## Hierarchical Reflection

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Citations: | 3 - 3 self |

### BibTeX

@MISC{Cruz-filipe_hierarchicalreflection,

author = {Luís Cruz-filipe},

title = {Hierarchical Reflection},

year = {}

}

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### Abstract

Abstract. The technique of reflection is a way to automate proof construction in type theoretical proof assistants. Reflection is based on the definition of a type of syntactic expressions that gets interpreted in the domain of discourse. By allowing the interpretation function to be partial or even a relation one gets a more general method known as ``partial reflection''. In this paper we show how one can take advantage of the partiality of the interpretation to uniformly define a family of tactics for equational reasoning that will work in different algebraic structures. The tactics then follow the hierarchy of those algebraic structures in a natural way.

### Citations

90 | The semantics of reflected proof
- Allen, Constable, et al.
- 1990
(Show Context)
Citation Context ... the logic of the theorem prover, by formalizing relevant meta-theory. Reflection is a common approach for proof automation in type theoretical systems like NuPRL and Coq, as described for example in =-=[1]-=- and [10] respectively. Another name for reflection is “the two-level approach”. In Nijmegen we formalized the Fundamental Theorems of Algebra and Calculus in Coq, and then extended these formalizatio... |

36 | A Tactic Language for the System Coq
- Delahaye
- 2000
(Show Context)
Citation Context ...sion in A (dotted arrows) is not definable in the type theory, and needs to be implemented outside of it. In a system like Coq it will be implemented in ML or in the tactic language Ltac described in =-=[6]-=- and [4, Chapter 9]. Things get more interesting when the syntactic expressions in E contain partial operations, like division. In that case the interpretation [[e]]ρ will not always be defined. To ad... |

29 | Induction-recursion and initial algebras
- Dybjer, Setzer
- 2003
(Show Context)
Citation Context ...ve definitions are not supported by the Coq system, and for a good reason: induction-recursion makes a system significantly stronger. In set theory it corresponds to the existence of a Mahlo cardinal =-=[8]-=-. The solution from [10] for doing partial reflection without induction-recursion is to replace the interpretation function with an inductively defined interpretation relation. ][ρ ⊆ E × A The relatio... |

28 |
Investigations into intensional type theory
- Streicher
- 1993
(Show Context)
Citation Context ... a hierarchical way. Finally in Section 6 we present a possibility to have even tighter integration in a hierarchical reflection tactic, which unfortunately turns out to require the so-called K axiom =-=[14]-=-. 2 Reflection and Partial Reflection In this section we will briefly summarize [10]. That paper describes a generalization of the technique of reflection there called partial reflection. One can give... |

20 | F.: C-CoRN, the constructive coq repository at Nijmegen
- Cruz-Filipe, Geuvers, et al.
- 2004
(Show Context)
Citation Context ... approach”. In Nijmegen we formalized the Fundamental Theorems of Algebra and Calculus in Coq, and then extended these formalizations into a structured library of mathematics named the C-CoRN library =-=[3, 5]-=-. For this library we defined a reflection tactic called rational that automatically establishes equalities of rational expressions in a field by bringing both to the same side of the equal sign and t... |

14 | The algebraic hierarchy of the FTA Project
- Geuvers, Pollack, et al.
- 2002
(Show Context)
Citation Context ...cal Reflection The normalization procedure described in Section 3 was used to define a tactic which would prove algebraic equalities in an arbitrary field in the context of the Algebraic Hierarchy of =-=[9]-=-. In this hierarchy, fields are formalized as rings with an extra operation (division) which satisfies some properties; rings, in turn, are themselves Abelian groups where a multiplication is defined ... |

13 |
Field: une procédure de décision pour les nombres réels en Coq
- Delahaye, Mayero
- 2001
(Show Context)
Citation Context ...tion and division, operations that do not make sense in a group. 1.3 Related Work In the C-CoRN setoid framework, rational is the equivalent of the standard Coq tactic field for Leibniz equality (see =-=[7]-=- and [4, Chapter 8.11]). Both tactics were developed at about the same time. The field tactic is a generalization of the Coq ring tactic [4, Chapter 19], so with the field and ring tactics the duplica... |

11 | Equational reasoning via partial reflection
- Geuvers, Wiedijk, et al.
- 2000
(Show Context)
Citation Context ...ic of the theorem prover, by formalizing relevant meta-theory. Reflection is a common approach for proof automation in type theoretical systems like NuPRL and Coq, as described for example in [1] and =-=[10]-=- respectively. Another name for reflection is “the two-level approach”. In Nijmegen we formalized the Fundamental Theorems of Algebra and Calculus in Coq, and then extended these formalizations into a... |

9 | Equational reasoning via partial re - Geuvers, Wiedijk, et al. - 2000 |

6 | The HOL Light manual (1.1 - Harrison - 2000 |

6 | Formalizing abstract algebra in type theory with dependent records
- Yu, Nogin, et al.
(Show Context)
Citation Context ...ques that allow code reuse for tactics in MetaPRL, although the ideas therein are different from ours. Since the library of this system also includes an algebraic hierarchy built using subtyping (see =-=[15]-=-), it seems reasonable to expect that the work we describe could be easily adapted to that framework.s1.4 Contribution Hierarchical Reflection 3 We show that it is possible to have one unified mechani... |

5 | The Semantics of Re Proof - Allen, Constable, et al. - 1990 |

4 |
The Groupoid Interpretation of Type Theory
- Homan, Streicher
- 1996
(Show Context)
Citation Context ...n dependent pairs where what one needs is equality between the second components of those pairs. In Coq this is not derivable without the so-called K axiom, which states uniqueness of equality proofs =-=[13]-=-. forall (A:Set) (x:A) (p:(x=x)), p = refl_equal A x We did not want to assume an axiom to be able to have our tactic prove equalities in algebraic structures that are clearly provable without this ax... |

1 |
and Femke van Raamsdonk. Constructor subtyping in the Calculus of Inductive Constructions
- Barthe
- 2000
(Show Context)
Citation Context ... to prove these lemmas using inversion, there might be an alternative way to prove them that avoids this problem. A different approach to the same problem would be to use the constructor subtyping of =-=[2]-=-. This would allow one to define e.g. the interpretation relation for rings ][ R ρ by adding one constructor to that for groups ][ G ρ ; proving the relevant lemmas for the broader relation would then... |