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doc-src/preface.tex

author | nipkow |

Sun Mar 02 15:02:06 2008 +0100 (2008-03-02) | |

changeset 26191 | ae537f315b34 |

parent 14379 | ea10a8c3e9cf |

permissions | -rw-r--r-- |

Generalized Zorn and added well-ordering theorem

1 \chapter*{Preface}

2 \markboth{Preface}{Preface} %or Preface ?

3 %%\addcontentsline{toc}{chapter}{Preface}

5 Most theorem provers support a fixed logic, such as first-order or

6 equational logic. They bring sophisticated proof procedures to bear upon

7 the conjectured formula. The resolution prover Otter~\cite{wos-bledsoe} is

8 an impressive example.

10 {\sc alf}~\cite{alf}, Coq~\cite{coq} and Nuprl~\cite{constable86} each

11 support a fixed logic too. These are higher-order type theories,

12 explicitly concerned with computation and capable of expressing

13 developments in constructive mathematics. They are far removed from

14 classical first-order logic.

16 A diverse collection of logics --- type theories, process calculi,

17 $\lambda$-calculi --- may be found in the Computer Science literature.

18 Such logics require proof support. Few proof procedures are known for

19 them, but the theorem prover can at least automate routine steps.

21 A {\bf generic} theorem prover is one that supports a variety of logics.

22 Some generic provers are noteworthy for their user interfaces

23 \cite{dawson90,mural,sawamura92}. Most of them work by implementing a

24 syntactic framework that can express typical inference rules. Isabelle's

25 distinctive feature is its representation of logics within a fragment of

26 higher-order logic, called the meta-logic. The proof theory of

27 higher-order logic may be used to demonstrate that the representation is

28 correct~\cite{paulson89}. The approach has much in common with the

29 Edinburgh Logical Framework~\cite{harper-jacm} and with

30 Felty's~\cite{felty93} use of $\lambda$Prolog to implement logics.

32 An inference rule in Isabelle is a generalized Horn clause. Rules are

33 joined to make proofs by resolving such clauses. Logical variables in

34 goals can be instantiated incrementally. But Isabelle is not a resolution

35 theorem prover like Otter. Isabelle's clauses are drawn from a richer

36 language and a fully automatic search would be impractical. Isabelle does

37 not resolve clauses automatically, but under user direction. You can

38 conduct single-step proofs, use Isabelle's built-in proof procedures, or

39 develop new proof procedures using tactics and tacticals.

41 Isabelle's meta-logic is higher-order, based on the simply typed

42 $\lambda$-calculus. So resolution cannot use ordinary unification, but

43 higher-order unification~\cite{huet75}. This complicated procedure gives

44 Isabelle strong support for many logical formalisms involving variable

45 binding.

47 The diagram below illustrates some of the logics distributed with Isabelle.

48 These include first-order logic (intuitionistic and classical), the sequent

49 calculus, higher-order logic, Zermelo-Fraenkel set theory~\cite{suppes72},

50 a version of Constructive Type Theory~\cite{nordstrom90}, several modal

51 logics, and a Logic for Computable Functions~\cite{paulson87}. Several

52 experimental logics are being developed, such as linear logic.

54 \centerline{\epsfbox{gfx/Isa-logics.eps}}

57 \section*{How to read this book}

58 Isabelle is a complex system, but beginners can get by with a few commands

59 and a basic knowledge of how Isabelle works. Some knowledge of

60 Standard~\ML{} is essential because \ML{} is Isabelle's user interface.

61 Advanced Isabelle theorem proving can involve writing \ML{} code, possibly

62 with Isabelle's sources at hand. My book on~\ML{}~\cite{paulson91} covers

63 much material connected with Isabelle, including a simple theorem prover.

65 The Isabelle documentation is divided into three parts, which serve

66 distinct purposes:

67 \begin{itemize}

68 \item {\em Introduction to Isabelle\/} describes the basic features of

69 Isabelle. This part is intended to be read through. If you are

70 impatient to get started, you might skip the first chapter, which

71 describes Isabelle's meta-logic in some detail. The other chapters

72 present on-line sessions of increasing difficulty. It also explains how

73 to derive rules define theories, and concludes with an extended example:

74 a Prolog interpreter.

76 \item {\em The Isabelle Reference Manual\/} provides detailed information

77 about Isabelle's facilities, excluding the object-logics. This part

78 would make boring reading, though browsing might be useful. Mostly you

79 should use it to locate facts quickly.

81 \item {\em Isabelle's Object-Logics\/} describes the various logics

82 distributed with Isabelle. The chapters are intended for reference only;

83 they overlap somewhat so that each chapter can be read in isolation.

84 \end{itemize}

85 This book should not be read from start to finish. Instead you might read

86 a couple of chapters from {\em Introduction to Isabelle}, then try some

87 examples referring to the other parts, return to the {\em Introduction},

88 and so forth. Starred sections discuss obscure matters and may be skipped

89 on a first reading.

93 \section*{Releases of Isabelle}

94 Isabelle was first distributed in 1986. The 1987 version introduced a

95 higher-order meta-logic with an improved treatment of quantifiers. The

96 1988 version added limited polymorphism and support for natural deduction.

97 The 1989 version included a parser and pretty printer generator. The 1992

98 version introduced type classes, to support many-sorted and higher-order

99 logics. The 1993 version provides greater support for theories and is

100 much faster.

102 Isabelle is still under development. Projects under consideration include

103 better support for inductive definitions, some means of recording proofs, a

104 graphical user interface, and developments in the standard object-logics.

105 I hope but cannot promise to maintain upwards compatibility.

107 Isabelle can be downloaded from .

108 \begin{quote}

109 {\tt http://www.cl.cam.ac.uk/Research/HVG/Isabelle/dist/}

110 \end{quote}

111 The electronic distribution list {\tt isabelle-users\at cl.cam.ac.uk}

112 provides a forum for discussing problems and applications involving

113 Isabelle. To join, send me a message via {\tt lcp\at cl.cam.ac.uk}.

114 Please notify me of any errors you find in this book.

116 \section*{Acknowledgements}

117 Tobias Nipkow has made immense contributions to Isabelle, including the

118 parser generator, type classes, the simplifier, and several object-logics.

119 He also arranged for several of his students to help. Carsten Clasohm

120 implemented the theory database; Markus Wenzel implemented macros; Sonia

121 Mahjoub and Karin Nimmermann also contributed.

123 Nipkow and his students wrote much of the documentation underlying this

124 book. Nipkow wrote the first versions of \S\ref{sec:defining-theories},

125 \S\ref{sec:ref-defining-theories}, Chap.\ts\ref{Defining-Logics},

126 Chap.\ts\ref{simp-chap} and App.\ts\ref{app:TheorySyntax}\@. Carsten

127 Clasohm contributed to Chap.\ts\ref{theories}. Markus Wenzel contributed

128 to Chap.\ts\ref{chap:syntax}. Nipkow also provided the quotation at

129 the front.

131 David Aspinall, Sara Kalvala, Ina Kraan, Chris Owens, Zhenyu Qian, Norbert

132 V{\"o}lker and Markus Wenzel suggested changes and corrections to the

133 documentation.

135 Martin Coen, Rajeev Gor\'e, Philippe de Groote and Philippe No\"el helped

136 to develop Isabelle's standard object-logics. David Aspinall performed

137 some useful research into theories and implemented an Isabelle Emacs mode.

138 Isabelle was developed using Dave Matthews's Standard~{\sc ml} compiler,

139 Poly/{\sc ml}.

141 The research has been funded by numerous SERC grants dating from the Alvey

142 programme (grants GR/E0355.7, GR/G53279, GR/H40570) and by ESPRIT (projects

143 3245: Logical Frameworks and 6453: Types).