## Calculational Reasoning Revisited -- An Isabelle/Isar experience (2001)

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Venue: | THEOREM PROVING IN HIGHER ORDER LOGICS: TPHOLS 2001 |

Citations: | 17 - 6 self |

### BibTeX

@INPROCEEDINGS{Bauer01calculationalreasoning,

author = {Gertrud Bauer and Markus Wenzel},

title = {Calculational Reasoning Revisited -- An Isabelle/Isar experience},

booktitle = {THEOREM PROVING IN HIGHER ORDER LOGICS: TPHOLS 2001},

year = {2001},

pages = {75--90},

publisher = {Springer-Verlag}

}

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### Abstract

We discuss the general concept of calculational reasoning within Isabelle/Isar, which provides a framework for high-level natural deduction proofs that may be written in a human-readable fashion. Setting out from a few basic logical concepts of the underlying meta-logical framework of Isabelle, such as higher-order unification and resolution, calculational commands are added to the basic Isar proof language in a flexible and non-intrusive manner. Thus calculational proof style may be combined with the remaining natural deduction proof language in a liberal manner, resulting in many useful proof patterns. A case-study on formalizing Computational Tree Logic (CTL) in simply-typed set-theory demonstrates common calculational idioms in practice.

### Citations

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539 |
Symbolic Model Checking: An Approach to the State Explosion Problem
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(Show Context)
Citation Context ...s of CTL In order to demonstrate how the idiomatic expressions of calculational reasoning are used in practice, we present a case-study of formalizing basic concepts of Computational Tree Logic (CTL) =-=[10, 9] wit-=-hin the simply-typed set theory of HOL. 2 The proofs are mostly by algebraic reasoning over basic set operations. 4.1 CTL formulae By using the common technique of “shallow embedding”, a CTL formu... |

433 | Isabelle : a generic theorem prover
- Paulson, Nipkow
- 1994
(Show Context)
Citation Context ...s, we shall see how the above example works out within the Isar proof language. 1 First of all, we observe that it could be easily finished by a single stroke of an automated proof method of Isabelle =-=[13]. th-=-eorem � {X , Y , {}} = � {X , Y } by auto In fact, many calculations in the Mizar library are rather trivial from the perspective of automated proof tools available in Isabelle, HOL, PVS etc., whi... |

84 | A Generic Interpretative Approach to Readable Formal Proof Documents
- Wenzel
(Show Context)
Citation Context ...ansitivity behind the scenes. The present paper discusses quite general concepts of calculational reasoning that may be expressed within the Isabelle/Isar framework for human-readable proof documents =-=[19, 20]-=-. Isar provides a high-level view on natural deduction, but is open to incorporate additional derived language elements such as those for calculational reasoning. Thus techniques of natural deduction ... |

44 |
Program Construction and Verification
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- 1986
(Show Context)
Citation Context ...to build tools for logical manipulations. The popular Math � pad tool supports manipulation of algebraic expressions in a systematic way; it has recently even acquired means for formal proof checkin=-=g [18]-=-, using PVS as the backend.sThe Mizar system [17, 12, 22] focuses on formal proof in common mathematics style in the first place. It also offers a mechanism for iterated equality reasoning, which shal... |

28 |
A mizar mode for HOL
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- 1996
(Show Context)
Citation Context ... “. . .” refers to right-hand side of the most recent explicit fact statement. This enables the user to include relevant parts of the previous statement in a succinct manner. The “Mizar mode for=-= HOL” [8] provides a -=-similar element, while Mizar [12] uses the “.=” construct. We may now write the previous examples of calculational sequences as follows. have x1 = x2 〈proof 〉 also have . . . = x3 〈proof 〉... |

19 |
Window inference in the hol system
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- 1991
(Show Context)
Citation Context ...s in the past. Simons proposes tools to support calculational reasoning within tactical proof scripts [14]. Grundy provides an even more general transformational infrastructure for “window inference=-=” [7]. Harr-=-ison’s “Mizar mode for HOL” simulates a number of concepts of declarative theorem proving on top of the tactic-based hol-light system [8], including calculational reasoning for mixed transitivit... |

17 | Structured calculational proof
- Back, Grundy, et al.
- 1996
(Show Context)
Citation Context ...y that they are doing calculational reasoning. In fact, calculational reasoning has been occasionally proposed as simple means to rephrase mathematical proof into a slightly more formal setting (e.g. =-=[2, 1]-=-), which does not necessarily include machine-checking of proofs, of course. Observing that logical equivalence and implication may be just as well used in calculations, some have even set out to do a... |

12 |
Predicate calculus and program semantics. Texts and monographs in computer science
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- 1990
(Show Context)
Citation Context ...of course. Observing that logical equivalence and implication may be just as well used in calculations, some have even set out to do away with traditional natural-deduction style reasoning altogether =-=[5], -=-although that discipline does not appeal to everyone. Nevertheless, calculational reasoning offers a relatively simple conceptual basis to build tools for logical manipulations. The popular Math � p... |

10 | DECLARE: A prototype declarative proof system for higher order logic
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(Show Context)
Citation Context ...ve Mizar [17, 12] supports a fixed format for iterative equations, with implicit application of both transitivity and general substitution rules. Syme’s DECLARE system for declarative theorem provin=-=g [15, 16]-=- does not address calculations at all. Zammit outlines a generalized version of Mizarstyle calculations for SPL [23], but observes that these have not been required for the examples at hand, so it has... |

10 |
Some features of the Mizar language. Presented at a workshop in
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- 1993
(Show Context)
Citation Context ...ular Math � pad tool supports manipulation of algebraic expressions in a systematic way; it has recently even acquired means for formal proof checking [18], using PVS as the backend.sThe Mizar syste=-=m [17, 12, 22]-=- focuses on formal proof in common mathematics style in the first place. It also offers a mechanism for iterated equality reasoning, which shall serve here as an example for calculations within a form... |

9 |
On the implementation of an extensible declarative proof language
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- 1999
(Show Context)
Citation Context ...neral substitution rules. Syme’s DECLARE system for declarative theorem proving [15, 16] does not address calculations at all. Zammit outlines a generalized version of Mizarstyle calculations for SP=-=L [23]-=-, but observes that these have not been required for the examples at hand, so it has not been implemented. For users of Isabelle/Isar, calculational reasoning has become a useful tool for everyday app... |

7 | M: Computer-Assisted Mathematics at Work The Hahn-Banach Theorem
- Bauer, Wenzel
(Show Context)
Citation Context ...ncluding finally or ultimately command, which just offers the result with forward-chaining indicated (cf. §2.4). The next command may be any kind of goal, such have, show, or even the powerful obtain=-= [3, 20]. An-=-y such claim has to be followed by a proof. The most basic one is “.”, meaning that the goal statement actually reiterates the calculational result directly. Another useful idiom is to feed the re... |

7 |
Lecture notes on verification of digital and hybrid systems
- McMillan
(Show Context)
Citation Context ...s of CTL In order to demonstrate how the idiomatic expressions of calculational reasoning are used in practice, we present a case-study of formalizing basic concepts of Computational Tree Logic (CTL) =-=[10, 9] wit-=-hin the simply-typed set theory of HOL. 2 The proofs are mostly by algebraic reasoning over basic set operations. 4.1 CTL formulae By using the common technique of “shallow embedding”, a CTL formu... |

7 | Three tactic theorem proving
- Syme
- 1999
(Show Context)
Citation Context ...ve Mizar [17, 12] supports a fixed format for iterative equations, with implicit application of both transitivity and general substitution rules. Syme’s DECLARE system for declarative theorem provin=-=g [15, 16]-=- does not address calculations at all. Zammit outlines a generalized version of Mizarstyle calculations for SPL [23], but observes that these have not been required for the examples at hand, so it has... |

6 | An Outline of PC Mizar
- Muzalewski
- 1993
(Show Context)
Citation Context ...ular Math � pad tool supports manipulation of algebraic expressions in a systematic way; it has recently even acquired means for formal proof checking [18], using PVS as the backend.sThe Mizar syste=-=m [17, 12, 22]-=- focuses on formal proof in common mathematics style in the first place. It also offers a mechanism for iterated equality reasoning, which shall serve here as an example for calculations within a form... |

4 |
Mizar: An impression. Unpublished paper
- Wiedijk
- 1999
(Show Context)
Citation Context ...ular Math � pad tool supports manipulation of algebraic expressions in a systematic way; it has recently even acquired means for formal proof checking [18], using PVS as the backend.sThe Mizar syste=-=m [17, 12, 22]-=- focuses on formal proof in common mathematics style in the first place. It also offers a mechanism for iterated equality reasoning, which shall serve here as an example for calculations within a form... |

2 |
Proof presentation for Isabelle
- Simons
- 1997
(Show Context)
Citation Context ...PVS etc. lack support for calculational reasoning altogether. This has been addressed several times in the past. Simons proposes tools to support calculational reasoning within tactical proof scripts =-=[14]. Grundy p-=-rovides an even more general transformational infrastructure for “window inference” [7]. Harrison’s “Mizar mode for HOL” simulates a number of concepts of declarative theorem proving on top ... |

1 | Structured derivations: A method for doing highschool mathematics carefully
- Back, Wright
- 1999
(Show Context)
Citation Context ...y that they are doing calculational reasoning. In fact, calculational reasoning has been occasionally proposed as simple means to rephrase mathematical proof into a slightly more formal setting (e.g. =-=[2, 1]-=-), which does not necessarily include machine-checking of proofs, of course. Observing that logical equivalence and implication may be just as well used in calculations, some have even set out to do a... |

1 |
The Isabelle/Isar Reference Manual, 2000. Part of the Isabelle distribution, http://isabelle.in.tum.de/doc/isar-ref.pdf
- Wenzel
(Show Context)
Citation Context ...ansitivity behind the scenes. The present paper discusses quite general concepts of calculational reasoning that may be expressed within the Isabelle/Isar framework for human-readable proof documents =-=[19, 20]-=-. Isar provides a high-level view on natural deduction, but is open to incorporate additional derived language elements such as those for calculational reasoning. Thus techniques of natural deduction ... |

1 |
Some aspects of Unix file-system security. Isabelle/Isar proof document
- Wenzel
- 2001
(Show Context)
Citation Context ..., or even context commands such as assume. See §4.3 for some uses of “moreover note”, and §4.4 for “also note”. Combinations with obtain are very useful in typical computer-science applicati=-=ons (e.g. [21]-=-) where results about representations of syntactic entities are incrementally put together. 9sThe use of assume within a calculation represents the most basic case of combining calculational reasoning... |