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\chapter{Introduction}
Several logics come with Isabelle. Many of them are sufficiently developed
to serve as comfortable reasoning environments. They are also good
starting points for defining new logics. Each logic is distributed with
sample proofs, some of which are described in this document.
\begin{quote}
{\ttindexbold{FOL}} is many-sorted first-order logic with natural
deduction. It comes in both constructive and classical versions.
{\ttindexbold{ZF}} is axiomatic set theory, using the Zermelo-Fraenkel
axioms~\cite{suppes72}. It is built upon classical~\FOL{}.
{\ttindexbold{HOL}} is the higher-order logic of Church~\cite{church40},
which is also implemented by Gordon's~{\sc hol} system~\cite{gordon88a}. This
object-logic should not be confused with Isabelle's meta-logic, which is
also a form of higher-order logic.
{\ttindexbold{CTT}} is a version of Martin-L\"of's Constructive Type
Theory~\cite{nordstrom90}, with extensional equality. Universes are not
included.
{\ttindexbold{LK}} is another version of first-order logic, a classical
sequent calculus. Sequents have the form $A@1,\ldots,A@m\turn
B@1,\ldots,B@n$; rules are applied using associative matching.
{\ttindexbold{Modal}} implements the modal logics $T$, $S4$, and~$S43$. It
is built upon~\LK{}.
{\ttindexbold{Cube}} is Barendregt's $\lambda$-cube.
{\ttindexbold{LCF}} is a version of Scott's Logic for Computable Functions,
which is also implemented by the~{\sc lcf} system~\cite{paulson87}.
\end{quote}
The logics {\tt Modal}, {\tt Cube} and {\tt LCF} are currently undocumented.
This manual assumes that you have read the {\em Introduction to Isabelle\/}
and have some experience of using Isabelle to perform interactive proofs.
It refers to packages defined in the {\em Reference Manual}, which you
should keep at hand.
\section{Syntax definitions}
This manual defines each logic's syntax using a context-free grammar.
These grammars obey the following conventions:
\begin{itemize}
\item identifiers denote nonterminal symbols
\item {\tt typewriter} font denotes terminal symbols
\item parentheses $(\ldots)$ express grouping
\item constructs followed by a Kleene star, such as $id^*$ and $(\ldots)^*$
can be repeated~0 or more times
\item alternatives are separated by a vertical bar,~$|$
\item the symbol for alphanumeric identifier is~{\it id\/}
\item the symbol for scheme variables is~{\it var}
\end{itemize}
To reduce the number of nonterminals and grammar rules required, Isabelle's
syntax module employs {\bf precedences}. Each grammar rule is given by a
mixfix declaration, which has a precedence, and each argument place has a
precedence. This general approach handles infix operators that associate
either to the left or to the right, as well as prefix and binding
operators.
In a syntactically valid expression, an operator's arguments never involve
an operator of lower precedence unless brackets are used. Consider
first-order logic, where $\exists$ has lower precedence than $\disj$,
which has lower precedence than $\conj$. There, $P\conj Q \disj R$
abbreviates $(P\conj Q) \disj R$ rather than $P\conj (Q\disj R)$. Also,
$\exists x.P\disj Q$ abbreviates $\exists x.(P\disj Q)$ rather than
$(\exists x.P)\disj Q$. Note especially that $P\disj(\exists x.Q)$
becomes syntactically invalid if the brackets are removed.
A {\bf binder} is a symbol associated with a constant of type
$(\sigma\To\tau)\To\tau'$. For instance, we may declare~$\forall$ as a
binder for the constant~$All$, which has type $(\alpha\To o)\To o$. This
defines the syntax $\forall x.t$ to mean $\forall(\lambda x.t)$. We can
also write $\forall x@1\ldots x@m.t$ to abbreviate $\forall x@1. \ldots
\forall x@m.t$; this is possible for any constant provided that $\tau$ and
$\tau'$ are the same type. \HOL's description operator $\epsilon x.P(x)$
has type $(\alpha\To bool)\To\alpha$ and can bind only one variable, except
when $\alpha$ is $bool$. \ZF's bounded quantifier $\forall x\in A.P(x)$
cannot be declared as a binder because it has type $[i, i\To o]\To o$. The
syntax for binders allows type constraints on bound variables, as in
\[ \forall (x{::}\alpha) \; y{::}\beta. R(x,y) \]
To avoid excess detail, the logic descriptions adopt a semi-formal style.
Infix operators and binding operators are listed in separate tables, which
include their precedences. Grammars do not give numeric precedences;
instead, the rules appear in order of decreasing precedence. This should
suffice for most purposes; for detailed precedence information, consult the
syntax definitions in the {\tt.thy} files. Chapter~\ref{Defining-Logics}
describes Isabelle's syntax module, including its use of precedences.
Each nonterminal symbol is associated with some Isabelle type. For
example, the {\bf formulae} of first-order logic have type~$o$. Every
Isabelle expression of type~$o$ is therefore a formula. These include
atomic formulae such as $P$, where $P$ is a variable of type~$o$, and more
generally expressions such as $P(t,u)$, where $P$, $t$ and~$u$ have
suitable types. Therefore, ``expression of type~$o$'' is listed as a
separate possibility in the grammar for formulae.
Infix operators are represented internally by constants with the prefix
\hbox{\tt"op "}. For instance, implication is the constant
\hbox{\tt"op~-->"}. This permits infixes to be used in non-infix contexts
(just as with \ML's~{\tt op} keyword). You also need to know the name of
the internal constant if you are writing code that inspects terms.
\section{Proof procedures}
Most object-logics come with simple proof procedures. These are reasonably
powerful for interactive use, though often simplistic and incomplete. You
can do single-step proofs using \verb|resolve_tac| and
\verb|assume_tac|, referring to the inference rules of the logic by {\sc
ml} identifiers.
Call a rule {\em safe\/} if applying it to a provable goal always yields
provable subgoals. If a rule is safe then it can be applied automatically
to a goal without destroying our chances of finding a proof. For instance,
all the rules of the classical sequent calculus {\sc lk} are safe.
Intuitionistic logic includes some unsafe rules, like disjunction
introduction ($P\disj Q$ can be true when $P$ is false) and existential
introduction ($\ex{x}P(x)$ can be true when $P(a)$ is false for certain
$a$). Universal elimination is unsafe if the formula $\all{x}P(x)$ is
deleted after use.
Proof procedures use safe rules whenever possible, delaying the application
of unsafe rules. Those safe rules are preferred that generate the fewest
subgoals. Safe rules are (by definition) deterministic, while the unsafe
rules require search. The design of a suitable set of rules can be as
important as the strategy for applying them.
Many of the proof procedures use backtracking. Typically they attempt to
solve subgoal~$i$ by repeatedly applying a certain tactic to it. This
tactic, which is known as a {\it step tactic}, resolves a selection of
rules with subgoal~$i$. This may replace one subgoal by many; but the
search persists until there are fewer subgoals in total than at the start.
Backtracking happens when the search reaches a dead end: when the step
tactic fails. Alternative outcomes are then searched by a depth-first or
best-first strategy.