doc-src/Logics/FOL.tex
 changeset 287 6b62a6ddbe15 parent 157 8258c26ae084 child 313 a45ae7b38672
--- a/doc-src/Logics/FOL.tex	Mon Mar 21 11:02:57 1994 +0100
+++ b/doc-src/Logics/FOL.tex	Mon Mar 21 11:41:41 1994 +0100
@@ -1,5 +1,5 @@
%% $Id$
-\chapter{First-order logic}
+\chapter{First-Order logic}
The directory~\ttindexbold{FOL} contains theories for first-order logic
based on Gentzen's natural deduction systems (which he called {\sc nj} and
{\sc nk}).  Intuitionistic logic is defined first; then classical logic is
@@ -34,7 +34,7 @@
exists a unique pair $(x,y)$ satisfying~$P(x,y)$.

Some intuitionistic derived rules are shown in
-Figure~\ref{fol-int-derived}, again with their \ML\ names.  These include
+Fig.\ts\ref{fol-int-derived}, again with their \ML\ names.  These include
rules for the defined symbols $\neg$, $\bimp$ and $\exists!$.  Natural
deduction typically involves a combination of forwards and backwards
reasoning, particularly with the destruction rules $(\conj E)$,
@@ -46,8 +46,8 @@
conj_impE}, etc., support the intuitionistic proof procedure
(see~\S\ref{fol-int-prover}).

-See the files \ttindexbold{FOL/ifol.thy}, \ttindexbold{FOL/ifol.ML} and
-\ttindexbold{FOL/intprover.ML} for complete listings of the rules and
+See the files {\tt FOL/ifol.thy}, {\tt FOL/ifol.ML} and
+{\tt FOL/intprover.ML} for complete listings of the rules and
derived rules.

\begin{figure}
@@ -194,7 +194,7 @@

\section{Generic packages}
\FOL{} instantiates most of Isabelle's generic packages;
-see \ttindexbold{FOL/ROOT.ML} for details.
+see {\tt FOL/ROOT.ML} for details.
\begin{itemize}
\item
Because it includes a general substitution rule, \FOL{} instantiates the
@@ -205,7 +205,7 @@
set for intuitionistic first-order logic, while \ttindexbold{FOL_ss} is the
simplification set for classical logic.  Both equality ($=$) and the
biconditional ($\bimp$) may be used for rewriting.  See the file
-\ttindexbold{FOL/simpdata.ML} for a complete listing of the simplification
+{\tt FOL/simpdata.ML} for a complete listing of the simplification
rules.
\item
It instantiates the classical reasoning module.  See~\S\ref{fol-cla-prover}
@@ -228,7 +228,7 @@
\]
The theorem prover for intuitionistic logic does not use~{\tt impE}.\@
Instead, it simplifies implications using derived rules
-(Figure~\ref{fol-int-derived}).  It reduces the antecedents of implications
+(Fig.\ts\ref{fol-int-derived}).  It reduces the antecedents of implications
to atoms and then uses Modus Ponens: from $P\imp Q$ and $P$ deduce~$Q$.
The rules \ttindex{conj_impE} and \ttindex{disj_impE} are
straightforward: $(P\conj Q)\imp S$ is equivalent to $P\imp (Q\imp S)$, and
@@ -254,7 +254,7 @@
\end{ttbox}
Most of these belong to the structure \ttindexbold{Int}.  They are
similar or identical to tactics (with the same names) provided by
-Isabelle's classical module (see {\em The Isabelle Reference Manual\/}).
+Isabelle's classical module (see the {\em Reference Manual\/}).

\begin{description}
\item[\ttindexbold{mp_tac} {\it i}]
@@ -328,7 +328,7 @@
Natural deduction in classical logic is not really all that natural.
{\FOL} derives classical introduction rules for $\disj$ and~$\exists$, as
well as classical elimination rules for~$\imp$ and~$\bimp$, and the swap
-rule (see Figure~\ref{fol-cla-derived}).
+rule (see Fig.\ts\ref{fol-cla-derived}).

The classical reasoning module is set up for \FOL, as the
structure~\ttindexbold{Cla}.  This structure is open, so \ML{} identifiers
@@ -361,7 +361,7 @@
generally unsuitable for depth-first search.
\end{description}
\noindent
-See the file \ttindexbold{FOL/fol.ML} for derivations of the
+See the file {\tt FOL/fol.ML} for derivations of the
classical rules, and the {\em Reference Manual} for more discussion of
classical proof methods.

@@ -537,6 +537,7 @@
{\out Level 5}
{\out ~ ~ ((P --> Q) | (Q --> P))}
{\out  1. [| P; Q --> False; Q; P --> False |] ==> P}
+\ttbreak
by (assume_tac 1);
{\out Level 6}
{\out ~ ~ ((P --> Q) | (Q --> P))}
@@ -835,13 +836,17 @@
{\out Level 0}
{\out if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))}
{\out  1. if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))}
-\ttbreak
+\end{ttbox}
+This time, simply unfold using the definition of $if$:
+\begin{ttbox}
by (rewrite_goals_tac [if_def]);
{\out Level 1}
{\out if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))}
{\out  1. (P & Q | ~ P & R) & A | ~ (P & Q | ~ P & R) & B <->}
{\out     P & (Q & A | ~ Q & B) | ~ P & (R & A | ~ R & B)}
-\ttbreak
+\end{ttbox}
+We are left with a subgoal in pure first-order logic:
+\begin{ttbox}
by (fast_tac FOL_cs 1);
{\out Level 2}
{\out if(if(P,Q,R),A,B) <-> if(P,if(Q,A,B),if(R,A,B))}