proof by induction can be realized by primitive recursion, which is already definable in Fω. 1

### Citations

711 | A framework for defining logics
- Harper, Honsel, et al.
- 1987
(Show Context)
Citation Context ...werful type theory, yet it can be formulated very concisely. It encompasses Girard’s system Fω (see [13, 14]) and the type theory of LF, the Edinburgh Logical Framework (see Harper, Honsell & Plotkin =-=[15]-=-) and may be considered the result of combining these two type theories (see Barendregt [2]). The formulation we present here is a very brief summary of the concrete syntax, notation, and inference sy... |

489 | The Calculus of Constructions - Coquand, Huet - 1988 |

377 |
Types, abstraction and parametric polymorphism
- Reynolds
- 1983
(Show Context)
Citation Context ...C) which have the same computational content as our definitions. Another alternative would be to strengthen the notion of equality. We conjecture that one can use Reynolds’ condition of parametricity =-=[26]-=- to recover uniqueness of representations at least in the Fω fragment. convenient to simply use ∗ to encompass all of them. We thus use the terms “proposition” and “specification” interchangeably.sInd... |

293 |
Interprétation fonctionelle et élimination des coupures dans l’arithmetique d’ordre supérieur
- Girard
- 1972
(Show Context)
Citation Context ...t be able to extract from such proofs is already definable in pure Fω—we just would not be able to show in CoC without induction that it satisfies its specification. Closely related is work by Girard =-=[13, 14]-=-, Fortune, Leivant & O’Donnell [12], and Leivant [17, 18] who are concerned with the relationship between higher-order logic and polymorphic λ-calculi. Mendler [19, 20] studied inductive types in the ... |

231 |
Une extension de l’interpretation de Gödel, à l’analyse, et son application l’élimination des coupures dans l’analyse et la théorie des types
- Girard
- 1971
(Show Context)
Citation Context ...t be able to extract from such proofs is already definable in pure Fω—we just would not be able to show in CoC without induction that it satisfies its specification. Closely related is work by Girard =-=[13, 14]-=-, Fortune, Leivant & O’Donnell [12], and Leivant [17, 18] who are concerned with the relationship between higher-order logic and polymorphic λ-calculi. Mendler [19, 20] studied inductive types in the ... |

160 |
de Bruijn. Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem
- G
- 1972
(Show Context)
Citation Context ...ry brief summary of the concrete syntax, notation, and inference system given in [8]. We use M, N, . . . for terms in general and x, y, z for variables (abstractly, though, they are de Bruijn indices =-=[10]-=-, where the occurrences of x in (λx:M) N and [x:M] N are binding occurrences). We have M ::= x | (λx:M) N | (M N) | [x:M] N | ∗ Following [8] we call [x:M] N a product. ∗ is the universe of all types,... |

84 | Using typed lambda calculus to implement formal systems on a machine
- Avron, Honsell, et al.
- 1992
(Show Context)
Citation Context ...em become types or type families, logical connectives and quantifiers and inference rules become typed constants. See Harper, Honsell & Plotkin [15] for a description of LF and Avron, Honsell & Mason =-=[1]-=- for LF representations of a variety of logics. These signatures resemble inductive type definitions, but upon closer inspection the analogy fails. Consider the following two problematic declarations ... |

74 |
Constructions: A higher order proof system for mechanizing mathematics
- Coquand, Huet
(Show Context)
Citation Context ... such types in Fω is currently under way in the framework of the Ergo project at Carnegie Mellon University. 2 The Calculus of Constructions The Calculus of Constructions (CoC) of Coquand & Huet (see =-=[7, 6, 16, 8]-=-) is a very powerful type theory, yet it can be formulated very concisely. It encompasses Girard’s system Fω (see [13, 14]) and the type theory of LF, the Edinburgh Logical Framework (see Harper, Hons... |

71 |
Three approaches to type structure
- Reynolds
- 1985
(Show Context)
Citation Context ... non-trivial induction. As we will see later in Example 21 the representation of this parameterized type in our framework is somewhat different from the representation, for example, given by Reynolds =-=[27]-=-. indtype list : ∗ → ∗ with nil : [A:∗] listA cons : [A:∗] A → listA → listA end Ordinal notations, the next example, are not algebraic for a different reason: the argument to one of the constructors ... |

68 |
Inductive Definition in Type Theory
- Mendler
- 1987
(Show Context)
Citation Context ...osely related is work by Girard [13, 14], Fortune, Leivant & O’Donnell [12], and Leivant [17, 18] who are concerned with the relationship between higher-order logic and polymorphic λ-calculi. Mendler =-=[19, 20]-=- studied inductive types in the setting of the second-order polymorphic λ-calculus and the NuPrl type theory. He adds to the system F a new scheme for defining recursive types. The system is extended ... |

54 |
Polymorphic rewriting conserves algebraic strong normalization
- Breazu-Tannen, Gallier
- 1991
(Show Context)
Citation Context ...h the addition of induction one will in general be able to prove many more specifications. Other conservative extension results for polymorphic λ-calculi have been obtained by Breazu-Tannen & Gallier =-=[3]-=-. Definition 33 (Induction principle indα for inductively defined α) Let α be an inductively defined type as before. We define indα, the induction principle over α by indα : [A:[z1:Q1] . . .[zm:Qm] α ... |

49 |
Extraction de programmes dans le Calcul des Constructions
- Paulin-Mohring
- 1989
(Show Context)
Citation Context ...rsion, which is already definable in Fω. 1 Introduction The motivation for the this paper comes from two sources: work on the extraction of programs from proofs in the Calculus of Constructions (CoC) =-=[23, 24]-=- and work on the implementation of LEAP [25], an explicitly polymorphic ML-like programming language (here we only consider the pure Fω fragment of LEAP). The former emphasizes the logical aspects of ... |

46 |
Automatic synthesis of typed Λprograms on term algebras. Theoretical Computer Science 39:135–154
- Böhm, Berarducci
- 1985
(Show Context)
Citation Context ... for the Calculus of Constructions is presented by Coquand and Paulin-Mohring [9] and for Martin-Löf’s type theory by Dybjer [11]. On the purely computational level, we generalize Böhm & Berarducci’s =-=[4]-=- construction of functions on term algebras in the second-order polymorphic λ-calculus (F2) to Fω. One of their results does not generalize in unmodified form beyond algebraic types: not every closed ... |

44 |
and Gérard Huet. The calculus of constructions
- Coquand
- 1988
(Show Context)
Citation Context ... such types in Fω is currently under way in the framework of the Ergo project at Carnegie Mellon University. 2 The Calculus of Constructions The Calculus of Constructions (CoC) of Coquand & Huet (see =-=[7, 6, 16, 8]-=-) is a very powerful type theory, yet it can be formulated very concisely. It encompasses Girard’s system Fω (see [13, 14]) and the type theory of LF, the Edinburgh Logical Framework (see Harper, Hons... |

44 | Extracting F ! programs from proofs in the Calculus of Constructions - Paulin-Mohring - 1989 |

41 |
Une The'orie Des Constructions
- Coquand
- 1985
(Show Context)
Citation Context ... such types in Fω is currently under way in the framework of the Ergo project at Carnegie Mellon University. 2 The Calculus of Constructions The Calculus of Constructions (CoC) of Coquand & Huet (see =-=[7, 6, 16, 8]-=-) is a very powerful type theory, yet it can be formulated very concisely. It encompasses Girard’s system Fω (see [13, 14]) and the type theory of LF, the Edinburgh Logical Framework (see Harper, Hons... |

36 |
The expressiveness of simple and second-order type structures
- Fortune, Leivant, et al.
- 1983
(Show Context)
Citation Context ...s already definable in pure Fω—we just would not be able to show in CoC without induction that it satisfies its specification. Closely related is work by Girard [13, 14], Fortune, Leivant & O’Donnell =-=[12]-=-, and Leivant [17, 18] who are concerned with the relationship between higher-order logic and polymorphic λ-calculi. Mendler [19, 20] studied inductive types in the setting of the second-order polymor... |

36 |
Contracting proofs to programs
- Leivant
- 1990
(Show Context)
Citation Context ...e in pure Fω—we just would not be able to show in CoC without induction that it satisfies its specification. Closely related is work by Girard [13, 14], Fortune, Leivant & O’Donnell [12], and Leivant =-=[17, 18]-=- who are concerned with the relationship between higher-order logic and polymorphic λ-calculi. Mendler [19, 20] studied inductive types in the setting of the second-order polymorphic λ-calculus and th... |

33 |
Reasoning about functional programs and complexity classes associated with type disciplines
- Leivant
- 1983
(Show Context)
Citation Context ...e in pure Fω—we just would not be able to show in CoC without induction that it satisfies its specification. Closely related is work by Girard [13, 14], Fortune, Leivant & O’Donnell [12], and Leivant =-=[17, 18]-=- who are concerned with the relationship between higher-order logic and polymorphic λ-calculi. Mendler [19, 20] studied inductive types in the setting of the second-order polymorphic λ-calculus and th... |

31 |
Extracting Fω’s programs from proofs in the Calculus of Constructions
- Paulin-Mohring
- 1989
(Show Context)
Citation Context ...rsion, which is already definable in Fω. 1 Introduction The motivation for the this paper comes from two sources: work on the extraction of programs from proofs in the Calculus of Constructions (CoC) =-=[23, 24]-=- and work on the implementation of LEAP [25], an explicitly polymorphic ML-like programming language (here we only consider the pure Fω fragment of LEAP). The former emphasizes the logical aspects of ... |

26 | A framework for de ning logics - Harper, Honsell, et al. - 1992 |

21 | About primitive recursive algorithms - Colson - 1991 |

19 |
On representation of data in lambda calculus
- Parigot
- 1989
(Show Context)
Citation Context ...ing language based on an enriched Fω one would probably need to choose primitive recursion, since its implementation through iteration is provably inefficient in some cases (see Colson [5] or Parigot =-=[22]-=-). Definition 26 (Definition by iteration) Let an α be an inductively defined data type as in Section 3. Given a β : Q and functions h1:P β 1 , . . .,hn:P β n . Then the function f : [z1:Q1] . . .[zm:... |

16 |
Second-order logical relations
- Mitchell, Meyer
- 1985
(Show Context)
Citation Context ...ion that completeness fails because βη-equality is too weak to identify indistinguishable terms, under some reasonable assumptions about when terms should be indistinguishable (see Mitchell and Meyer =-=[21]-=-). Computationally this failure of completeness is not a problem, and the logical characterization of an inductive type in terms of an induction axiom is satisfactory from the logical point of view (t... |

9 | A.Berarducci, Automatic Synthesis of Typed λ-Programs - Böhm - 1985 |

8 | Metacircularity in the polymorphic lambdacalculus - Pfenning, Lee - 1991 |

7 |
An inversion principle for Martin-Lof's type theory. Talk presented at the Workshop on Programming Logic
- Dybjer
- 1989
(Show Context)
Citation Context ...his paper are slightly different but equivalent. Work along Mendler’s lines for the Calculus of Constructions is presented by Coquand and Paulin-Mohring [9] and for Martin-Löf’s type theory by Dybjer =-=[11]-=-. On the purely computational level, we generalize Böhm & Berarducci’s [4] construction of functions on term algebras in the second-order polymorphic λ-calculus (F2) to Fω. One of their results does n... |

6 | Inductive De nition in Type Theory - Mendler - 1987 |

4 |
second-order lambda calculi with recursive types
- First-
- 1986
(Show Context)
Citation Context ...osely related is work by Girard [13, 14], Fortune, Leivant & O’Donnell [12], and Leivant [17, 18] who are concerned with the relationship between higher-order logic and polymorphic λ-calculi. Mendler =-=[19, 20]-=- studied inductive types in the setting of the second-order polymorphic λ-calculus and the NuPrl type theory. He adds to the system F a new scheme for defining recursive types. The system is extended ... |

1 |
The forest of lambda calculi with types. Talk given at the Workshop on Semantics of Lambda Calculus and Category Theory
- Barendregt
- 1988
(Show Context)
Citation Context ...Fω (see [13, 14]) and the type theory of LF, the Edinburgh Logical Framework (see Harper, Honsell & Plotkin [15]) and may be considered the result of combining these two type theories (see Barendregt =-=[2]-=-). The formulation we present here is a very brief summary of the concrete syntax, notation, and inference system given in [8]. We use M, N, . . . for terms in general and x, y, z for variables (abstr... |

1 |
Inductively defined types. Talk presented at the Workshop on Programming Logic
- Coquand, Paulin-Mohring
- 1989
(Show Context)
Citation Context ... primitive recursion in Mendler’s work and in this paper are slightly different but equivalent. Work along Mendler’s lines for the Calculus of Constructions is presented by Coquand and Paulin-Mohring =-=[9]-=- and for Martin-Löf’s type theory by Dybjer [11]. On the purely computational level, we generalize Böhm & Berarducci’s [4] construction of functions on term algebras in the second-order polymorphic λ-... |

1 |
Formal structures for computation and deduction. Lecture notes for a graduate course at
- Huet
- 1986
(Show Context)
Citation Context |

1 | Inductively de ned types. Talk presented at the Workshop on Programming Logic - Coquand, Paulin-Mohring - 1989 |