## A Calculus for and Termination of Rippling (1996)

Venue: | Journal of Automated Reasoning |

Citations: | 41 - 2 self |

### BibTeX

@ARTICLE{Basin96acalculus,

author = {David A. Basin and Toby Walsh},

title = {A Calculus for and Termination of Rippling},

journal = {Journal of Automated Reasoning},

year = {1996},

volume = {16},

pages = {147--180}

}

### Years of Citing Articles

### OpenURL

### Abstract

. Rippling is a type of rewriting developed for inductive theorem proving that uses annotations to direct search. Rippling has many desirable properties: for example, it is highly goal directed, usually involves little search, and always terminates. In this paper we give a new and more general formalization of rippling. We introduce a simple calculus for rewriting annotated terms, close in spirit to first-order rewriting, and prove that it has the formal properties desired of rippling. Next we develop criteria for proving the termination of such annotated rewriting, and introduce orders on annotated terms that lead to termination. In addition, we show how to make rippling more flexible by adapting the termination orders to the problem domain. Our work has practical as well as theoretical advantages: it has led to a very simple implementation of rippling that has been integrated in the Edinburgh CLAM system. Key words: Mathematical Induction, Inductive Theorem Proving, Term Rewriting. ...

### Citations

530 |
A computational logic
- Boyer, Moore
- 1979
(Show Context)
Citation Context ...ction conclusion and hypothesis while leaving their common structure preserved; this is in contrast to rewriting based on normalization, which is used in other inductive theorem provers such as NQTHM =-=[4]-=-. Rippling also involves little search, since annotations severely restrict rewriting. Funded by the German Ministry for Research and Technology under grant ITS 9102. y Supported by a Human Capital an... |

453 | Termination of rewriting
- Dershowitz
- 1985
(Show Context)
Citation Context ...This is without loss of generality as the proofs below do not depend on the order of wave-holes. 4.1. Ground Rewriting We first consider rewriting using ground rewrite rules. As is typical (e.g., see =-=[11] Section I-=-I), we distinguish between two kinds of variables: those in rewrite rules and those in terms. We treat the later kind, "term variables", as constants. We begin by redefining subterm replacem... |

265 | Orderings for term rewriting systems
- DERSHOWITZ
- 1982
(Show Context)
Citation Context ...s position but only its form. Hence we want this to be decreasing under some ordering on the contents of wave-fronts. There are many such orderings; here we take ?wf to be the recursive path ordering =-=[12]-=- on the terms in the wave-front where !? has a higher precedence than :: and all other function symbols have an equivalent but lower priority. The measure of the LHS of (27) is now greater than that o... |

162 | A.: Rippling: A Heuristic for Guiding Inductive Proofs
- Bundy, Stevens, et al.
- 1993
(Show Context)
Citation Context ...as been integrated in the Edinburgh CLAM system. Key words: Mathematical Induction, Inductive Theorem Proving, Term Rewriting. 1. Introduction Rippling is a form of rewriting developed by Bundy et al =-=[6, 8]-=- which uses annotations to restrict rewriting and to guide the derivation towards a particular goal. Rippling applies naturally in inductive theorem proving where the induction conclusion typically di... |

126 |
The Oyster-Clam system
- Bundy, Harmelen, et al.
- 1990
(Show Context)
Citation Context ... in forward directed theorem proving. Wave-rules can also ripple wave-fronts inwards, or change the orientation of wave-fronts from out to in (but not in to out). (U !? V ) !? W # ! U !? ( V !? W # ) =-=(7) ( U -=-!? V " ) !? W ! U !? ( V !? W # ) (8) A simple example of a proof guided by rippling is the proof of the associativity of multiplication (x \Theta y) \Theta z = x \Theta (y \Theta z) : (9) The pr... |

97 | Productive use of failure in inductive proof
- Ireland, Bundy
- 1995
(Show Context)
Citation Context ...by Walsh Indeed, the use of annotation places such strong restrictions on the search space that it is often possible to analyze failed rippling proofs and to suggest missing lemmas or generalizations =-=[15]-=-. In this paper we give a new account of rippling and its properties that is both substantially simpler and more general than previous accounts. Conceptually, the starting point of our work is the for... |

93 | Experiments with proof plans for induction
- Bundy, Harmelen, et al.
- 1991
(Show Context)
Citation Context ...vity, distributivity, and cancellation (!? is infix append on lists). ( U + V " ) \Theta W ! U \Theta W + V \Theta W " (3) (U !? V ) " !? W ! U !? (V !? W ) " (4) U !? ( V !? W &qu=-=ot; ) ! (U !? V ) !? W " (5) U + V " =-=- W + Z " ! U = WsV = Z " (6) (4) and (5) demonstrate that rules like associativity can be wave-rules in both directions; the precondition on rippling that annotations in wave-rules match tho... |

61 |
Guiding inductive proofs
- Hutter
- 1990
(Show Context)
Citation Context ...eptable under their definition are acceptable under ours. Moreover simple examples are wave-rules under our formalism but not theirs, e.g., the base-case of addition 0 + x " ! x. 8.2. INKA Hutter=-=, in [14, 13]-=-, describes a calculus for rippling implemented in the INKA system [3]. Hutter rigorously develops an algebra of annotated terms, called Cterms. These are terms in an extended signature where function... |

44 |
Using meta-level inference for selective application of multiple rewrite rule sets in algebraic manipulation
- Bundy, Welham
- 1981
(Show Context)
Citation Context ...and to guide equational reasoning [10]. However, new domains, especially non-inductive ones, require new orderings to guide proof construction. Here we sketch an application based on the PRESS system =-=[9]-=-. 3 To solve algebraic equations, PRESS uses a set of methods which apply rewrite rules. The three main methods are: isolation, collection, and attraction. Below are examples of rewrite rules used by ... |

37 |
Extensions to the rippling-out tactic for guiding inductive proofs
- Bundy, Harmelen, et al.
- 1990
(Show Context)
Citation Context ...as been integrated in the Edinburgh CLAM system. Key words: Mathematical Induction, Inductive Theorem Proving, Term Rewriting. 1. Introduction Rippling is a form of rewriting developed by Bundy et al =-=[6, 8]-=- which uses annotations to restrict rewriting and to guide the derivation towards a particular goal. Rippling applies naturally in inductive theorem proving where the induction conclusion typically di... |

32 |
The Karlsruhe induction theorem proving system
- Biundo, Hummel, et al.
- 1986
(Show Context)
Citation Context ...xamples are wave-rules under our formalism but not theirs, e.g., the base-case of addition 0 + x " ! x. 8.2. INKA Hutter, in [14, 13], describes a calculus for rippling implemented in the INKA sy=-=stem [3]. Hutter r-=-igorously develops an algebra of annotated terms, called Cterms. These are terms in an extended signature where functions and variables each carry a "color", which represents annotation, or ... |

27 | Difference unification
- Basin, Walsh
- 1992
(Show Context)
Citation Context ...marily to prove inductive theorems, it has recently been applied to other problem domains. For example, it has been used to sum series [16], to prove limit theorems [17], and to perform normalization =-=[1]-=-. In rippling, as in conventional rewriting, the termination ordering can be made domain dependent. We illustrate this idea by two new orderings. A practical contribution of our work is that it greatl... |

24 | The use of proof plans to sum series
- Walsh, Nunes, et al.
- 1992
(Show Context)
Citation Context ... extend the power of rippling. Although rippling was designed primarily to prove inductive theorems, it has recently been applied to other problem domains. For example, it has been used to sum series =-=[16]-=-, to prove limit theorems [17], and to perform normalization [1]. In rippling, as in conventional rewriting, the termination ordering can be made domain dependent. We illustrate this idea by two new o... |

14 | A methodology for equational reasoning
- Cleve, Hutter
- 1994
(Show Context)
Citation Context ...blem Solving Rippling has found several novel uses besides inductive theorem proving. For example, it has been used to sum series [16], to prove limit theorems [17], and to guide equational reasoning =-=[10]-=-. However, new domains, especially non-inductive ones, require new orderings to guide proof construction. Here we sketch an application based on the PRESS system [9]. 3 To solve algebraic equations, P... |

11 | D.: Coloured rippling: An extension of a theorem proving heuristic
- Yoshida, Bundy, et al.
- 1994
(Show Context)
Citation Context ... Although rippling was designed primarily to prove inductive theorems, it has recently been applied to other problem domains. For example, it has been used to sum series [16], to prove limit theorems =-=[17]-=-, and to perform normalization [1]. In rippling, as in conventional rewriting, the termination ordering can be made domain dependent. We illustrate this idea by two new orderings. A practical contribu... |

9 | Termination Orderings for Rippling
- Basin, Walsh
- 1994
(Show Context)
Citation Context ...e efficiently. In particular when all annotation is simple (single wave-holes) it is possible to orient the right-hand side in linear time (in the size of the term). An algorithm for this is given in =-=[2]-=-. 6.2. Sinks and Colors One kind of annotation we have not discussed in our measures or parsing is sinks. This is deliberate as we can safely ignore sinks in both the measure and the parser. Sinks onl... |

4 |
Colouring terms to control equational reasoning. An Expanded Version of PhD Thesis: Mustergesteuerte Strategien fur Beweisen von Gleichheiten (Universitat Karlsruhe
- Hutter
- 1991
(Show Context)
Citation Context ...eptable under their definition are acceptable under ours. Moreover simple examples are wave-rules under our formalism but not theirs, e.g., the base-case of addition 0 + x " ! x. 8.2. INKA Hutter=-=, in [14, 13]-=-, describes a calculus for rippling implemented in the INKA system [3]. Hutter rigorously develops an algebra of annotated terms, called Cterms. These are terms in an extended signature where function... |