## Formalising formulas-as-types-as-objects (2000)

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Venue: | Types for Proofs and Programs |

Citations: | 2 - 0 self |

### BibTeX

@INPROCEEDINGS{Haiyan00formalisingformulas-as-types-as-objects,

author = {Qiao Haiyan},

title = {Formalising formulas-as-types-as-objects},

booktitle = {Types for Proofs and Programs},

year = {2000},

pages = {174--194},

publisher = {Springer}

}

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### Abstract

Abstract. We describe a formalisation of the Curry-Howard-Lawvere correspondence between the natural deduction system for minimal logic, the typed lambda calculus and Cartesian closed categories. We formalise the type of natural deduction proof trees as a family of sets Γ ⊢ A indexed by the current assumption list Γ and the conclusion A and organise numerous useful lemmas about proof trees categorically. We prove categorical properties about proof trees up to (syntactic) identity as well as up to βη-convertibility. We prove that our notion of proof trees is equivalent in an appropriate sense to more traditional representations of lambda terms. The formalisation is carried out in the proof assistant ALF for Martin-Löf type theory. 1

### Citations

261 |
Programming in Martin-Löf’s Type Theory: An Introduction
- Nordstrom, Petersson, et al.
- 1990
(Show Context)
Citation Context ...isomorphic categories with families from such terms module βη-conversion. The rest of the paper is organised as follows. In section 2 we give the formal definitions (inside Martin-Löf type theory/ALF =-=[16,14]-=- ) of minimal logic. In section 3 we introduce the notion of a simple category with families. In section 4 we show some properties of minimal logic structured as properties of three P-categories with ... |

88 | The HOL logic
- Pitts
- 1993
(Show Context)
Citation Context ...n of model for type-dependency. Similar notions have also been considered: Cartmell’s contextual categories [5], Seely’s locally Cartesian closed categories [20], and Pitts’ categories with fibration =-=[18]-=-. Cwfs are closer to the traditional syntax of dependent types, and at the same time they are completely algebraic and have a nice categorical description. A cwf has a family Ty(Γ ) of types in contex... |

66 | Inductive families
- Dybjer
- 1994
(Show Context)
Citation Context ...d (N ′ typa, =). 5.2 Typable de Bruijn Raw Terms First we define λ-terms as an inductive family indexed by N. Λn represents λ-terms with at most n free variables with the following introduction rules =-=[9]-=-: n : N i :N ′ n varr(i) :Λn lterm ∈(N) Set varI ∈(n ∈ N; i ∈ N’(n)) lterm(n) lamI ∈(n ∈ N; lterm(succ(n))) lterm(n) apI ∈(n ∈ N; lterm(n); lterm(n)) lterm(n) n : N t : Λ s(n) λr(t) :Λn Then we define... |

58 |
Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic 32
- Cartmell
- 1986
(Show Context)
Citation Context ...ilies (cwf) [10]. Cwfs were introduced by Dybjer [10] to be an appropriate categorical notion of model for type-dependency. Similar notions have also been considered: Cartmell’s contextual categories =-=[5]-=-, Seely’s locally Cartesian closed categories [20], and Pitts’ categories with fibration [18]. Cwfs are closer to the traditional syntax of dependent types, and at the same time they are completely al... |

56 |
Locally cartesian closed categories and type theory
- Seely
- 1984
(Show Context)
Citation Context ...[10] to be an appropriate categorical notion of model for type-dependency. Similar notions have also been considered: Cartmell’s contextual categories [5], Seely’s locally Cartesian closed categories =-=[20]-=-, and Pitts’ categories with fibration [18]. Cwfs are closer to the traditional syntax of dependent types, and at the same time they are completely algebraic and have a nice categorical description. A... |

44 | Intuitionistic model constructions and normalization proofs
- Coquand, Dybjer
- 1997
(Show Context)
Citation Context ...ined in the next section, between the representation of proof trees and the representation of typed λ-calculus we just defined above. 5.3 The Map Strip and Its Inverse We define a function strip (see =-=[7]-=-) which maps a proof tree t of type Γ ⊢ A to a raw term by stripping its type information. In the other direction, we define a function prft which maps a raw term b such that Γ ⊢ b :: A to a proof tre... |

40 | Syntax and semantics of dependent types
- Hofmann
- 1996
(Show Context)
Citation Context ...that p ◦ θ = γ and q[θ] =a. Simple cwfs can be axiomatised by taking the axiomatisation of general cwfs from Dybjer [10] and removing type-dependency. More information on cwfs can be found in Hofmann =-=[11]-=-. We will refer to a cwf as a structure (C, Ty, Tm). P-cwfs. When we formalise category theory in Martin-Löf type theory we shall follow ˘ Cubrić, Dybjer, and Scott [8] and use P-categories. A P-categ... |

36 | Internal type theory
- Dybjer
(Show Context)
Citation Context ... to βη-convertibility. It turns out thatsFormalising Formulas-as-Types-as-Objects 179 these properties can be organised in an elegant categorical way, using the notion of category with families (cwf) =-=[10]-=-. Cwfs were introduced by Dybjer [10] to be an appropriate categorical notion of model for type-dependency. Similar notions have also been considered: Cartmell’s contextual categories [5], Seely’s loc... |

29 | From Semantics to Rules: A Machine Assisted Analysis
- Coquand
- 1993
(Show Context)
Citation Context ...term) and (F, ∼, Type, Term) are isomorphic. 6 Conclusion Many authors have proved theorems about simply and dependently typed lambda calculi inside type theory proof assistants, for example, Coquand =-=[6]-=-, Bove [4] (in ALF), McKinna and Pollack [15], Altenkirch [1] (in LEGO), and B. Barras [3], Huet [12], Saïbi [19] (in Coq). A variety of approaches have been used. The aim of our work is not only to s... |

22 | Categorical reconstruction of a reduction free normalization proof
- Altenkirch, Hofmann, et al.
- 1995
(Show Context)
Citation Context ... does not depend on Γ . We call such cwfs “simple”. Similar notions are Obtulowicz’s Church algebraic theory [17], Jacobs’ indexed categories [13], Altenkirch, Hofmann and Streicher’s contextual CCCs =-=[2]-=-, and etc. Definition 1 A simple cwf consists of the following parts: – A base category C. Its objects are called contexts and its morphisms are called substitutions. We write ∆ → Γ for the hom-set of... |

22 | Normalization and the yoneda embedding
- Cubric, Dybier, et al.
- 1997
(Show Context)
Citation Context ...rtinLöf type theory. 1 Introduction The background of the present paper is as follows. D. Čubrić, P. Dybjer and P. Scott discovered an elegant categorical decision method for equality in the free ccc =-=[8]-=-. This method was based on extracting an algorithm from some basic categorical facts related to the Yoneda lemma. To this end it was necessary to develop a certain version of constructive category the... |

20 | Residual theory in λ-calculus: A formal development
- Huet
- 2013
(Show Context)
Citation Context ...mply and dependently typed lambda calculi inside type theory proof assistants, for example, Coquand [6], Bove [4] (in ALF), McKinna and Pollack [15], Altenkirch [1] (in LEGO), and B. Barras [3], Huet =-=[12]-=-, Saïbi [19] (in Coq). A variety of approaches have been used. The aim of our work is not only to suggest a formalization which we claim is particularly natural from a type-theoretic point of view, bu... |

12 |
Some type theory and lambda calculus formalised
- McKinna, Pollack
- 1999
(Show Context)
Citation Context ... 6 Conclusion Many authors have proved theorems about simply and dependently typed lambda calculi inside type theory proof assistants, for example, Coquand [6], Bove [4] (in ALF), McKinna and Pollack =-=[15]-=-, Altenkirch [1] (in LEGO), and B. Barras [3], Huet [12], Saïbi [19] (in Coq). A variety of approaches have been used. The aim of our work is not only to suggest a formalization which we claim is part... |

9 |
A Formalization of the Strong Normalisation Proof for System F
- Altenkirch
- 1993
(Show Context)
Citation Context ...y authors have proved theorems about simply and dependently typed lambda calculi inside type theory proof assistants, for example, Coquand [6], Bove [4] (in ALF), McKinna and Pollack [15], Altenkirch =-=[1]-=- (in LEGO), and B. Barras [3], Huet [12], Saïbi [19] (in Coq). A variety of approaches have been used. The aim of our work is not only to suggest a formalization which we claim is particularly natural... |

7 |
Simply typed and untyped lambda calculus revisited
- Jacobs
- 1992
(Show Context)
Citation Context ...ependent types, it suffices to consider cwfs, where Ty(Γ ) does not depend on Γ . We call such cwfs “simple”. Similar notions are Obtulowicz’s Church algebraic theory [17], Jacobs’ indexed categories =-=[13]-=-, Altenkirch, Hofmann and Streicher’s contextual CCCs [2], and etc. Definition 1 A simple cwf consists of the following parts: – A base category C. Its objects are called contexts and its morphisms ar... |

7 |
Categorical, functorial, and algebraic aspects of the. type-free lambda calculus Banach Center Publications 9
- Obtulowicz, Wiweger
- 1982
(Show Context)
Citation Context .... Since we here do not consider dependent types, it suffices to consider cwfs, where Ty(Γ ) does not depend on Γ . We call such cwfs “simple”. Similar notions are Obtulowicz’s Church algebraic theory =-=[17]-=-, Jacobs’ indexed categories [13], Altenkirch, Hofmann and Streicher’s contextual CCCs [2], and etc. Definition 1 A simple cwf consists of the following parts: – A base category C. Its objects are cal... |

6 |
Coq en Coq
- Barras
- 1996
(Show Context)
Citation Context ...s about simply and dependently typed lambda calculi inside type theory proof assistants, for example, Coquand [6], Bove [4] (in ALF), McKinna and Pollack [15], Altenkirch [1] (in LEGO), and B. Barras =-=[3]-=-, Huet [12], Saïbi [19] (in Coq). A variety of approaches have been used. The aim of our work is not only to suggest a formalization which we claim is particularly natural from a type-theoretic point ... |

3 | A machine-assisted proof that well typed expressions cannot go wrong
- Bove
- 1998
(Show Context)
Citation Context ...(F, ∼, Type, Term) are isomorphic. 6 Conclusion Many authors have proved theorems about simply and dependently typed lambda calculi inside type theory proof assistants, for example, Coquand [6], Bove =-=[4]-=- (in ALF), McKinna and Pollack [15], Altenkirch [1] (in LEGO), and B. Barras [3], Huet [12], Saïbi [19] (in Coq). A variety of approaches have been used. The aim of our work is not only to suggest a f... |

2 |
The ALF Proof Editor and Its Proof Engine. Types for Proofs and Programs
- Magnusson, Nordström
- 1994
(Show Context)
Citation Context ...isomorphic categories with families from such terms module βη-conversion. The rest of the paper is organised as follows. In section 2 we give the formal definitions (inside Martin-Löf type theory/ALF =-=[16,14]-=- ) of minimal logic. In section 3 we introduce the notion of a simple category with families. In section 4 we show some properties of minimal logic structured as properties of three P-categories with ... |

1 |
Formalisation of a λ-Calculus with Explicit Substitutions in Coq
- Saïbi
- 1994
(Show Context)
Citation Context ...endently typed lambda calculi inside type theory proof assistants, for example, Coquand [6], Bove [4] (in ALF), McKinna and Pollack [15], Altenkirch [1] (in LEGO), and B. Barras [3], Huet [12], Saïbi =-=[19]-=- (in Coq). A variety of approaches have been used. The aim of our work is not only to suggest a formalization which we claim is particularly natural from a type-theoretic point of view, but also to ab... |