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doc-src/Logics/intro.tex

author | paulson |

Fri Feb 16 18:00:47 1996 +0100 (1996-02-16) | |

changeset 1512 | ce37c64244c0 |

parent 343 | 8d77f767bd26 |

child 3139 | 671a5f2cac6a |

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

Elimination of fully-functorial style.

Type tactic changed to a type abbrevation (from a datatype).

Constructor tactic and function apply deleted.

Type tactic changed to a type abbrevation (from a datatype).

Constructor tactic and function apply deleted.

1 %% $Id$

2 \chapter{Basic Concepts}

3 Several logics come with Isabelle. Many of them are sufficiently developed

4 to serve as comfortable reasoning environments. They are also good

5 starting points for defining new logics. Each logic is distributed with

6 sample proofs, some of which are described in this document.

8 \begin{ttdescription}

9 \item[\thydx{FOL}] is many-sorted first-order logic with natural

10 deduction. It comes in both constructive and classical versions.

12 \item[\thydx{ZF}] is axiomatic set theory, using the Zermelo-Fraenkel

13 axioms~\cite{suppes72}. It is built upon classical~\FOL{}.

15 \item[\thydx{CCL}] is Martin Coen's Classical Computational Logic,

16 which is the basis of a preliminary method for deriving programs from

17 proofs~\cite{coen92}. It is built upon classical~\FOL{}.

19 \item[\thydx{LCF}] is a version of Scott's Logic for Computable

20 Functions, which is also implemented by the~{\sc lcf}

21 system~\cite{paulson87}. It is built upon classical~\FOL{}.

23 \item[\thydx{HOL}] is the higher-order logic of Church~\cite{church40},

24 which is also implemented by Gordon's~{\sc hol} system~\cite{mgordon-hol}.

25 This object-logic should not be confused with Isabelle's meta-logic, which is

26 also a form of higher-order logic.

28 \item[\thydx{HOLCF}] is an alternative version of {\sc lcf}, defined

29 as an extension of {\tt HOL}\@.

31 \item[\thydx{CTT}] is a version of Martin-L\"of's Constructive Type

32 Theory~\cite{nordstrom90}, with extensional equality. Universes are not

33 included.

35 \item[\thydx{LK}] is another version of first-order logic, a classical

36 sequent calculus. Sequents have the form $A@1,\ldots,A@m\turn

37 B@1,\ldots,B@n$; rules are applied using associative matching.

39 \item[\thydx{Modal}] implements the modal logics $T$, $S4$,

40 and~$S43$. It is built upon~\LK{}.

42 \item[\thydx{Cube}] is Barendregt's $\lambda$-cube.

43 \end{ttdescription}

44 The logics {\tt CCL}, {\tt LCF}, {\tt HOLCF}, {\tt Modal} and {\tt Cube}

45 are currently undocumented.

47 You should not read this before reading {\em Introduction to Isabelle\/}

48 and performing some Isabelle proofs. Consult the {\em Reference Manual}

49 for more information on tactics, packages, etc.

52 \section{Syntax definitions}

53 The syntax of each logic is presented using a context-free grammar.

54 These grammars obey the following conventions:

55 \begin{itemize}

56 \item identifiers denote nonterminal symbols

57 \item {\tt typewriter} font denotes terminal symbols

58 \item parentheses $(\ldots)$ express grouping

59 \item constructs followed by a Kleene star, such as $id^*$ and $(\ldots)^*$

60 can be repeated~0 or more times

61 \item alternatives are separated by a vertical bar,~$|$

62 \item the symbol for alphanumeric identifiers is~{\it id\/}

63 \item the symbol for scheme variables is~{\it var}

64 \end{itemize}

65 To reduce the number of nonterminals and grammar rules required, Isabelle's

66 syntax module employs {\bf priorities},\index{priorities} or precedences.

67 Each grammar rule is given by a mixfix declaration, which has a priority,

68 and each argument place has a priority. This general approach handles

69 infix operators that associate either to the left or to the right, as well

70 as prefix and binding operators.

72 In a syntactically valid expression, an operator's arguments never involve

73 an operator of lower priority unless brackets are used. Consider

74 first-order logic, where $\exists$ has lower priority than $\disj$,

75 which has lower priority than $\conj$. There, $P\conj Q \disj R$

76 abbreviates $(P\conj Q) \disj R$ rather than $P\conj (Q\disj R)$. Also,

77 $\exists x.P\disj Q$ abbreviates $\exists x.(P\disj Q)$ rather than

78 $(\exists x.P)\disj Q$. Note especially that $P\disj(\exists x.Q)$

79 becomes syntactically invalid if the brackets are removed.

81 A {\bf binder} is a symbol associated with a constant of type

82 $(\sigma\To\tau)\To\tau'$. For instance, we may declare~$\forall$ as a

83 binder for the constant~$All$, which has type $(\alpha\To o)\To o$. This

84 defines the syntax $\forall x.t$ to mean $All(\lambda x.t)$. We can

85 also write $\forall x@1\ldots x@m.t$ to abbreviate $\forall x@1. \ldots

86 \forall x@m.t$; this is possible for any constant provided that $\tau$ and

87 $\tau'$ are the same type. \HOL's description operator $\epsilon x.P(x)$

88 has type $(\alpha\To bool)\To\alpha$ and can bind only one variable, except

89 when $\alpha$ is $bool$. \ZF's bounded quantifier $\forall x\in A.P(x)$

90 cannot be declared as a binder because it has type $[i, i\To o]\To o$. The

91 syntax for binders allows type constraints on bound variables, as in

92 \[ \forall (x{::}\alpha) \; (y{::}\beta). R(x,y) \]

94 To avoid excess detail, the logic descriptions adopt a semi-formal style.

95 Infix operators and binding operators are listed in separate tables, which

96 include their priorities. Grammar descriptions do not include numeric

97 priorities; instead, the rules appear in order of decreasing priority.

98 This should suffice for most purposes; for full details, please consult the

99 actual syntax definitions in the {\tt.thy} files.

101 Each nonterminal symbol is associated with some Isabelle type. For

102 example, the formulae of first-order logic have type~$o$. Every

103 Isabelle expression of type~$o$ is therefore a formula. These include

104 atomic formulae such as $P$, where $P$ is a variable of type~$o$, and more

105 generally expressions such as $P(t,u)$, where $P$, $t$ and~$u$ have

106 suitable types. Therefore, `expression of type~$o$' is listed as a

107 separate possibility in the grammar for formulae.

110 \section{Proof procedures}\label{sec:safe}

111 Most object-logics come with simple proof procedures. These are reasonably

112 powerful for interactive use, though often simplistic and incomplete. You

113 can do single-step proofs using \verb|resolve_tac| and

114 \verb|assume_tac|, referring to the inference rules of the logic by {\sc

115 ml} identifiers.

117 For theorem proving, rules can be classified as {\bf safe} or {\bf unsafe}.

118 A rule is safe if applying it to a provable goal always yields provable

119 subgoals. If a rule is safe then it can be applied automatically to a goal

120 without destroying our chances of finding a proof. For instance, all the

121 rules of the classical sequent calculus {\sc lk} are safe. Universal

122 elimination is unsafe if the formula $\all{x}P(x)$ is deleted after use.

123 Other unsafe rules include the following:

124 \[ \infer[({\disj}I1)]{P\disj Q}{P} \qquad

125 \infer[({\imp}E)]{Q}{P\imp Q & P} \qquad

126 \infer[({\exists}I)]{\exists x.P}{P[t/x]}

127 \]

129 Proof procedures use safe rules whenever possible, delaying the application

130 of unsafe rules. Those safe rules are preferred that generate the fewest

131 subgoals. Safe rules are (by definition) deterministic, while the unsafe

132 rules require search. The design of a suitable set of rules can be as

133 important as the strategy for applying them.

135 Many of the proof procedures use backtracking. Typically they attempt to

136 solve subgoal~$i$ by repeatedly applying a certain tactic to it. This

137 tactic, which is known as a {\bf step tactic}, resolves a selection of

138 rules with subgoal~$i$. This may replace one subgoal by many; the

139 search persists until there are fewer subgoals in total than at the start.

140 Backtracking happens when the search reaches a dead end: when the step

141 tactic fails. Alternative outcomes are then searched by a depth-first or

142 best-first strategy.