## Ordinal Arithmetic: A Case Study for Rippling in a Higher Order Domain (2001)

### Cached

### Download Links

- [www.cs.nott.ac.uk]
- [www.informatics.ed.ac.uk]
- DBLP

### Other Repositories/Bibliography

Venue: | In TPHOLs’01, volume 2152 of LNCS |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Dennis01ordinalarithmetic:,

author = {Louise A. Dennis and Alan Smaill},

title = {Ordinal Arithmetic: A Case Study for Rippling in a Higher Order Domain},

booktitle = {In TPHOLs’01, volume 2152 of LNCS},

year = {2001},

pages = {185--200},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper reports a case study in the use of proof planning in the context of higher order syntax. Rippling is a heuristic for guiding rewriting steps in induction that has been used successfully in proof planning inductive proofs using first order representations. Ordinal arithmetic provides a natural set of higher order examples on which transfinite induction may be attempted using rippling. Previously Boyer-Moore style automation could not be applied to such domains. We demonstrate that a higher-order extension of the rippling heuristic is sufficient to plan such proofs automatically. Accordingly, ordinal arithmetic has been implemented in Clam, a higher order proof planning system for induction, and standard undergraduate text book problems have been successfully planned. We show the synthesis of a fixpoint for normal ordinal functions which demonstrates how our automation could be extended to produce more interesting results than the textbook examples tried so far.

### Citations

940 |
Term Rewriting and All That
- Baader, Nipkow
- 1998
(Show Context)
Citation Context ...appears between them as an embedding 6 tree with dashed lines { the address label of the nodes is also shown. The dotted arrows illustrate how the embedding tree links the two terms. lambda x x [2] + =-=[1]-=- f [1] f [2,1,1] y [2,1,2] x [2,2] lambda x lambda y [1,2] application application [] application [] [1,2] [2,1,1] [2,1,2] Fig. 3. An Embedding The embedding tree for this is (node [1, 2] [(leaf [1, 1... |

395 |
A Computational Logic Handbook
- Boyer, Moore
- 1988
(Show Context)
Citation Context ... spirit to our presentation is the introduction of ordinals in the Coq system, though again user guidance is assumed in building proofs as a new datatype. The system, ACL2, described by Boyer & Moore =-=[3] mak-=-es use of induction over ordinals to strengthen the proof system. However, this feature is hidden in the system's \black box" treatment used to check termination of user-dened 13 functions, and i... |

265 | The use of explicit plans to guide inductive proofs
- Bundy
- 1988
(Show Context)
Citation Context ...ains higher order aspects. Not all theorems in standard arithmetic hold for ordinal numbers making it genuinely distinct from thesnite case. 3 Proof Planning Proof planning wassrst suggested by Bundy =-=[4]-=-. A proof plan is a proof of a theorem, at some level of abstraction, presented as a tree. A proof plan is generated using AI-style planning techniques. The planning operators used by a proof planner ... |

162 | A.: Rippling: A Heuristic for Guiding Inductive Proofs
- Bundy, Stevens, et al.
- 1993
(Show Context)
Citation Context ... We will discuss the wave method in more detail since it was a new formulation of this method we wished to investigate. The wave method embodies the rippling heuristic. Rippling wassrst introduced in =-=[5]-=-. We use the theory as presented by Smaill & Green [20] who proposed a version that naturally coped with higher-order features. Rippling steps apply rewrite rules to a target term which is associated ... |

147 |
LEGO Proof Development System: User's Manual
- Luo, Pollack
- 1992
(Show Context)
Citation Context ...ered linear orders. As such they are a generalisation of the usual natural numbers. We work with so-called ordinal notations, as in [14]; this formalisation was carried out in the Lego theorem prover =-=[15-=-] working in a constructive type theory. These notations give us a language that allows us to speak about some (but not all) of the ordinals. An inductively dened type for ordinal notations is given b... |

126 |
Naive Set Theory
- Halmos
- 1960
(Show Context)
Citation Context ...ithmetic identities for natural numbers still hold for ordinals (commutativity of addition fails for example), and we have to be more careful about the arguments used for recursive denitions. Halmos [=-=12]-=- gives a good account of the ideas involved. In brief, ordinal arithmetic presents itself as an area where induction applies but one which contains higher order aspects. Not all theorems in standard a... |

81 |
Axiomatic Set Theory
- Suppes
- 1960
(Show Context)
Citation Context ...hings, for its use in aiding termination proofs (see [7]), and in classifying proof-theoretic complexity. The system was used to attempt to plan proofs of examples appearing in a number of text books =-=[21, 9, 13]-=- with encouraging results. We were also able to synthesize asxpoint for a normal function. The emphasis is on the automated control of proof search, and we aim for both a declarative account of this c... |

78 | IMPS: An interactive mathematical proof system
- Farmer, Guttman, et al.
- 1990
(Show Context)
Citation Context ...on for natural numbers except that the recursion is dened on a non-standard argument: x + 0 = x; (7) x + s(y) = s(x + y); (8) These become the rewrite rules X + 0 (:) X; (9) X + s(Y ) (:) s(X + Y ); (=-=-=-10) in Clam . These rules can be used in either direction during rippling but only from left to right in symbolic evaluation. Clam allows a user to state exactly which lemmas and denitions they wish t... |

58 |
Elements of Set Theory
- Enderton
- 1977
(Show Context)
Citation Context ...hings, for its use in aiding termination proofs (see [7]), and in classifying proof-theoretic complexity. The system was used to attempt to plan proofs of examples appearing in a number of text books =-=[21, 9, 13]-=- with encouraging results. We were also able to synthesize asxpoint for a normal function. The emphasis is on the automated control of proof search, and we aim for both a declarative account of this c... |

58 | System description: proof planning in higher-order logic with Lambda-Clam
- Richardson, Smaill, et al.
- 1998
(Show Context)
Citation Context ...lan is that for induction with the associated rippling heuristic (a form of rewriting constrained to be terminating by meta-logical annotations). This was implemented in the Clam proof planner. Clam [=-=1-=-9] is a higher order descendant of Clam and was the chosen system for this case study. 3.1 Proof Planning in Clam Proof planning in Clam works as follows: A goal is presented to the system. This goal ... |

42 | The automation of proof by mathematical induction
- Bundy
(Show Context)
Citation Context ...tautology checking, generalisation of common subterms and also symbolic evaluation and the induction strategy (ind strat). Within the induction strategy, the induction method performs ripple analysis =-=[-=-6] to choose an induction scheme (from a selection specied in Clam's theories) and produces subgoals for base and step cases. The base cases are passed out to the repeat of the top level strategy. The... |

41 | A Calculus for and Termination of Rippling
- BASIN, WALSH
- 1996
(Show Context)
Citation Context ...olic Evaluation Other Methods... Induction_top_meth (Base Cases) (Step Cases) Ind_strat Induction_meth step_case Fertilise wave_method Fig. 1. The Proof Strategy for Induction Rippling is terminating =-=-=-[2]. Rippling either moves dierences outwards in the term structure so that they can be cancelled away or inwards so that the dierences surround a universally quantied variable (or sink ). If it is po... |

41 | Set theory for verification: II. Induction and recursion
- Paulson
- 1995
(Show Context)
Citation Context ...gherorder proof planning systems in the ordinal domain. 14s6 Related and Further Work Different mechanisations of reasoning about ordinals and cardinals have been carried out previously. For example, =-=[13]-=- introduces ordinals in the course of a development of set theory. While providing the foundational assurance of a development from first principles, this work assumes a fair amount of user interactio... |

40 |
Une Théorie des Constructions
- Coquand
- 1985
(Show Context)
Citation Context ... constructive type theory. These notations give us a language that allows us to speak about some (but not all) of the ordinals. An inductively dened type for ordinal notations is given by Coquand in [=-=8]-=- as follows: datatype ordinal = 0 | s of ordinal | sup of (nat -> ordinal) where we understand the Sup operation to refer to the least upper bound of the ordinals in the range of its argument. This de... |

20 | Higher-order annotated terms for proof search
- Smaill, Green
- 1996
(Show Context)
Citation Context ...it was a new formulation of this method we wished to investigate. The wave method embodies the rippling heuristic. Rippling wassrst introduced in [5]. We use the theory as presented by Smaill & Green =-=[20]-=- who proposed a version that naturally coped with higher-order features. Rippling steps apply rewrite rules to a target term which is associated with a skeleton and an embedding that relates the skele... |

16 | A logic programming approach to implementing higher-order term rewriting
- Felty
- 1992
(Show Context)
Citation Context ...on to the target term (e.g. rippling rewrites an induction conclusion which has an induction hypothesis embedded in it). In the present context, we make use of higher order rewriting, in the style of =-=[11-=-]. After rewriting a new embedding of the skeleton into the rewritten term is calculated. There is a measure on embeddings and any rewriting step must reduce this embedding measure (written ass ). Thi... |

16 | abczewski. Mechanizing set theory. Cardinal arithmetic and the axiom of choice
- Paulson, Gr
- 1996
(Show Context)
Citation Context .... While providing the foundational assurance of a development fromsrst principles, this work assumes a fair amount of user interaction in building up proofs. A further development in this style is in =-=[18]-=-. Closer in spirit to our presentation is the introduction of ordinals in the Coq system, though again user guidance is assumed in building proofs as a new datatype. The system, ACL2, described by Boy... |

7 |
1991], Satis of systems of ordinal notations with the subterm property is decidable
- Jouannaud, Okada
(Show Context)
Citation Context ...son. Ordinals can be thought of as (equivalence classes of) well-ordered linear orders. As such they are a generalisation of the usual natural numbers. We work with so-called ordinal notations, as in =-=[14]-=-; this formalisation was carried out in the Lego theorem prover [15] working in a constructive type theory. These notations give us a language that allows us to speak about some (but not all) of the o... |

6 |
Sets and Axioms: The Apparatus of Mathematics
- Numbers
- 1982
(Show Context)
Citation Context ...hings, for its use in aiding termination proofs (see [7]), and in classifying proof-theoretic complexity. The system was used to attempt to plan proofs of examples appearing in a number of text books =-=[21, 9, 13]-=- with encouraging results. We were also able to synthesize asxpoint for a normal function. The emphasis is on the automated control of proof search, and we aim for both a declarative account of this c... |

5 | An ordinal calculus for proving termination in term rewriting
- Cichon, Touzet
- 1996
(Show Context)
Citation Context ...ports on using Clam to plan proofs about ordinal arithmetic, making use of higher order features. Ordinal arithmetic is of interest, among other things, for its use in aiding termination proofs (see [=-=7]-=-), and in classifying proof-theoretic complexity. The system was used to attempt to plan proofs of examples appearing in a number of text books [21, 9, 13] with encouraging results. We were also able ... |

4 |
Set theory for veri II. induction and recursion
- Paulson
- 1995
(Show Context)
Citation Context ...y of Problems First we present the background to the problem area. Traditionally the theory of ordinals is presented within set theory; a machine-checked presentation following this route is given in =-=[17]-=-. We instead present a computationally more direct theory of ordinals in which to reason. Ordinals can be thought of as (equivalence classes of) well-ordered linear orders. As such they are a generali... |

3 |
PVS : An integrated approach to specification and verification
- Owre, Rushby, et al.
- 1992
(Show Context)
Citation Context ... strengthen the proof system. However, this feature is hidden in the system’s “black box” treatment used to check termination of user-defined functions, and is not directly accessible by the use=-=r. PVS[12] -=-also contains a construction of the ordinals up to ɛ 0 and a well-foundedness proof for the associated order based on the development in ACL2. There are a number of extensions we would like to make t... |

1 |
PVS : An integrated approach to speci and veri
- Owre, Rushby, et al.
- 1992
(Show Context)
Citation Context ...rengthen the proof system. However, this feature is hidden in the system's \black box" treatment used to check termination of user-dened 13 functions, and is not directly accessible by the user. =-=PVS [16-=-] also contains a construction of the ordinals up to 0 and a well-foundedness proof for the associated order based on the development in ACL2. check this { is the wellfoundedness simply asserted? The... |