## From semantics to rules: A machine assisted analysis (1999)

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Venue: | Proceedings of CSL '93, LNCS 832 |

Citations: | 29 - 0 self |

### BibTeX

@INPROCEEDINGS{Coquand99fromsemantics,

author = {Catarina Coquand},

title = {From semantics to rules: A machine assisted analysis},

booktitle = {Proceedings of CSL '93, LNCS 832},

year = {1999},

pages = {91--105},

publisher = {Springer-Verlag}

}

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### Abstract

this paper is similar to the one in [2]. In this paper they define a normalization function for simply typed

### Citations

391 | Explicit substitutions
- Abadi, Cardelli, et al.
- 1991
(Show Context)
Citation Context ... For proving the correctness of these implementations it is not trivial to translate the implementation to the theoretical presentation. A - calculus with explicit substitutions has been suggested in =-=[1, 12]-=- to bring the theoretical presentation closer to the usual implementations. ffl Explicit substitutions also correspond to let-expressions in functional language. ffl There has been a recent interest i... |

261 |
Programming in Martin-Löf’s Type Theory: An Introduction
- Nordstrom, Petersson, et al.
- 1990
(Show Context)
Citation Context ...rphic, this means that all type information is in the term. For readability this information has been taken away in most of the terms. We will use the intensional equality Id(A; a; b) as described in =-=[15]-=- with its constructor id(A; a) 2 Id(A; a; a) (the A will be dropped) and the set 2 Bool with its standard elements and operations. We also introduce a set, T (b) depending on Bool, which has one eleme... |

157 | An intuitionistic theory of types: Predicative part - Martin-Löf - 1975 |

116 | Inheritance as implicit coercion - Breazu-Tannen, Coquand, et al. - 1991 |

85 | Pattern matching with dependent types
- Coquand
- 1992
(Show Context)
Citation Context ...o element in T (false). All the work presented below has been done in ALF and all the proofs of the theorems have been type checked. The proofs are using the pattern matching introduced by Coquand in =-=[8]-=- which is not a part of Martin-Lof's framework, but has been essential for carrying out this proof; in fact this study motivated the mechanism. ALF does not yet check that the theorems are proved usin... |

60 | Program extraction from normalization proofs
- Berger
- 1993
(Show Context)
Citation Context ...et of conversion rules for this calculus. The treatment of variables is also different in our approach. The inversion function is also presented in [6] where it is discussed for combinatory logic. In =-=[3]-=- a more detailed study of how to extract the normalization function from a normalization proof using Tait's method is presented. One important difference with other works on explicit substitution is t... |

50 |
An inverse of the evaluation functional for typed -calculus
- Berger, Schwichtenberg
- 1991
(Show Context)
Citation Context ... and [[N ]] are equal, then quote([[M ]]) and quote([[N ]]) are the same and hence M and N are convertible with each other. The normalization function presented in this paper is similar to the one in =-=[2]-=-. In this paper they define a normalization function for simply typed -calculus with full fij reduction to obtain a refinement of Friedman's completeness theorem. The main difference compared with the... |

44 | Intuitionistic model constructions and normalization proofs
- Coquand, Dybjer
- 1997
(Show Context)
Citation Context ...nd substitutions and that we give the the complete set of conversion rules for this calculus. The treatment of variables is also different in our approach. The inversion function is also presented in =-=[6]-=- where it is discussed for combinatory logic. In [3] a more detailed study of how to extract the normalization function from a normalization proof using Tait's method is presented. One important diffe... |

44 | Kripke-style models for typed lambda calculus
- Mitchell, Moggi
- 1991
(Show Context)
Citation Context ...mation that might be used in a computation. 4.2 The semantics Since we want to deal with full conversion on open terms and the j-rule, we choose to describe the semantics in a Kripke style model. See =-=[7, 14, 10]-=- for a 7 discussion on Kripke models. A Kripke model is a set of possible worlds, W, with a partial ordering, , of extensions of worlds. We also have a family of ground sets G(w) over possible worlds ... |

31 |
Logical Relations and the Typed -Calculus
- Statman
- 1985
(Show Context)
Citation Context ...7 13 we know that if Eq([[M ]]id; [[N ]]id), then Id(nf(M); nf(N)) and then by theorem8 and symmetry and transitivity of = we know that M =N . To prove the theorem we define a Kripke logical relation =-=[14, 18]-=-: M 2 \Gamma ` A u 2 \Gamma k\Gamma A CV \Gamma;A (M;u) 2 Set this relation will correspond to Tait's computability predicate. A derivation of base type is intuitively CV-related to a semantic object ... |

29 |
Relating theories of the lambda-calculus
- Scott
- 1980
(Show Context)
Citation Context ...should be equal to first lifting the arguments and then do the application. We will say that a semantic object is 8 uniform U if the application and monotonicity function commutes for this object. In =-=[17]-=- there is a further discussions on this commutativity. The predicates Eq and U will be mutually defined u; v 2 w k\Gamma A Eqw;A (u; v) 2 Set u 2 w k\Gamma A U w;A (u) 2 Set They will both be defined ... |

28 |
Semantics of Type Theory
- Streicher
- 1991
(Show Context)
Citation Context ...e usual notation with -terms is not ambiguous; there may be several derivations of the same typing judgement, but they are all convertible with each other. The idea of this proof comes from Streicher =-=[19]-=- (chapter IV). 5.1 Definition of terms We mutually define the set of terms, T 2 Set, and substitutions, S 2 Set as x 2 Name v(x) 2 T x 2 Name t 2 T lam(x; t) 2 T t 1 2 T t 2 2 T app(t 1 ; t 2 ) 2 T s ... |

20 | A proof of strong normalization for the theory of constructions using a Kripke-like interpretation
- Coquand, Gallier
- 1990
(Show Context)
Citation Context ...mation that might be used in a computation. 4.2 The semantics Since we want to deal with full conversion on open terms and the j-rule, we choose to describe the semantics in a Kripke style model. See =-=[7, 14, 10]-=- for a 7 discussion on Kripke models. A Kripke model is a set of possible worlds, W, with a partial ordering, , of extensions of worlds. We also have a family of ground sets G(w) over possible worlds ... |

20 | Categorical abstract machines for higher-order typed lambdacalculi - Ritter - 1994 |

17 | Formulation of Martin-Löf’s theory of types with explicit substitutions
- Tasistro
- 1993
(Show Context)
Citation Context ...ution is then derivable. In [1] they also have a substitution " which corresponds to a shift on substitutions; this substitution is here defined assc where c 2 [\Gamma; x : A]s\Gamma. In the calc=-=ulus [12, 20]-=- we have as primitives also the thinning rules: M 2 \Gamma ` A c 2 \Deltas\Gamma M 2 \Delta ` A fl 2 \Delta!\Gamma c 2 \Thetas\Delta fl 2 \Theta!\Gamma fl 2 \Delta!\Gamma c 2 \Gammas\Theta fl 2 \Delta... |

16 |
The new implementation of ALF
- Magnusson
- 1992
(Show Context)
Citation Context ...ion closer to the usual implementations. ffl Explicit substitutions also correspond to let-expressions in functional language. ffl There has been a recent interest in using them in proofs, eg. in ALF =-=[11]-=- this is implemented. The present proof makes use of explicit substitutions. 4.1 Definitions of the calculus We define the set of simply typed derivation of -terms with named variables in the ordinary... |

3 | Analysis of simply typed lambda-calculus in ALF - Coquand - 1993 |

2 |
On the Axiom of Extensionality
- Gandy
- 1956
(Show Context)
Citation Context ...ally defined u; v 2 w k\Gamma A Eqw;A (u; v) 2 Set u 2 w k\Gamma A U w;A (u) 2 Set They will both be defined by induction on the types, the idea of this way of defining extensionality is presented in =-=[9]-=-. Two semantic objects of base type are equal if they are intensionally equal and two semantic objects of function type are equal if the application of them to a uniform semantic object is extensional... |

1 |
Kripke Semantical analysis of intutionistic logic 1 Formal systems and recursive functions
- A
- 1965
(Show Context)
Citation Context ...mation that might be used in a computation. 4.2 The semantics Since we want to deal with full conversion on open terms and the j-rule, we choose to describe the semantics in a Kripke style model. See =-=[7, 14, 10]-=- for a 7 discussion on Kripke models. A Kripke model is a set of possible worlds, W, with a partial ordering, , of extensions of worlds. We also have a family of ground sets G(w) over possible worlds ... |

1 |
Substitution calculus. Handwritten notes, Goteborg
- Martin-Lof
- 1992
(Show Context)
Citation Context ... For proving the correctness of these implementations it is not trivial to translate the implementation to the theoretical presentation. A - calculus with explicit substitutions has been suggested in =-=[1, 12]-=- to bring the theoretical presentation closer to the usual implementations. ffl Explicit substitutions also correspond to let-expressions in functional language. ffl There has been a recent interest i... |

1 | Semantical analysis of specification logic. Logics of programs - Tennant - 1985 |