## Iterative circular coinduction for CoCasl in Isabelle/HOL (2005)

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Venue: | FUNDAMENTAL APPROACHES TO SOFTWARE ENGINEERING, VOLUME 3442 OF LECT. NOTES COMPUT. SCI |

Citations: | 6 - 1 self |

### BibTeX

@INPROCEEDINGS{Hausmann05iterativecircular,

author = {Daniel Hausmann and Till Mossakowski and Lutz Schröder},

title = {Iterative circular coinduction for CoCasl in Isabelle/HOL},

booktitle = {FUNDAMENTAL APPROACHES TO SOFTWARE ENGINEERING, VOLUME 3442 OF LECT. NOTES COMPUT. SCI},

year = {2005},

pages = {341--356},

publisher = {Springer}

}

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### Abstract

Coalgebra has in recent years been recognized as the framework of choice for the treatment of reactive systems at an appropriate level of generality. Proofs about the reactive behavior of a coalgebraic system typically rely on the method of coinduction. In comparison to ‘traditional ’ coinduction, which has the disadvantage of requiring the invention of a bisimulation relation, the method of circular coinduction allows a higher degree of automation. As part of an effort to provide proof support for the algebraic-coalgebraic specification language CoCasl, we develop a new coinductive proof strategy which iteratively constructs a bisimulation relation, thus arriving at a new variant of circular coinduction. Based on this result, we design and implement tactics for the theorem prover Isabelle which allow for both automatic and semiautomatic coinductive proofs. The flexibility of this approach is demonstrated by means of examples of (semi-)automatic proofs of consequences of Co-Casl specifications, automatically translated into Isabelle theories by means of the Bremen heterogeneous Casl tool set Hets.

### Citations

729 |
Isabelle/HOL: A Proof Assistant for Higher-Order Logic
- Nipkow, Paulson, et al.
- 2003
(Show Context)
Citation Context ...pes and coinductive process types. This work forms part of an effort to provide proof support for CoCasl. To this end, an existing embedding of Casl into the semiautomatic theorem prover Isabelle/HOL =-=[9]-=- has been extended to CoCasl, so that proofs about CoCasl specifications can now be conducted in a well-developed higher order logical environment. This embedding is the basis for the development of a... |

43 |
Universal Coalgebra: A Theory of Systems, Theoretical Computer Science 249
- Rutten
- 2000
(Show Context)
Citation Context ...ranslated into Isabelle theories by means of the Bremen heterogeneous Casl tool set Hets. Introduction Coalgebra is emerging as a standard unifying framework for the specification of reactive systems =-=[11]-=-, complementing the use of universal algebra for the specification of the functional correctness of programs. Following this paradigm, several coalgebraic specification languages have recently been de... |

18 | Conditional Circular Coinductive Rewriting with Case Analysis
- Goguen, Lin, et al.
- 2003
(Show Context)
Citation Context ...roved semiautomatically because explicit case distinction needs to be performed in the course of the proof (an approach for further automation of proofs which involve case distinction is described in =-=[4]-=-). Using this theorem in simplification, the goals tick = flip(tock) (where tick and tock are the two alternating bitstreams) and flip(flip(b)) = b can be proved automatically. Another point where the... |

14 |
Casl user manual
- Bidoit, Mosses
- 2004
(Show Context)
Citation Context ...tion Language CCSL, which is geared towards object oriented programs, and the algebraic-coalgebraic specification language CoCasl [7], which extends the standard algebraic specification language Casl =-=[1, 8]-=- and thus allows not only the specification of both functional and reactive requirements, but also the intercombination of inductive datatypes and coinductive process types. This work forms part of an... |

14 |
Algebraic-coalgebraic specification
- Mossakowski, Schröder, et al.
- 2006
(Show Context)
Citation Context ...ages have recently been designed, e.g. the Coalgebraic Class Specification Language CCSL, which is geared towards object oriented programs, and the algebraic-coalgebraic specification language CoCasl =-=[7]-=-, which extends the standard algebraic specification language Casl [1, 8] and thus allows not only the specification of both functional and reactive requirements, but also the intercombination of indu... |

11 | Using a generalisation critic to find bisimulations for coinductive proofs
- Dennis, Bundy, et al.
- 1997
(Show Context)
Citation Context ...tions written in full first-order (and even higher-order) logic, not just conditional equations. Moreover, while the degree of automation that we achieve is comparable to that of BOBJ [10] and CoClam =-=[3]-=-, the availability ofssemiautomatic tactics means that user interaction may help to complete proofs that fail with a completely automatic proof procedure (in particular, missing lemmas appear as open ... |

8 |
HetCasl - heterogeneous specification. Language summary
- Mossakowski
- 2004
(Show Context)
Citation Context ...ation: A more general proof goal is stated, which one may then, in turn, attempt to solve with the algorithm. 4 Iterative Coinduction in Isabelle/HOL As part of the Bremen heterogeneous tool set Hets =-=[6, 5]-=-, a translation of CoCasl specifications into Isabelle/HOL theories has been implemented in order to allow for the interactive proving of properties of reactive systems (see e.g. Figure 2). This inclu... |

7 | Matching logic
- Rosu, Schulte
- 2009
(Show Context)
Citation Context ... the problem that standard coinduction requires the invention of a bisimulation relation. A variant of coinduction that lends itself more easily to mechanization is the method of circular coinduction =-=[10]-=-, which works by ‘reducing the claim to itself’ adhering to certain restrictions in the permissible proof steps. Here, we introduce an implementation of a related proof method where the bisimulation i... |

2 |
Proof Planning Coinduction
- Dennis
- 1999
(Show Context)
Citation Context ... of the Bisimulation A first approach to the construction of a bisimulation R in coinductive proofs is as follows. Given a proof goal ∀X. t1 = t2, 1. Let R = {(x, y) | ∃X. x = t1 ∧ y = t2} (following =-=[2]-=-, we call R the current trial bisimulation). 2. Try to prove ObsΘ(R) ⊂ ∆ and TransΘ(R) ⊂ R (i.e. try to show that R is a Θ-bisimulation). 3. If this succeeds, the proof is finished. However, this appr... |

1 |
specification and the heterogeneous tool set, Habilitation thesis (draft
- Heterogeneous
- 2004
(Show Context)
Citation Context ...ation: A more general proof goal is stated, which one may then, in turn, attempt to solve with the algorithm. 4 Iterative Coinduction in Isabelle/HOL As part of the Bremen heterogeneous tool set Hets =-=[6, 5]-=-, a translation of CoCasl specifications into Isabelle/HOL theories has been implemented in order to allow for the interactive proving of properties of reactive systems (see e.g. Figure 2). This inclu... |