## Sequent Style Proof Terms for HOL

### BibTeX

@MISC{Ridge_sequentstyle,

author = {Tom Ridge},

title = {Sequent Style Proof Terms for HOL},

year = {}

}

### OpenURL

### Abstract

In this work we present proof terms for a Gentzen sequent style presentation of HOL. Existing implementations of proof terms for HOL are natural deduction style systems. Sequent style proof terms have many advantages over natural deduction style proof terms. For example, we can translate proof terms directly into tactics, which we can execute at the tactic level of HOL implementations. We describe several applications of our work, such as an implementation of theory interpretation, and an approach to optimising proof terms by rewriting.

### Citations

164 | Basic Proof Theory
- Troelstra, Schwichtenberg
- 2000
(Show Context)
Citation Context ...crete development: the theorems hold for fixed constants and cannot 3 Actually, classical first order theorem provers will typically produce proofs in a multiple conclusion sequent system such as G3c =-=[ST96]-=-, but the translation from G3c to our system is straight forward: one typically considers multiple conclusions as negated hypotheses.sbe arbitrarily instantiated in different contexts. Theory interpre... |

57 | Some lambda calculus and type theory formalized
- McKinna, Pollack
- 1999
(Show Context)
Citation Context ...lle system has not been mechanised, although certainly this is an option. On the other hand, if one views the Isabelle implementation as a subsystem of pure type systems, then mechanisations do exist =-=[MP99]-=-. Whilst our system has not been mechanised, a certain subsystem of derived rules and proof terms for first order logic has [Pfe00], and proofs about Cut elimination for this system have been carried ... |

54 | Structural cut elimination I. intuitionistic and classical logic
- Pfenning
- 2000
(Show Context)
Citation Context ...prover supplied with the HOL Light distribution, a proof term with size 11 is produced, as expected. Mechanised Subsystem for First Order Logic Pfenning mechanises proofs of Cut elimination in Elf in =-=[Pfe00]-=-. To accomplish this, he first encodes first order sequent derivations as proof terms in LF. This system of sequent style FOL proof terms matches our derived system. For our derived system, Pfenning t... |

37 | Proof Terms for Simply Typed Higher Order Logic
- Berghofer, Nipkow
- 2000
(Show Context)
Citation Context ... for HOL. This system has been implemented in HOL Light, although the system itself is applicable to all HOL implementations. The main innovation compared with previous systems of proof terms [Won95] =-=[BN00]-=- is that we present proof terms for a sequent system (L-style) rather than a natural deduction system (N-style). As a consequence, there are many applications that our system of proof terms support, t... |

36 | Theory Interpretation in Simple Type Theory
- Farmer
- 1994
(Show Context)
Citation Context ...s in the scenario have been implemented in some form. The main applications, which represent novel uses of proof terms and are supporting contributions, are an implementation of theory interpretation =-=[Far94]-=- by proof term transformation, and an implementation of proof optimisation by rewriting. An overview of the paper is as follows. A main requirement for our proof terms is that they be L-style proof te... |

23 | MC2: A module calculus for pure type systems
- Courant
- 2001
(Show Context)
Citation Context ...tial work to be done. For example, having implemented theory interpretation, we now have a good idea of what a module system for HOL should look like. Technically, one can take many of the ideas from =-=[Cou97]-=-. From a user’s point of view, the module system will resemble typical module systems for the programming languages ML and OCaML [Ler04]. The barrier to implementing such a system is the way context i... |

22 |
Notes on Logic and Set Theory
- Johnstone
- 1987
(Show Context)
Citation Context ...odes. Various recent papers [MB00] [JL04] transform proof terms, typically in order to reuse proofs in different settings. In fact, these techniques are all instances of theory interpretation [Far94] =-=[Joh87]-=-. We give a sketch of theory interpretation. The basic idea is to develop a theory axiomatically, and then transfer the results to other theories which satisfy the axioms. In this way, theory interpre... |

16 |
Proof transformations for equational theories
- Nipkow
- 1990
(Show Context)
Citation Context ...g to reimplement higher order matching and related routines. This approach has some connections with program optimisation. We also note that already Nipkow investigated rewriting equational proofs in =-=[Nip90]-=-. Berghofer extends this work to Isabelle proof terms in [Ber03].sHOL4 Isabelle LHOL+ N-style/L-style N-style N-style L-style Standard N Y Y Extensible Y? Y? Y Mechanised Y? Y (related system) Y (deri... |

15 | Recording and checking HOL proofs
- Wong
- 1995
(Show Context)
Citation Context ...of terms for HOL. This system has been implemented in HOL Light, although the system itself is applicable to all HOL implementations. The main innovation compared with previous systems of proof terms =-=[Won95]-=- [BN00] is that we present proof terms for a sequent system (L-style) rather than a natural deduction system (N-style). As a consequence, there are many applications that our system of proof terms sup... |

13 |
Representing higher-order logic proofs in HOL
- Wright
- 1995
(Show Context)
Citation Context ...m with other existing implementations, whose features we tabulate in Fig. 5. In the figure, the column headed “HOL4” refers to the implementation of proof terms based on work originating in Cambridge =-=[vW94]-=- [Won95]. Some versions of HOL Light also use this system. The column headed “Isabelle” refers to the implementation of proof terms for Isabelle by Berghofer [BN00] [Ber03]. The last column headed “LH... |

8 | Theorem reuse by proof term transformation
- Johnsen, Lüth
- 2004
(Show Context)
Citation Context ...instance of a rule. By a nonstructural transformation we mean a transformation that does not alter the shape of the tree, although it may alter the contents of the nodes. Various recent papers [MB00] =-=[JL04]-=- transform proof terms, typically in order to reuse proofs in different settings. In fact, these techniques are all instances of theory interpretation [Far94] [Joh87]. We give a sketch of theory inter... |

6 | Calculus of Constructions and Higher Order Logic, volume 8 of Cahiers du Centre de Logique
- Geuvers
(Show Context)
Citation Context ...via cut-elimination) are identified. . . If one is looking for these kind of applications, it is much more promising to use the ‘coding’ of a logic in a relatively weak framework like Automath or LF. =-=[Geu94]-=- Coding a logic in this way leads to an embedding of proofs in a lambda calculus like language. A fine example of this approach is Pfenning’s encoding of sequent calculi in Elf [Pfe00]. Pfenning then ... |

3 |
Programs and Executable Specifications in Higher Order Logic
- Proofs
- 2004
(Show Context)
Citation Context ...s approach has some connections with program optimisation. We also note that already Nipkow investigated rewriting equational proofs in [Nip90]. Berghofer extends this work to Isabelle proof terms in =-=[Ber03]-=-.sHOL4 Isabelle LHOL+ N-style/L-style N-style N-style L-style Standard N Y Y Extensible Y? Y? Y Mechanised Y? Y (related system) Y (derived system) Binding N Y Y Reduction N Y N LF style embedding N N... |

3 | Changing data structures in type theory: A study of natural numbers
- Magaud, Bertot
- 1999
(Show Context)
Citation Context ...nts an instance of a rule. By a nonstructural transformation we mean a transformation that does not alter the shape of the tree, although it may alter the contents of the nodes. Various recent papers =-=[MB00]-=- [JL04] transform proof terms, typically in order to reuse proofs in different settings. In fact, these techniques are all instances of theory interpretation [Far94] [Joh87]. We give a sketch of theor... |

1 |
Theory interpretation by proof term transformation
- Ridge
- 2005
(Show Context)
Citation Context ... that proof term transformation can be expensive, and the tools are not yet ready for the non-expert user. We describe theory interpretation, and an earlier version of this work, in greater detail in =-=[Rid05]-=-. Structural Transformation Structural transformations are those proof term transformations that potentially alter the shape of the proof. Cut elimination is an example of a structural transformation.... |