src/Doc/Intro/document/advanced.tex
author haftmann
Sun Oct 08 22:28:22 2017 +0200 (23 months ago)
changeset 66816 212a3334e7da
parent 48985 5386df44a037
permissions -rw-r--r--
more fundamental definition of div and mod on int
lcp@284
     1
\part{Advanced Methods}
lcp@105
     2
Before continuing, it might be wise to try some of your own examples in
lcp@105
     3
Isabelle, reinforcing your knowledge of the basic functions.
lcp@105
     4
wenzelm@3103
     5
Look through {\em Isabelle's Object-Logics\/} and try proving some
wenzelm@3103
     6
simple theorems.  You probably should begin with first-order logic
paulson@5205
     7
(\texttt{FOL} or~\texttt{LK}).  Try working some of the examples provided,
paulson@5205
     8
and others from the literature.  Set theory~(\texttt{ZF}) and
paulson@5205
     9
Constructive Type Theory~(\texttt{CTT}) form a richer world for
wenzelm@3103
    10
mathematical reasoning and, again, many examples are in the
paulson@5205
    11
literature.  Higher-order logic~(\texttt{HOL}) is Isabelle's most
paulson@3485
    12
elaborate logic.  Its types and functions are identified with those of
wenzelm@3103
    13
the meta-logic.
lcp@105
    14
lcp@105
    15
Choose a logic that you already understand.  Isabelle is a proof
lcp@105
    16
tool, not a teaching tool; if you do not know how to do a particular proof
lcp@105
    17
on paper, then you certainly will not be able to do it on the machine.
lcp@105
    18
Even experienced users plan large proofs on paper.
lcp@105
    19
lcp@105
    20
We have covered only the bare essentials of Isabelle, but enough to perform
lcp@105
    21
substantial proofs.  By occasionally dipping into the {\em Reference
lcp@105
    22
Manual}, you can learn additional tactics, subgoal commands and tacticals.
lcp@105
    23
lcp@105
    24
lcp@105
    25
\section{Deriving rules in Isabelle}
lcp@105
    26
\index{rules!derived}
lcp@105
    27
A mathematical development goes through a progression of stages.  Each
lcp@105
    28
stage defines some concepts and derives rules about them.  We shall see how
lcp@105
    29
to derive rules, perhaps involving definitions, using Isabelle.  The
lcp@310
    30
following section will explain how to declare types, constants, rules and
lcp@105
    31
definitions.
lcp@105
    32
lcp@105
    33
lcp@296
    34
\subsection{Deriving a rule using tactics and meta-level assumptions} 
lcp@296
    35
\label{deriving-example}
lcp@310
    36
\index{examples!of deriving rules}\index{assumptions!of main goal}
lcp@296
    37
lcp@307
    38
The subgoal module supports the derivation of rules, as discussed in
paulson@5205
    39
\S\ref{deriving}.  When the \ttindex{Goal} command is supplied a formula of
paulson@5205
    40
the form $\List{\theta@1; \ldots; \theta@k} \Imp \phi$, there are two
paulson@5205
    41
possibilities:
paulson@5205
    42
\begin{itemize}
paulson@5205
    43
\item If all of the premises $\theta@1$, \ldots, $\theta@k$ are simple
paulson@5205
    44
  formulae{} (they do not involve the meta-connectives $\Forall$ or
paulson@5205
    45
  $\Imp$) then the command sets the goal to be 
paulson@5205
    46
  $\List{\theta@1; \ldots; \theta@k} \Imp \phi$ and returns the empty list.
paulson@5205
    47
\item If one or more premises involves the meta-connectives $\Forall$ or
paulson@5205
    48
  $\Imp$, then the command sets the goal to be $\phi$ and returns a list
paulson@5205
    49
  consisting of the theorems ${\theta@i\;[\theta@i]}$, for $i=1$, \ldots,~$k$.
paulson@14148
    50
  These meta-level assumptions are also recorded internally, allowing
paulson@5205
    51
  \texttt{result} (which is called by \texttt{qed}) to discharge them in the
paulson@5205
    52
  original order.
paulson@5205
    53
\end{itemize}
paulson@5205
    54
Rules that discharge assumptions or introduce eigenvariables have complex
paulson@14148
    55
premises, and the second case applies.  In this section, many of the
paulson@14148
    56
theorems are subject to meta-level assumptions, so we make them visible by by setting the 
paulson@14148
    57
\ttindex{show_hyps} flag:
paulson@14148
    58
\begin{ttbox} 
paulson@14148
    59
set show_hyps;
paulson@14148
    60
{\out val it = true : bool}
paulson@14148
    61
\end{ttbox}
lcp@105
    62
paulson@14148
    63
Now, we are ready to derive $\conj$ elimination.  Until now, calling \texttt{Goal} has
paulson@5205
    64
returned an empty list, which we have ignored.  In this example, the list
paulson@5205
    65
contains the two premises of the rule, since one of them involves the $\Imp$
paulson@5205
    66
connective.  We bind them to the \ML\ identifiers \texttt{major} and {\tt
paulson@5205
    67
  minor}:\footnote{Some ML compilers will print a message such as {\em binding
paulson@5205
    68
    not exhaustive}.  This warns that \texttt{Goal} must return a 2-element
paulson@5205
    69
  list.  Otherwise, the pattern-match will fail; ML will raise exception
paulson@5205
    70
  \xdx{Match}.}
lcp@105
    71
\begin{ttbox}
paulson@5205
    72
val [major,minor] = Goal
lcp@105
    73
    "[| P&Q;  [| P; Q |] ==> R |] ==> R";
lcp@105
    74
{\out Level 0}
lcp@105
    75
{\out R}
lcp@105
    76
{\out  1. R}
lcp@105
    77
{\out val major = "P & Q  [P & Q]" : thm}
lcp@105
    78
{\out val minor = "[| P; Q |] ==> R  [[| P; Q |] ==> R]" : thm}
lcp@105
    79
\end{ttbox}
lcp@105
    80
Look at the minor premise, recalling that meta-level assumptions are
paulson@5205
    81
shown in brackets.  Using \texttt{minor}, we reduce $R$ to the subgoals
lcp@105
    82
$P$ and~$Q$:
lcp@105
    83
\begin{ttbox}
lcp@105
    84
by (resolve_tac [minor] 1);
lcp@105
    85
{\out Level 1}
lcp@105
    86
{\out R}
lcp@105
    87
{\out  1. P}
lcp@105
    88
{\out  2. Q}
lcp@105
    89
\end{ttbox}
lcp@105
    90
Deviating from~\S\ref{deriving}, we apply $({\conj}E1)$ forwards from the
lcp@105
    91
assumption $P\conj Q$ to obtain the theorem~$P\;[P\conj Q]$.
lcp@105
    92
\begin{ttbox}
lcp@105
    93
major RS conjunct1;
lcp@105
    94
{\out val it = "P  [P & Q]" : thm}
lcp@105
    95
\ttbreak
lcp@105
    96
by (resolve_tac [major RS conjunct1] 1);
lcp@105
    97
{\out Level 2}
lcp@105
    98
{\out R}
lcp@105
    99
{\out  1. Q}
lcp@105
   100
\end{ttbox}
lcp@105
   101
Similarly, we solve the subgoal involving~$Q$.
lcp@105
   102
\begin{ttbox}
lcp@105
   103
major RS conjunct2;
lcp@105
   104
{\out val it = "Q  [P & Q]" : thm}
lcp@105
   105
by (resolve_tac [major RS conjunct2] 1);
lcp@105
   106
{\out Level 3}
lcp@105
   107
{\out R}
lcp@105
   108
{\out No subgoals!}
lcp@105
   109
\end{ttbox}
lcp@105
   110
Calling \ttindex{topthm} returns the current proof state as a theorem.
wenzelm@3103
   111
Note that it contains assumptions.  Calling \ttindex{qed} discharges
wenzelm@3103
   112
the assumptions --- both occurrences of $P\conj Q$ are discharged as
wenzelm@3103
   113
one --- and makes the variables schematic.
lcp@105
   114
\begin{ttbox}
lcp@105
   115
topthm();
lcp@105
   116
{\out val it = "R  [P & Q, P & Q, [| P; Q |] ==> R]" : thm}
wenzelm@3103
   117
qed "conjE";
lcp@105
   118
{\out val conjE = "[| ?P & ?Q; [| ?P; ?Q |] ==> ?R |] ==> ?R" : thm}
lcp@105
   119
\end{ttbox}
lcp@105
   120
lcp@105
   121
lcp@105
   122
\subsection{Definitions and derived rules} \label{definitions}
lcp@310
   123
\index{rules!derived}\index{definitions!and derived rules|(}
lcp@310
   124
lcp@105
   125
Definitions are expressed as meta-level equalities.  Let us define negation
lcp@105
   126
and the if-and-only-if connective:
lcp@105
   127
\begin{eqnarray*}
lcp@105
   128
  \neg \Var{P}          & \equiv & \Var{P}\imp\bot \\
lcp@105
   129
  \Var{P}\bimp \Var{Q}  & \equiv & 
lcp@105
   130
                (\Var{P}\imp \Var{Q}) \conj (\Var{Q}\imp \Var{P})
lcp@105
   131
\end{eqnarray*}
lcp@331
   132
\index{meta-rewriting}%
lcp@105
   133
Isabelle permits {\bf meta-level rewriting} using definitions such as
lcp@105
   134
these.  {\bf Unfolding} replaces every instance
lcp@331
   135
of $\neg \Var{P}$ by the corresponding instance of ${\Var{P}\imp\bot}$.  For
lcp@105
   136
example, $\forall x.\neg (P(x)\conj \neg R(x,0))$ unfolds to
lcp@105
   137
\[ \forall x.(P(x)\conj R(x,0)\imp\bot)\imp\bot.  \]
lcp@105
   138
{\bf Folding} a definition replaces occurrences of the right-hand side by
lcp@105
   139
the left.  The occurrences need not be free in the entire formula.
lcp@105
   140
lcp@105
   141
When you define new concepts, you should derive rules asserting their
lcp@105
   142
abstract properties, and then forget their definitions.  This supports
lcp@331
   143
modularity: if you later change the definitions without affecting their
lcp@105
   144
abstract properties, then most of your proofs will carry through without
lcp@105
   145
change.  Indiscriminate unfolding makes a subgoal grow exponentially,
lcp@105
   146
becoming unreadable.
lcp@105
   147
lcp@105
   148
Taking this point of view, Isabelle does not unfold definitions
lcp@105
   149
automatically during proofs.  Rewriting must be explicit and selective.
lcp@105
   150
Isabelle provides tactics and meta-rules for rewriting, and a version of
paulson@5205
   151
the \texttt{Goal} command that unfolds the conclusion and premises of the rule
lcp@105
   152
being derived.
lcp@105
   153
lcp@105
   154
For example, the intuitionistic definition of negation given above may seem
lcp@105
   155
peculiar.  Using Isabelle, we shall derive pleasanter negation rules:
lcp@105
   156
\[  \infer[({\neg}I)]{\neg P}{\infer*{\bot}{[P]}}   \qquad
lcp@105
   157
    \infer[({\neg}E)]{Q}{\neg P & P}  \]
lcp@296
   158
This requires proving the following meta-formulae:
wenzelm@3103
   159
$$ (P\Imp\bot)    \Imp \neg P   \eqno(\neg I) $$
wenzelm@3103
   160
$$ \List{\neg P; P} \Imp Q.       \eqno(\neg E) $$
lcp@105
   161
lcp@105
   162
lcp@296
   163
\subsection{Deriving the $\neg$ introduction rule}
paulson@5205
   164
To derive $(\neg I)$, we may call \texttt{Goal} with the appropriate formula.
paulson@5205
   165
Again, the rule's premises involve a meta-connective, and \texttt{Goal}
paulson@5205
   166
returns one-element list.  We bind this list to the \ML\ identifier \texttt{prems}.
lcp@105
   167
\begin{ttbox}
paulson@5205
   168
val prems = Goal "(P ==> False) ==> ~P";
lcp@105
   169
{\out Level 0}
lcp@105
   170
{\out ~P}
lcp@105
   171
{\out  1. ~P}
lcp@105
   172
{\out val prems = ["P ==> False  [P ==> False]"] : thm list}
lcp@105
   173
\end{ttbox}
lcp@310
   174
Calling \ttindex{rewrite_goals_tac} with \tdx{not_def}, which is the
lcp@105
   175
definition of negation, unfolds that definition in the subgoals.  It leaves
lcp@105
   176
the main goal alone.
lcp@105
   177
\begin{ttbox}
lcp@105
   178
not_def;
lcp@105
   179
{\out val it = "~?P == ?P --> False" : thm}
lcp@105
   180
by (rewrite_goals_tac [not_def]);
lcp@105
   181
{\out Level 1}
lcp@105
   182
{\out ~P}
lcp@105
   183
{\out  1. P --> False}
lcp@105
   184
\end{ttbox}
lcp@310
   185
Using \tdx{impI} and the premise, we reduce subgoal~1 to a triviality:
lcp@105
   186
\begin{ttbox}
lcp@105
   187
by (resolve_tac [impI] 1);
lcp@105
   188
{\out Level 2}
lcp@105
   189
{\out ~P}
lcp@105
   190
{\out  1. P ==> False}
lcp@105
   191
\ttbreak
lcp@105
   192
by (resolve_tac prems 1);
lcp@105
   193
{\out Level 3}
lcp@105
   194
{\out ~P}
lcp@105
   195
{\out  1. P ==> P}
lcp@105
   196
\end{ttbox}
lcp@296
   197
The rest of the proof is routine.  Note the form of the final result.
lcp@105
   198
\begin{ttbox}
lcp@105
   199
by (assume_tac 1);
lcp@105
   200
{\out Level 4}
lcp@105
   201
{\out ~P}
lcp@105
   202
{\out No subgoals!}
lcp@296
   203
\ttbreak
wenzelm@3103
   204
qed "notI";
lcp@105
   205
{\out val notI = "(?P ==> False) ==> ~?P" : thm}
lcp@105
   206
\end{ttbox}
lcp@310
   207
\indexbold{*notI theorem}
lcp@105
   208
paulson@5205
   209
There is a simpler way of conducting this proof.  The \ttindex{Goalw}
paulson@5205
   210
command starts a backward proof, as does \texttt{Goal}, but it also
lcp@296
   211
unfolds definitions.  Thus there is no need to call
lcp@296
   212
\ttindex{rewrite_goals_tac}:
lcp@105
   213
\begin{ttbox}
paulson@5205
   214
val prems = Goalw [not_def]
lcp@105
   215
    "(P ==> False) ==> ~P";
lcp@105
   216
{\out Level 0}
lcp@105
   217
{\out ~P}
lcp@105
   218
{\out  1. P --> False}
lcp@105
   219
{\out val prems = ["P ==> False  [P ==> False]"] : thm list}
lcp@105
   220
\end{ttbox}
lcp@105
   221
lcp@105
   222
lcp@296
   223
\subsection{Deriving the $\neg$ elimination rule}
paulson@5205
   224
Let us derive the rule $(\neg E)$.  The proof follows that of~\texttt{conjE}
lcp@296
   225
above, with an additional step to unfold negation in the major premise.
paulson@5205
   226
The \texttt{Goalw} command is best for this: it unfolds definitions not only
paulson@5205
   227
in the conclusion but the premises.
lcp@105
   228
\begin{ttbox}
paulson@5205
   229
Goalw [not_def] "[| ~P;  P |] ==> R";
lcp@105
   230
{\out Level 0}
paulson@5205
   231
{\out [| ~ P; P |] ==> R}
paulson@5205
   232
{\out  1. [| P --> False; P |] ==> R}
paulson@5205
   233
\end{ttbox}
paulson@5205
   234
As the first step, we apply \tdx{FalseE}:
paulson@5205
   235
\begin{ttbox}
lcp@105
   236
by (resolve_tac [FalseE] 1);
lcp@105
   237
{\out Level 1}
paulson@5205
   238
{\out [| ~ P; P |] ==> R}
paulson@5205
   239
{\out  1. [| P --> False; P |] ==> False}
lcp@284
   240
\end{ttbox}
paulson@5205
   241
%
lcp@284
   242
Everything follows from falsity.  And we can prove falsity using the
lcp@284
   243
premises and Modus Ponens:
lcp@284
   244
\begin{ttbox}
paulson@5205
   245
by (eresolve_tac [mp] 1);
lcp@105
   246
{\out Level 2}
paulson@5205
   247
{\out [| ~ P; P |] ==> R}
paulson@5205
   248
{\out  1. P ==> P}
paulson@5205
   249
\ttbreak
paulson@5205
   250
by (assume_tac 1);
lcp@105
   251
{\out Level 3}
paulson@5205
   252
{\out [| ~ P; P |] ==> R}
lcp@105
   253
{\out No subgoals!}
paulson@5205
   254
\ttbreak
wenzelm@3103
   255
qed "notE";
lcp@105
   256
{\out val notE = "[| ~?P; ?P |] ==> ?R" : thm}
lcp@105
   257
\end{ttbox}
paulson@5205
   258
lcp@105
   259
lcp@105
   260
\medskip
paulson@5205
   261
\texttt{Goalw} unfolds definitions in the premises even when it has to return
paulson@5205
   262
them as a list.  Another way of unfolding definitions in a theorem is by
paulson@5205
   263
applying the function \ttindex{rewrite_rule}.
lcp@105
   264
paulson@5205
   265
\index{definitions!and derived rules|)}
lcp@105
   266
lcp@105
   267
lcp@284
   268
\section{Defining theories}\label{sec:defining-theories}
lcp@105
   269
\index{theories!defining|(}
lcp@310
   270
wenzelm@3103
   271
Isabelle makes no distinction between simple extensions of a logic ---
wenzelm@3103
   272
like specifying a type~$bool$ with constants~$true$ and~$false$ ---
wenzelm@3103
   273
and defining an entire logic.  A theory definition has a form like
lcp@105
   274
\begin{ttbox}
lcp@105
   275
\(T\) = \(S@1\) + \(\cdots\) + \(S@n\) +
lcp@105
   276
classes      {\it class declarations}
lcp@105
   277
default      {\it sort}
lcp@331
   278
types        {\it type declarations and synonyms}
wenzelm@3103
   279
arities      {\it type arity declarations}
lcp@105
   280
consts       {\it constant declarations}
wenzelm@3103
   281
syntax       {\it syntactic constant declarations}
wenzelm@3103
   282
translations {\it ast translation rules}
wenzelm@3103
   283
defs         {\it meta-logical definitions}
lcp@105
   284
rules        {\it rule declarations}
lcp@105
   285
end
lcp@105
   286
ML           {\it ML code}
lcp@105
   287
\end{ttbox}
lcp@105
   288
This declares the theory $T$ to extend the existing theories
wenzelm@3103
   289
$S@1$,~\ldots,~$S@n$.  It may introduce new classes, types, arities
wenzelm@3103
   290
(of existing types), constants and rules; it can specify the default
wenzelm@3103
   291
sort for type variables.  A constant declaration can specify an
wenzelm@3103
   292
associated concrete syntax.  The translations section specifies
wenzelm@3103
   293
rewrite rules on abstract syntax trees, handling notations and
paulson@5205
   294
abbreviations.  \index{*ML section} The \texttt{ML} section may contain
wenzelm@3103
   295
code to perform arbitrary syntactic transformations.  The main
wenzelm@3212
   296
declaration forms are discussed below.  There are some more sections
wenzelm@3212
   297
not presented here, the full syntax can be found in
wenzelm@3212
   298
\iflabelundefined{app:TheorySyntax}{an appendix of the {\it Reference
wenzelm@3212
   299
    Manual}}{App.\ts\ref{app:TheorySyntax}}.  Also note that
wenzelm@3212
   300
object-logics may add further theory sections, for example
wenzelm@9695
   301
\texttt{typedef}, \texttt{datatype} in HOL.
lcp@105
   302
wenzelm@3103
   303
All the declaration parts can be omitted or repeated and may appear in
wenzelm@3103
   304
any order, except that the {\ML} section must be last (after the {\tt
wenzelm@3103
   305
  end} keyword).  In the simplest case, $T$ is just the union of
wenzelm@3103
   306
$S@1$,~\ldots,~$S@n$.  New theories always extend one or more other
wenzelm@3103
   307
theories, inheriting their types, constants, syntax, etc.  The theory
paulson@3485
   308
\thydx{Pure} contains nothing but Isabelle's meta-logic.  The variant
wenzelm@3103
   309
\thydx{CPure} offers the more usual higher-order function application
wenzelm@9695
   310
syntax $t\,u@1\ldots\,u@n$ instead of $t(u@1,\ldots,u@n)$ in Pure.
lcp@105
   311
wenzelm@3103
   312
Each theory definition must reside in a separate file, whose name is
wenzelm@3103
   313
the theory's with {\tt.thy} appended.  Calling
wenzelm@3103
   314
\ttindexbold{use_thy}~{\tt"{\it T\/}"} reads the definition from {\it
wenzelm@3103
   315
  T}{\tt.thy}, writes a corresponding file of {\ML} code {\tt.{\it
wenzelm@3103
   316
    T}.thy.ML}, reads the latter file, and deletes it if no errors
wenzelm@3103
   317
occurred.  This declares the {\ML} structure~$T$, which contains a
paulson@5205
   318
component \texttt{thy} denoting the new theory, a component for each
wenzelm@3103
   319
rule, and everything declared in {\it ML code}.
lcp@105
   320
wenzelm@3103
   321
Errors may arise during the translation to {\ML} (say, a misspelled
wenzelm@3103
   322
keyword) or during creation of the new theory (say, a type error in a
paulson@5205
   323
rule).  But if all goes well, \texttt{use_thy} will finally read the file
wenzelm@3103
   324
{\it T}{\tt.ML} (if it exists).  This file typically contains proofs
paulson@3485
   325
that refer to the components of~$T$.  The structure is automatically
wenzelm@3103
   326
opened, so its components may be referred to by unqualified names,
paulson@5205
   327
e.g.\ just \texttt{thy} instead of $T$\texttt{.thy}.
lcp@105
   328
wenzelm@3103
   329
\ttindexbold{use_thy} automatically loads a theory's parents before
wenzelm@3103
   330
loading the theory itself.  When a theory file is modified, many
wenzelm@3103
   331
theories may have to be reloaded.  Isabelle records the modification
wenzelm@3103
   332
times and dependencies of theory files.  See
lcp@331
   333
\iflabelundefined{sec:reloading-theories}{the {\em Reference Manual\/}}%
lcp@331
   334
                 {\S\ref{sec:reloading-theories}}
lcp@296
   335
for more details.
lcp@296
   336
lcp@105
   337
lcp@1084
   338
\subsection{Declaring constants, definitions and rules}
lcp@310
   339
\indexbold{constants!declaring}\index{rules!declaring}
lcp@310
   340
lcp@1084
   341
Most theories simply declare constants, definitions and rules.  The {\bf
lcp@1084
   342
  constant declaration part} has the form
lcp@105
   343
\begin{ttbox}
clasohm@1387
   344
consts  \(c@1\) :: \(\tau@1\)
lcp@105
   345
        \vdots
clasohm@1387
   346
        \(c@n\) :: \(\tau@n\)
lcp@105
   347
\end{ttbox}
lcp@105
   348
where $c@1$, \ldots, $c@n$ are constants and $\tau@1$, \ldots, $\tau@n$ are
clasohm@1387
   349
types.  The types must be enclosed in quotation marks if they contain
paulson@5205
   350
user-declared infix type constructors like \texttt{*}.  Each
lcp@105
   351
constant must be enclosed in quotation marks unless it is a valid
lcp@105
   352
identifier.  To declare $c@1$, \ldots, $c@n$ as constants of type $\tau$,
lcp@105
   353
the $n$ declarations may be abbreviated to a single line:
lcp@105
   354
\begin{ttbox}
clasohm@1387
   355
        \(c@1\), \ldots, \(c@n\) :: \(\tau\)
lcp@105
   356
\end{ttbox}
lcp@105
   357
The {\bf rule declaration part} has the form
lcp@105
   358
\begin{ttbox}
lcp@105
   359
rules   \(id@1\) "\(rule@1\)"
lcp@105
   360
        \vdots
lcp@105
   361
        \(id@n\) "\(rule@n\)"
lcp@105
   362
\end{ttbox}
lcp@105
   363
where $id@1$, \ldots, $id@n$ are \ML{} identifiers and $rule@1$, \ldots,
lcp@284
   364
$rule@n$ are expressions of type~$prop$.  Each rule {\em must\/} be
paulson@14148
   365
enclosed in quotation marks.  Rules are simply axioms; they are 
paulson@14148
   366
called \emph{rules} because they are mainly used to specify the inference
paulson@14148
   367
rules when defining a new logic.
lcp@284
   368
wenzelm@3103
   369
\indexbold{definitions} The {\bf definition part} is similar, but with
paulson@5205
   370
the keyword \texttt{defs} instead of \texttt{rules}.  {\bf Definitions} are
wenzelm@3103
   371
rules of the form $s \equiv t$, and should serve only as
paulson@3485
   372
abbreviations.  The simplest form of a definition is $f \equiv t$,
paulson@3485
   373
where $f$ is a constant.  Also allowed are $\eta$-equivalent forms of
wenzelm@3106
   374
this, where the arguments of~$f$ appear applied on the left-hand side
wenzelm@3106
   375
of the equation instead of abstracted on the right-hand side.
lcp@1084
   376
wenzelm@3103
   377
Isabelle checks for common errors in definitions, such as extra
paulson@14148
   378
variables on the right-hand side and cyclic dependencies, that could
paulson@14148
   379
least to inconsistency.  It is still essential to take care:
paulson@14148
   380
theorems proved on the basis of incorrect definitions are useless,
paulson@14148
   381
your system can be consistent and yet still wrong.
wenzelm@3103
   382
wenzelm@3103
   383
\index{examples!of theories} This example theory extends first-order
wenzelm@3103
   384
logic by declaring and defining two constants, {\em nand} and {\em
wenzelm@3103
   385
  xor}:
lcp@284
   386
\begin{ttbox}
lcp@105
   387
Gate = FOL +
clasohm@1387
   388
consts  nand,xor :: [o,o] => o
lcp@1084
   389
defs    nand_def "nand(P,Q) == ~(P & Q)"
lcp@105
   390
        xor_def  "xor(P,Q)  == P & ~Q | ~P & Q"
lcp@105
   391
end
lcp@105
   392
\end{ttbox}
lcp@105
   393
nipkow@1649
   394
Declaring and defining constants can be combined:
nipkow@1649
   395
\begin{ttbox}
nipkow@1649
   396
Gate = FOL +
nipkow@1649
   397
constdefs  nand :: [o,o] => o
nipkow@1649
   398
           "nand(P,Q) == ~(P & Q)"
nipkow@1649
   399
           xor  :: [o,o] => o
nipkow@1649
   400
           "xor(P,Q)  == P & ~Q | ~P & Q"
nipkow@1649
   401
end
nipkow@1649
   402
\end{ttbox}
paulson@5205
   403
\texttt{constdefs} generates the names \texttt{nand_def} and \texttt{xor_def}
paulson@3485
   404
automatically, which is why it is restricted to alphanumeric identifiers.  In
nipkow@1649
   405
general it has the form
nipkow@1649
   406
\begin{ttbox}
nipkow@1649
   407
constdefs  \(id@1\) :: \(\tau@1\)
nipkow@1649
   408
           "\(id@1 \equiv \dots\)"
nipkow@1649
   409
           \vdots
nipkow@1649
   410
           \(id@n\) :: \(\tau@n\)
nipkow@1649
   411
           "\(id@n \equiv \dots\)"
nipkow@1649
   412
\end{ttbox}
nipkow@1649
   413
nipkow@1649
   414
nipkow@1366
   415
\begin{warn}
nipkow@1366
   416
A common mistake when writing definitions is to introduce extra free variables
nipkow@1468
   417
on the right-hand side as in the following fictitious definition:
nipkow@1366
   418
\begin{ttbox}
nipkow@1366
   419
defs  prime_def "prime(p) == (m divides p) --> (m=1 | m=p)"
nipkow@1366
   420
\end{ttbox}
paulson@5205
   421
Isabelle rejects this ``definition'' because of the extra \texttt{m} on the
paulson@3485
   422
right-hand side, which would introduce an inconsistency.  What you should have
nipkow@1366
   423
written is
nipkow@1366
   424
\begin{ttbox}
nipkow@1366
   425
defs  prime_def "prime(p) == ALL m. (m divides p) --> (m=1 | m=p)"
nipkow@1366
   426
\end{ttbox}
nipkow@1366
   427
\end{warn}
lcp@105
   428
lcp@105
   429
\subsection{Declaring type constructors}
nipkow@303
   430
\indexbold{types!declaring}\indexbold{arities!declaring}
lcp@284
   431
%
lcp@105
   432
Types are composed of type variables and {\bf type constructors}.  Each
lcp@284
   433
type constructor takes a fixed number of arguments.  They are declared
lcp@284
   434
with an \ML-like syntax.  If $list$ takes one type argument, $tree$ takes
lcp@284
   435
two arguments and $nat$ takes no arguments, then these type constructors
lcp@284
   436
can be declared by
lcp@105
   437
\begin{ttbox}
lcp@284
   438
types 'a list
lcp@284
   439
      ('a,'b) tree
lcp@284
   440
      nat
lcp@105
   441
\end{ttbox}
lcp@284
   442
lcp@284
   443
The {\bf type declaration part} has the general form
lcp@284
   444
\begin{ttbox}
lcp@284
   445
types   \(tids@1\) \(id@1\)
lcp@284
   446
        \vdots
wenzelm@841
   447
        \(tids@n\) \(id@n\)
lcp@284
   448
\end{ttbox}
lcp@284
   449
where $id@1$, \ldots, $id@n$ are identifiers and $tids@1$, \ldots, $tids@n$
lcp@284
   450
are type argument lists as shown in the example above.  It declares each
lcp@284
   451
$id@i$ as a type constructor with the specified number of argument places.
lcp@105
   452
lcp@105
   453
The {\bf arity declaration part} has the form
lcp@105
   454
\begin{ttbox}
lcp@105
   455
arities \(tycon@1\) :: \(arity@1\)
lcp@105
   456
        \vdots
lcp@105
   457
        \(tycon@n\) :: \(arity@n\)
lcp@105
   458
\end{ttbox}
lcp@105
   459
where $tycon@1$, \ldots, $tycon@n$ are identifiers and $arity@1$, \ldots,
lcp@105
   460
$arity@n$ are arities.  Arity declarations add arities to existing
lcp@296
   461
types; they do not declare the types themselves.
lcp@105
   462
In the simplest case, for an 0-place type constructor, an arity is simply
lcp@105
   463
the type's class.  Let us declare a type~$bool$ of class $term$, with
lcp@284
   464
constants $tt$ and~$ff$.  (In first-order logic, booleans are
lcp@284
   465
distinct from formulae, which have type $o::logic$.)
lcp@105
   466
\index{examples!of theories}
lcp@284
   467
\begin{ttbox}
lcp@105
   468
Bool = FOL +
lcp@284
   469
types   bool
lcp@105
   470
arities bool    :: term
clasohm@1387
   471
consts  tt,ff   :: bool
lcp@105
   472
end
lcp@105
   473
\end{ttbox}
lcp@296
   474
A $k$-place type constructor may have arities of the form
lcp@296
   475
$(s@1,\ldots,s@k)c$, where $s@1,\ldots,s@n$ are sorts and $c$ is a class.
lcp@296
   476
Each sort specifies a type argument; it has the form $\{c@1,\ldots,c@m\}$,
lcp@296
   477
where $c@1$, \dots,~$c@m$ are classes.  Mostly we deal with singleton
lcp@296
   478
sorts, and may abbreviate them by dropping the braces.  The arity
lcp@296
   479
$(term)term$ is short for $(\{term\})term$.  Recall the discussion in
lcp@296
   480
\S\ref{polymorphic}.
lcp@105
   481
lcp@105
   482
A type constructor may be overloaded (subject to certain conditions) by
lcp@296
   483
appearing in several arity declarations.  For instance, the function type
lcp@331
   484
constructor~$fun$ has the arity $(logic,logic)logic$; in higher-order
lcp@105
   485
logic, it is declared also to have arity $(term,term)term$.
lcp@105
   486
paulson@5205
   487
Theory \texttt{List} declares the 1-place type constructor $list$, gives
paulson@5205
   488
it the arity $(term)term$, and declares constants $Nil$ and $Cons$ with
lcp@296
   489
polymorphic types:%
paulson@5205
   490
\footnote{In the \texttt{consts} part, type variable {\tt'a} has the default
paulson@5205
   491
  sort, which is \texttt{term}.  See the {\em Reference Manual\/}
lcp@296
   492
\iflabelundefined{sec:ref-defining-theories}{}%
lcp@296
   493
{(\S\ref{sec:ref-defining-theories})} for more information.}
lcp@105
   494
\index{examples!of theories}
lcp@284
   495
\begin{ttbox}
lcp@105
   496
List = FOL +
lcp@284
   497
types   'a list
lcp@105
   498
arities list    :: (term)term
clasohm@1387
   499
consts  Nil     :: 'a list
clasohm@1387
   500
        Cons    :: ['a, 'a list] => 'a list
lcp@105
   501
end
lcp@105
   502
\end{ttbox}
lcp@284
   503
Multiple arity declarations may be abbreviated to a single line:
lcp@105
   504
\begin{ttbox}
lcp@105
   505
arities \(tycon@1\), \ldots, \(tycon@n\) :: \(arity\)
lcp@105
   506
\end{ttbox}
lcp@105
   507
wenzelm@3103
   508
%\begin{warn}
wenzelm@3103
   509
%Arity declarations resemble constant declarations, but there are {\it no\/}
wenzelm@3103
   510
%quotation marks!  Types and rules must be quoted because the theory
wenzelm@3103
   511
%translator passes them verbatim to the {\ML} output file.
wenzelm@3103
   512
%\end{warn}
lcp@105
   513
lcp@331
   514
\subsection{Type synonyms}\indexbold{type synonyms}
nipkow@303
   515
Isabelle supports {\bf type synonyms} ({\bf abbreviations}) which are similar
lcp@307
   516
to those found in \ML.  Such synonyms are defined in the type declaration part
nipkow@303
   517
and are fairly self explanatory:
nipkow@303
   518
\begin{ttbox}
clasohm@1387
   519
types gate       = [o,o] => o
clasohm@1387
   520
      'a pred    = 'a => o
clasohm@1387
   521
      ('a,'b)nuf = 'b => 'a
nipkow@303
   522
\end{ttbox}
nipkow@303
   523
Type declarations and synonyms can be mixed arbitrarily:
nipkow@303
   524
\begin{ttbox}
nipkow@303
   525
types nat
clasohm@1387
   526
      'a stream = nat => 'a
clasohm@1387
   527
      signal    = nat stream
nipkow@303
   528
      'a list
nipkow@303
   529
\end{ttbox}
wenzelm@3103
   530
A synonym is merely an abbreviation for some existing type expression.
wenzelm@3103
   531
Hence synonyms may not be recursive!  Internally all synonyms are
wenzelm@3103
   532
fully expanded.  As a consequence Isabelle output never contains
wenzelm@3103
   533
synonyms.  Their main purpose is to improve the readability of theory
wenzelm@3103
   534
definitions.  Synonyms can be used just like any other type:
nipkow@303
   535
\begin{ttbox}
clasohm@1387
   536
consts and,or :: gate
clasohm@1387
   537
       negate :: signal => signal
nipkow@303
   538
\end{ttbox}
nipkow@303
   539
lcp@348
   540
\subsection{Infix and mixfix operators}
lcp@310
   541
\index{infixes}\index{examples!of theories}
lcp@310
   542
lcp@310
   543
Infix or mixfix syntax may be attached to constants.  Consider the
lcp@310
   544
following theory:
lcp@284
   545
\begin{ttbox}
lcp@105
   546
Gate2 = FOL +
clasohm@1387
   547
consts  "~&"     :: [o,o] => o         (infixl 35)
clasohm@1387
   548
        "#"      :: [o,o] => o         (infixl 30)
lcp@1084
   549
defs    nand_def "P ~& Q == ~(P & Q)"    
lcp@105
   550
        xor_def  "P # Q  == P & ~Q | ~P & Q"
lcp@105
   551
end
lcp@105
   552
\end{ttbox}
lcp@310
   553
The constant declaration part declares two left-associating infix operators
lcp@310
   554
with their priorities, or precedences; they are $\nand$ of priority~35 and
lcp@310
   555
$\xor$ of priority~30.  Hence $P \xor Q \xor R$ is parsed as $(P\xor Q)
lcp@310
   556
\xor R$ and $P \xor Q \nand R$ as $P \xor (Q \nand R)$.  Note the quotation
lcp@310
   557
marks in \verb|"~&"| and \verb|"#"|.
lcp@105
   558
lcp@105
   559
The constants \hbox{\verb|op ~&|} and \hbox{\verb|op #|} are declared
lcp@105
   560
automatically, just as in \ML.  Hence you may write propositions like
lcp@105
   561
\verb|op #(True) == op ~&(True)|, which asserts that the functions $\lambda
lcp@105
   562
Q.True \xor Q$ and $\lambda Q.True \nand Q$ are identical.
lcp@105
   563
wenzelm@3212
   564
\medskip Infix syntax and constant names may be also specified
paulson@3485
   565
independently.  For example, consider this version of $\nand$:
wenzelm@3212
   566
\begin{ttbox}
wenzelm@3212
   567
consts  nand     :: [o,o] => o         (infixl "~&" 35)
wenzelm@3212
   568
\end{ttbox}
wenzelm@3212
   569
lcp@310
   570
\bigskip\index{mixfix declarations}
lcp@310
   571
{\bf Mixfix} operators may have arbitrary context-free syntaxes.  Let us
lcp@310
   572
add a line to the constant declaration part:
lcp@284
   573
\begin{ttbox}
clasohm@1387
   574
        If :: [o,o,o] => o       ("if _ then _ else _")
lcp@105
   575
\end{ttbox}
lcp@310
   576
This declares a constant $If$ of type $[o,o,o] \To o$ with concrete syntax {\tt
paulson@5205
   577
  if~$P$ then~$Q$ else~$R$} as well as \texttt{If($P$,$Q$,$R$)}.  Underscores
lcp@310
   578
denote argument positions.  
lcp@105
   579
paulson@5205
   580
The declaration above does not allow the \texttt{if}-\texttt{then}-{\tt
wenzelm@3103
   581
  else} construct to be printed split across several lines, even if it
wenzelm@3103
   582
is too long to fit on one line.  Pretty-printing information can be
wenzelm@3103
   583
added to specify the layout of mixfix operators.  For details, see
lcp@310
   584
\iflabelundefined{Defining-Logics}%
lcp@310
   585
    {the {\it Reference Manual}, chapter `Defining Logics'}%
lcp@310
   586
    {Chap.\ts\ref{Defining-Logics}}.
lcp@310
   587
lcp@310
   588
Mixfix declarations can be annotated with priorities, just like
lcp@105
   589
infixes.  The example above is just a shorthand for
lcp@284
   590
\begin{ttbox}
clasohm@1387
   591
        If :: [o,o,o] => o       ("if _ then _ else _" [0,0,0] 1000)
lcp@105
   592
\end{ttbox}
lcp@310
   593
The numeric components determine priorities.  The list of integers
lcp@310
   594
defines, for each argument position, the minimal priority an expression
lcp@310
   595
at that position must have.  The final integer is the priority of the
lcp@105
   596
construct itself.  In the example above, any argument expression is
lcp@310
   597
acceptable because priorities are non-negative, and conditionals may
lcp@310
   598
appear everywhere because 1000 is the highest priority.  On the other
lcp@310
   599
hand, the declaration
lcp@284
   600
\begin{ttbox}
clasohm@1387
   601
        If :: [o,o,o] => o       ("if _ then _ else _" [100,0,0] 99)
lcp@105
   602
\end{ttbox}
lcp@284
   603
defines concrete syntax for a conditional whose first argument cannot have
paulson@5205
   604
the form \texttt{if~$P$ then~$Q$ else~$R$} because it must have a priority
lcp@310
   605
of at least~100.  We may of course write
lcp@284
   606
\begin{quote}\tt
lcp@284
   607
if (if $P$ then $Q$ else $R$) then $S$ else $T$
lcp@156
   608
\end{quote}
lcp@310
   609
because expressions in parentheses have maximal priority.  
lcp@105
   610
lcp@105
   611
Binary type constructors, like products and sums, may also be declared as
lcp@105
   612
infixes.  The type declaration below introduces a type constructor~$*$ with
lcp@105
   613
infix notation $\alpha*\beta$, together with the mixfix notation
lcp@1084
   614
${<}\_,\_{>}$ for pairs.  We also see a rule declaration part.
lcp@310
   615
\index{examples!of theories}\index{mixfix declarations}
lcp@105
   616
\begin{ttbox}
lcp@105
   617
Prod = FOL +
lcp@284
   618
types   ('a,'b) "*"                           (infixl 20)
lcp@105
   619
arities "*"     :: (term,term)term
lcp@105
   620
consts  fst     :: "'a * 'b => 'a"
lcp@105
   621
        snd     :: "'a * 'b => 'b"
lcp@105
   622
        Pair    :: "['a,'b] => 'a * 'b"       ("(1<_,/_>)")
lcp@105
   623
rules   fst     "fst(<a,b>) = a"
lcp@105
   624
        snd     "snd(<a,b>) = b"
lcp@105
   625
end
lcp@105
   626
\end{ttbox}
lcp@105
   627
lcp@105
   628
\begin{warn}
paulson@5205
   629
  The name of the type constructor is~\texttt{*} and not \texttt{op~*}, as
wenzelm@3103
   630
  it would be in the case of an infix constant.  Only infix type
paulson@5205
   631
  constructors can have symbolic names like~\texttt{*}.  General mixfix
paulson@5205
   632
  syntax for types may be introduced via appropriate \texttt{syntax}
wenzelm@3103
   633
  declarations.
lcp@105
   634
\end{warn}
lcp@105
   635
lcp@105
   636
lcp@105
   637
\subsection{Overloading}
lcp@105
   638
\index{overloading}\index{examples!of theories}
lcp@105
   639
The {\bf class declaration part} has the form
lcp@105
   640
\begin{ttbox}
lcp@105
   641
classes \(id@1\) < \(c@1\)
lcp@105
   642
        \vdots
lcp@105
   643
        \(id@n\) < \(c@n\)
lcp@105
   644
\end{ttbox}
lcp@105
   645
where $id@1$, \ldots, $id@n$ are identifiers and $c@1$, \ldots, $c@n$ are
lcp@105
   646
existing classes.  It declares each $id@i$ as a new class, a subclass
lcp@105
   647
of~$c@i$.  In the general case, an identifier may be declared to be a
lcp@105
   648
subclass of $k$ existing classes:
lcp@105
   649
\begin{ttbox}
lcp@105
   650
        \(id\) < \(c@1\), \ldots, \(c@k\)
lcp@105
   651
\end{ttbox}
lcp@296
   652
Type classes allow constants to be overloaded.  As suggested in
lcp@307
   653
\S\ref{polymorphic}, let us define the class $arith$ of arithmetic
lcp@296
   654
types with the constants ${+} :: [\alpha,\alpha]\To \alpha$ and $0,1 {::}
lcp@296
   655
\alpha$, for $\alpha{::}arith$.  We introduce $arith$ as a subclass of
lcp@296
   656
$term$ and add the three polymorphic constants of this class.
lcp@310
   657
\index{examples!of theories}\index{constants!overloaded}
lcp@105
   658
\begin{ttbox}
lcp@105
   659
Arith = FOL +
lcp@105
   660
classes arith < term
clasohm@1387
   661
consts  "0"     :: 'a::arith                  ("0")
clasohm@1387
   662
        "1"     :: 'a::arith                  ("1")
clasohm@1387
   663
        "+"     :: ['a::arith,'a] => 'a       (infixl 60)
lcp@105
   664
end
lcp@105
   665
\end{ttbox}
lcp@105
   666
No rules are declared for these constants: we merely introduce their
lcp@105
   667
names without specifying properties.  On the other hand, classes
lcp@105
   668
with rules make it possible to prove {\bf generic} theorems.  Such
lcp@105
   669
theorems hold for all instances, all types in that class.
lcp@105
   670
lcp@105
   671
We can now obtain distinct versions of the constants of $arith$ by
lcp@105
   672
declaring certain types to be of class $arith$.  For example, let us
lcp@105
   673
declare the 0-place type constructors $bool$ and $nat$:
lcp@105
   674
\index{examples!of theories}
lcp@105
   675
\begin{ttbox}
lcp@105
   676
BoolNat = Arith +
lcp@348
   677
types   bool  nat
lcp@348
   678
arities bool, nat   :: arith
clasohm@1387
   679
consts  Suc         :: nat=>nat
lcp@284
   680
\ttbreak
lcp@105
   681
rules   add0        "0 + n = n::nat"
lcp@105
   682
        addS        "Suc(m)+n = Suc(m+n)"
lcp@105
   683
        nat1        "1 = Suc(0)"
lcp@105
   684
        or0l        "0 + x = x::bool"
lcp@105
   685
        or0r        "x + 0 = x::bool"
lcp@105
   686
        or1l        "1 + x = 1::bool"
lcp@105
   687
        or1r        "x + 1 = 1::bool"
lcp@105
   688
end
lcp@105
   689
\end{ttbox}
lcp@105
   690
Because $nat$ and $bool$ have class $arith$, we can use $0$, $1$ and $+$ at
lcp@105
   691
either type.  The type constraints in the axioms are vital.  Without
paulson@14148
   692
constraints, the $x$ in $1+x = 1$ (axiom \texttt{or1l})
paulson@14148
   693
would have type $\alpha{::}arith$
lcp@105
   694
and the axiom would hold for any type of class $arith$.  This would
lcp@284
   695
collapse $nat$ to a trivial type:
lcp@105
   696
\[ Suc(1) = Suc(0+1) = Suc(0)+1 = 1+1 = 1! \]
lcp@296
   697
lcp@105
   698
lcp@296
   699
\section{Theory example: the natural numbers}
lcp@296
   700
lcp@296
   701
We shall now work through a small example of formalized mathematics
lcp@105
   702
demonstrating many of the theory extension features.
lcp@105
   703
lcp@105
   704
lcp@105
   705
\subsection{Extending first-order logic with the natural numbers}
lcp@105
   706
\index{examples!of theories}
lcp@105
   707
lcp@284
   708
Section\ts\ref{sec:logical-syntax} has formalized a first-order logic,
lcp@284
   709
including a type~$nat$ and the constants $0::nat$ and $Suc::nat\To nat$.
lcp@284
   710
Let us introduce the Peano axioms for mathematical induction and the
lcp@310
   711
freeness of $0$ and~$Suc$:\index{axioms!Peano}
lcp@307
   712
\[ \vcenter{\infer[(induct)]{P[n/x]}{P[0/x] & \infer*{P[Suc(x)/x]}{[P]}}}
lcp@105
   713
 \qquad \parbox{4.5cm}{provided $x$ is not free in any assumption except~$P$}
lcp@105
   714
\]
lcp@105
   715
\[ \infer[(Suc\_inject)]{m=n}{Suc(m)=Suc(n)} \qquad
lcp@105
   716
   \infer[(Suc\_neq\_0)]{R}{Suc(m)=0}
lcp@105
   717
\]
lcp@105
   718
Mathematical induction asserts that $P(n)$ is true, for any $n::nat$,
lcp@105
   719
provided $P(0)$ holds and that $P(x)$ implies $P(Suc(x))$ for all~$x$.
lcp@105
   720
Some authors express the induction step as $\forall x. P(x)\imp P(Suc(x))$.
lcp@105
   721
To avoid making induction require the presence of other connectives, we
lcp@105
   722
formalize mathematical induction as
lcp@105
   723
$$ \List{P(0); \Forall x. P(x)\Imp P(Suc(x))} \Imp P(n). \eqno(induct) $$
lcp@105
   724
lcp@105
   725
\noindent
lcp@105
   726
Similarly, to avoid expressing the other rules using~$\forall$, $\imp$
lcp@105
   727
and~$\neg$, we take advantage of the meta-logic;\footnote
lcp@105
   728
{On the other hand, the axioms $Suc(m)=Suc(n) \bimp m=n$
lcp@105
   729
and $\neg(Suc(m)=0)$ are logically equivalent to those given, and work
lcp@105
   730
better with Isabelle's simplifier.} 
lcp@105
   731
$(Suc\_neq\_0)$ is
lcp@105
   732
an elimination rule for $Suc(m)=0$:
lcp@105
   733
$$ Suc(m)=Suc(n) \Imp m=n  \eqno(Suc\_inject) $$
lcp@105
   734
$$ Suc(m)=0      \Imp R    \eqno(Suc\_neq\_0) $$
lcp@105
   735
lcp@105
   736
\noindent
lcp@105
   737
We shall also define a primitive recursion operator, $rec$.  Traditionally,
lcp@105
   738
primitive recursion takes a natural number~$a$ and a 2-place function~$f$,
lcp@105
   739
and obeys the equations
lcp@105
   740
\begin{eqnarray*}
lcp@105
   741
  rec(0,a,f)            & = & a \\
lcp@105
   742
  rec(Suc(m),a,f)       & = & f(m, rec(m,a,f))
lcp@105
   743
\end{eqnarray*}
lcp@105
   744
Addition, defined by $m+n \equiv rec(m,n,\lambda x\,y.Suc(y))$,
lcp@105
   745
should satisfy
lcp@105
   746
\begin{eqnarray*}
lcp@105
   747
  0+n      & = & n \\
lcp@105
   748
  Suc(m)+n & = & Suc(m+n)
lcp@105
   749
\end{eqnarray*}
lcp@296
   750
Primitive recursion appears to pose difficulties: first-order logic has no
lcp@296
   751
function-valued expressions.  We again take advantage of the meta-logic,
lcp@296
   752
which does have functions.  We also generalise primitive recursion to be
lcp@105
   753
polymorphic over any type of class~$term$, and declare the addition
lcp@105
   754
function:
lcp@105
   755
\begin{eqnarray*}
lcp@105
   756
  rec   & :: & [nat, \alpha{::}term, [nat,\alpha]\To\alpha] \To\alpha \\
lcp@105
   757
  +     & :: & [nat,nat]\To nat 
lcp@105
   758
\end{eqnarray*}
lcp@105
   759
lcp@105
   760
lcp@105
   761
\subsection{Declaring the theory to Isabelle}
lcp@105
   762
\index{examples!of theories}
lcp@310
   763
Let us create the theory \thydx{Nat} starting from theory~\verb$FOL$,
lcp@105
   764
which contains only classical logic with no natural numbers.  We declare
lcp@307
   765
the 0-place type constructor $nat$ and the associated constants.  Note that
lcp@307
   766
the constant~0 requires a mixfix annotation because~0 is not a legal
lcp@307
   767
identifier, and could not otherwise be written in terms:
lcp@310
   768
\begin{ttbox}\index{mixfix declarations}
lcp@105
   769
Nat = FOL +
lcp@284
   770
types   nat
lcp@105
   771
arities nat         :: term
clasohm@1387
   772
consts  "0"         :: nat                              ("0")
clasohm@1387
   773
        Suc         :: nat=>nat
clasohm@1387
   774
        rec         :: [nat, 'a, [nat,'a]=>'a] => 'a
clasohm@1387
   775
        "+"         :: [nat, nat] => nat                (infixl 60)
lcp@296
   776
rules   Suc_inject  "Suc(m)=Suc(n) ==> m=n"
lcp@105
   777
        Suc_neq_0   "Suc(m)=0      ==> R"
lcp@296
   778
        induct      "[| P(0);  !!x. P(x) ==> P(Suc(x)) |]  ==> P(n)"
lcp@105
   779
        rec_0       "rec(0,a,f) = a"
lcp@105
   780
        rec_Suc     "rec(Suc(m), a, f) = f(m, rec(m,a,f))"
lcp@296
   781
        add_def     "m+n == rec(m, n, \%x y. Suc(y))"
lcp@105
   782
end
lcp@105
   783
\end{ttbox}
paulson@5205
   784
In axiom \texttt{add_def}, recall that \verb|%| stands for~$\lambda$.
paulson@5205
   785
Loading this theory file creates the \ML\ structure \texttt{Nat}, which
wenzelm@3103
   786
contains the theory and axioms.
lcp@296
   787
lcp@296
   788
\subsection{Proving some recursion equations}
paulson@5205
   789
Theory \texttt{FOL/ex/Nat} contains proofs involving this theory of the
lcp@105
   790
natural numbers.  As a trivial example, let us derive recursion equations
lcp@105
   791
for \verb$+$.  Here is the zero case:
lcp@284
   792
\begin{ttbox}
paulson@5205
   793
Goalw [add_def] "0+n = n";
lcp@105
   794
{\out Level 0}
lcp@105
   795
{\out 0 + n = n}
lcp@284
   796
{\out  1. rec(0,n,\%x y. Suc(y)) = n}
lcp@105
   797
\ttbreak
lcp@105
   798
by (resolve_tac [rec_0] 1);
lcp@105
   799
{\out Level 1}
lcp@105
   800
{\out 0 + n = n}
lcp@105
   801
{\out No subgoals!}
wenzelm@3103
   802
qed "add_0";
lcp@284
   803
\end{ttbox}
lcp@105
   804
And here is the successor case:
lcp@284
   805
\begin{ttbox}
paulson@5205
   806
Goalw [add_def] "Suc(m)+n = Suc(m+n)";
lcp@105
   807
{\out Level 0}
lcp@105
   808
{\out Suc(m) + n = Suc(m + n)}
lcp@284
   809
{\out  1. rec(Suc(m),n,\%x y. Suc(y)) = Suc(rec(m,n,\%x y. Suc(y)))}
lcp@105
   810
\ttbreak
lcp@105
   811
by (resolve_tac [rec_Suc] 1);
lcp@105
   812
{\out Level 1}
lcp@105
   813
{\out Suc(m) + n = Suc(m + n)}
lcp@105
   814
{\out No subgoals!}
wenzelm@3103
   815
qed "add_Suc";
lcp@284
   816
\end{ttbox}
lcp@105
   817
The induction rule raises some complications, which are discussed next.
lcp@105
   818
\index{theories!defining|)}
lcp@105
   819
lcp@105
   820
lcp@105
   821
\section{Refinement with explicit instantiation}
lcp@310
   822
\index{resolution!with instantiation}
lcp@310
   823
\index{instantiation|(}
lcp@310
   824
lcp@105
   825
In order to employ mathematical induction, we need to refine a subgoal by
lcp@105
   826
the rule~$(induct)$.  The conclusion of this rule is $\Var{P}(\Var{n})$,
lcp@105
   827
which is highly ambiguous in higher-order unification.  It matches every
lcp@105
   828
way that a formula can be regarded as depending on a subterm of type~$nat$.
lcp@105
   829
To get round this problem, we could make the induction rule conclude
lcp@105
   830
$\forall n.\Var{P}(n)$ --- but putting a subgoal into this form requires
lcp@105
   831
refinement by~$(\forall E)$, which is equally hard!
lcp@105
   832
paulson@5205
   833
The tactic \texttt{res_inst_tac}, like \texttt{resolve_tac}, refines a subgoal by
lcp@105
   834
a rule.  But it also accepts explicit instantiations for the rule's
lcp@105
   835
schematic variables.  
lcp@105
   836
\begin{description}
lcp@310
   837
\item[\ttindex{res_inst_tac} {\it insts} {\it thm} {\it i}]
lcp@105
   838
instantiates the rule {\it thm} with the instantiations {\it insts}, and
lcp@105
   839
then performs resolution on subgoal~$i$.
lcp@105
   840
lcp@310
   841
\item[\ttindex{eres_inst_tac}] 
lcp@310
   842
and \ttindex{dres_inst_tac} are similar, but perform elim-resolution
lcp@105
   843
and destruct-resolution, respectively.
lcp@105
   844
\end{description}
lcp@105
   845
The list {\it insts} consists of pairs $[(v@1,e@1), \ldots, (v@n,e@n)]$,
lcp@105
   846
where $v@1$, \ldots, $v@n$ are names of schematic variables in the rule ---
lcp@307
   847
with no leading question marks! --- and $e@1$, \ldots, $e@n$ are
lcp@105
   848
expressions giving their instantiations.  The expressions are type-checked
lcp@105
   849
in the context of a particular subgoal: free variables receive the same
lcp@105
   850
types as they have in the subgoal, and parameters may appear.  Type
lcp@105
   851
variable instantiations may appear in~{\it insts}, but they are seldom
paulson@5205
   852
required: \texttt{res_inst_tac} instantiates type variables automatically
lcp@105
   853
whenever the type of~$e@i$ is an instance of the type of~$\Var{v@i}$.
lcp@105
   854
lcp@105
   855
\subsection{A simple proof by induction}
lcp@310
   856
\index{examples!of induction}
lcp@105
   857
Let us prove that no natural number~$k$ equals its own successor.  To
lcp@105
   858
use~$(induct)$, we instantiate~$\Var{n}$ to~$k$; Isabelle finds a good
lcp@105
   859
instantiation for~$\Var{P}$.
lcp@284
   860
\begin{ttbox}
paulson@5205
   861
Goal "~ (Suc(k) = k)";
lcp@105
   862
{\out Level 0}
lcp@459
   863
{\out Suc(k) ~= k}
lcp@459
   864
{\out  1. Suc(k) ~= k}
lcp@105
   865
\ttbreak
lcp@105
   866
by (res_inst_tac [("n","k")] induct 1);
lcp@105
   867
{\out Level 1}
lcp@459
   868
{\out Suc(k) ~= k}
lcp@459
   869
{\out  1. Suc(0) ~= 0}
lcp@459
   870
{\out  2. !!x. Suc(x) ~= x ==> Suc(Suc(x)) ~= Suc(x)}
lcp@284
   871
\end{ttbox}
lcp@105
   872
We should check that Isabelle has correctly applied induction.  Subgoal~1
lcp@105
   873
is the base case, with $k$ replaced by~0.  Subgoal~2 is the inductive step,
lcp@105
   874
with $k$ replaced by~$Suc(x)$ and with an induction hypothesis for~$x$.
lcp@310
   875
The rest of the proof demonstrates~\tdx{notI}, \tdx{notE} and the
paulson@5205
   876
other rules of theory \texttt{Nat}.  The base case holds by~\ttindex{Suc_neq_0}:
lcp@284
   877
\begin{ttbox}
lcp@105
   878
by (resolve_tac [notI] 1);
lcp@105
   879
{\out Level 2}
lcp@459
   880
{\out Suc(k) ~= k}
lcp@105
   881
{\out  1. Suc(0) = 0 ==> False}
lcp@459
   882
{\out  2. !!x. Suc(x) ~= x ==> Suc(Suc(x)) ~= Suc(x)}
lcp@105
   883
\ttbreak
lcp@105
   884
by (eresolve_tac [Suc_neq_0] 1);
lcp@105
   885
{\out Level 3}
lcp@459
   886
{\out Suc(k) ~= k}
lcp@459
   887
{\out  1. !!x. Suc(x) ~= x ==> Suc(Suc(x)) ~= Suc(x)}
lcp@284
   888
\end{ttbox}
lcp@105
   889
The inductive step holds by the contrapositive of~\ttindex{Suc_inject}.
lcp@284
   890
Negation rules transform the subgoal into that of proving $Suc(x)=x$ from
lcp@284
   891
$Suc(Suc(x)) = Suc(x)$:
lcp@284
   892
\begin{ttbox}
lcp@105
   893
by (resolve_tac [notI] 1);
lcp@105
   894
{\out Level 4}
lcp@459
   895
{\out Suc(k) ~= k}
lcp@459
   896
{\out  1. !!x. [| Suc(x) ~= x; Suc(Suc(x)) = Suc(x) |] ==> False}
lcp@105
   897
\ttbreak
lcp@105
   898
by (eresolve_tac [notE] 1);
lcp@105
   899
{\out Level 5}
lcp@459
   900
{\out Suc(k) ~= k}
lcp@105
   901
{\out  1. !!x. Suc(Suc(x)) = Suc(x) ==> Suc(x) = x}
lcp@105
   902
\ttbreak
lcp@105
   903
by (eresolve_tac [Suc_inject] 1);
lcp@105
   904
{\out Level 6}
lcp@459
   905
{\out Suc(k) ~= k}
lcp@105
   906
{\out No subgoals!}
lcp@284
   907
\end{ttbox}
lcp@105
   908
lcp@105
   909
paulson@5205
   910
\subsection{An example of ambiguity in \texttt{resolve_tac}}
lcp@105
   911
\index{examples!of induction}\index{unification!higher-order}
paulson@5205
   912
If you try the example above, you may observe that \texttt{res_inst_tac} is
lcp@105
   913
not actually needed.  Almost by chance, \ttindex{resolve_tac} finds the right
lcp@105
   914
instantiation for~$(induct)$ to yield the desired next state.  With more
lcp@105
   915
complex formulae, our luck fails.  
lcp@284
   916
\begin{ttbox}
paulson@5205
   917
Goal "(k+m)+n = k+(m+n)";
lcp@105
   918
{\out Level 0}
lcp@105
   919
{\out k + m + n = k + (m + n)}
lcp@105
   920
{\out  1. k + m + n = k + (m + n)}
lcp@105
   921
\ttbreak
lcp@105
   922
by (resolve_tac [induct] 1);
lcp@105
   923
{\out Level 1}
lcp@105
   924
{\out k + m + n = k + (m + n)}
lcp@105
   925
{\out  1. k + m + n = 0}
lcp@105
   926
{\out  2. !!x. k + m + n = x ==> k + m + n = Suc(x)}
lcp@284
   927
\end{ttbox}
lcp@284
   928
This proof requires induction on~$k$.  The occurrence of~0 in subgoal~1
lcp@284
   929
indicates that induction has been applied to the term~$k+(m+n)$; this
lcp@284
   930
application is sound but will not lead to a proof here.  Fortunately,
lcp@284
   931
Isabelle can (lazily!) generate all the valid applications of induction.
lcp@284
   932
The \ttindex{back} command causes backtracking to an alternative outcome of
lcp@284
   933
the tactic.
lcp@284
   934
\begin{ttbox}
lcp@105
   935
back();
lcp@105
   936
{\out Level 1}
lcp@105
   937
{\out k + m + n = k + (m + n)}
lcp@105
   938
{\out  1. k + m + n = k + 0}
lcp@105
   939
{\out  2. !!x. k + m + n = k + x ==> k + m + n = k + Suc(x)}
lcp@284
   940
\end{ttbox}
lcp@284
   941
Now induction has been applied to~$m+n$.  This is equally useless.  Let us
lcp@284
   942
call \ttindex{back} again.
lcp@284
   943
\begin{ttbox}
lcp@105
   944
back();
lcp@105
   945
{\out Level 1}
lcp@105
   946
{\out k + m + n = k + (m + n)}
lcp@105
   947
{\out  1. k + m + 0 = k + (m + 0)}
lcp@284
   948
{\out  2. !!x. k + m + x = k + (m + x) ==>}
lcp@284
   949
{\out          k + m + Suc(x) = k + (m + Suc(x))}
lcp@284
   950
\end{ttbox}
lcp@105
   951
Now induction has been applied to~$n$.  What is the next alternative?
lcp@284
   952
\begin{ttbox}
lcp@105
   953
back();
lcp@105
   954
{\out Level 1}
lcp@105
   955
{\out k + m + n = k + (m + n)}
lcp@105
   956
{\out  1. k + m + n = k + (m + 0)}
lcp@105
   957
{\out  2. !!x. k + m + n = k + (m + x) ==> k + m + n = k + (m + Suc(x))}
lcp@284
   958
\end{ttbox}
lcp@105
   959
Inspecting subgoal~1 reveals that induction has been applied to just the
lcp@105
   960
second occurrence of~$n$.  This perfectly legitimate induction is useless
lcp@310
   961
here.  
lcp@310
   962
lcp@310
   963
The main goal admits fourteen different applications of induction.  The
lcp@310
   964
number is exponential in the size of the formula.
lcp@105
   965
lcp@105
   966
\subsection{Proving that addition is associative}
lcp@331
   967
Let us invoke the induction rule properly, using~{\tt
lcp@310
   968
  res_inst_tac}.  At the same time, we shall have a glimpse at Isabelle's
lcp@310
   969
simplification tactics, which are described in 
lcp@310
   970
\iflabelundefined{simp-chap}%
lcp@310
   971
    {the {\em Reference Manual}}{Chap.\ts\ref{simp-chap}}.
lcp@284
   972
lcp@310
   973
\index{simplification}\index{examples!of simplification} 
lcp@310
   974
wenzelm@9695
   975
Isabelle's simplification tactics repeatedly apply equations to a subgoal,
wenzelm@9695
   976
perhaps proving it.  For efficiency, the rewrite rules must be packaged into a
wenzelm@9695
   977
{\bf simplification set},\index{simplification sets} or {\bf simpset}.  We
wenzelm@9695
   978
augment the implicit simpset of FOL with the equations proved in the previous
wenzelm@9695
   979
section, namely $0+n=n$ and $\texttt{Suc}(m)+n=\texttt{Suc}(m+n)$:
lcp@284
   980
\begin{ttbox}
wenzelm@3114
   981
Addsimps [add_0, add_Suc];
lcp@284
   982
\end{ttbox}
lcp@105
   983
We state the goal for associativity of addition, and
lcp@105
   984
use \ttindex{res_inst_tac} to invoke induction on~$k$:
lcp@284
   985
\begin{ttbox}
paulson@5205
   986
Goal "(k+m)+n = k+(m+n)";
lcp@105
   987
{\out Level 0}
lcp@105
   988
{\out k + m + n = k + (m + n)}
lcp@105
   989
{\out  1. k + m + n = k + (m + n)}
lcp@105
   990
\ttbreak
lcp@105
   991
by (res_inst_tac [("n","k")] induct 1);
lcp@105
   992
{\out Level 1}
lcp@105
   993
{\out k + m + n = k + (m + n)}
lcp@105
   994
{\out  1. 0 + m + n = 0 + (m + n)}
lcp@284
   995
{\out  2. !!x. x + m + n = x + (m + n) ==>}
lcp@284
   996
{\out          Suc(x) + m + n = Suc(x) + (m + n)}
lcp@284
   997
\end{ttbox}
lcp@105
   998
The base case holds easily; both sides reduce to $m+n$.  The
wenzelm@3114
   999
tactic~\ttindex{Simp_tac} rewrites with respect to the current
wenzelm@3114
  1000
simplification set, applying the rewrite rules for addition:
lcp@284
  1001
\begin{ttbox}
wenzelm@3114
  1002
by (Simp_tac 1);
lcp@105
  1003
{\out Level 2}
lcp@105
  1004
{\out k + m + n = k + (m + n)}
lcp@284
  1005
{\out  1. !!x. x + m + n = x + (m + n) ==>}
lcp@284
  1006
{\out          Suc(x) + m + n = Suc(x) + (m + n)}
lcp@284
  1007
\end{ttbox}
lcp@331
  1008
The inductive step requires rewriting by the equations for addition
paulson@14148
  1009
and with the induction hypothesis, which is also an equation.  The
wenzelm@3114
  1010
tactic~\ttindex{Asm_simp_tac} rewrites using the implicit
wenzelm@3114
  1011
simplification set and any useful assumptions:
lcp@284
  1012
\begin{ttbox}
wenzelm@3114
  1013
by (Asm_simp_tac 1);
lcp@105
  1014
{\out Level 3}
lcp@105
  1015
{\out k + m + n = k + (m + n)}
lcp@105
  1016
{\out No subgoals!}
lcp@284
  1017
\end{ttbox}
lcp@310
  1018
\index{instantiation|)}
lcp@105
  1019
lcp@105
  1020
lcp@284
  1021
\section{A Prolog interpreter}
lcp@105
  1022
\index{Prolog interpreter|bold}
lcp@284
  1023
To demonstrate the power of tacticals, let us construct a Prolog
lcp@105
  1024
interpreter and execute programs involving lists.\footnote{To run these
paulson@5205
  1025
examples, see the file \texttt{FOL/ex/Prolog.ML}.} The Prolog program
lcp@105
  1026
consists of a theory.  We declare a type constructor for lists, with an
lcp@105
  1027
arity declaration to say that $(\tau)list$ is of class~$term$
lcp@105
  1028
provided~$\tau$ is:
lcp@105
  1029
\begin{eqnarray*}
lcp@105
  1030
  list  & :: & (term)term
lcp@105
  1031
\end{eqnarray*}
lcp@105
  1032
We declare four constants: the empty list~$Nil$; the infix list
lcp@105
  1033
constructor~{:}; the list concatenation predicate~$app$; the list reverse
lcp@284
  1034
predicate~$rev$.  (In Prolog, functions on lists are expressed as
lcp@105
  1035
predicates.)
lcp@105
  1036
\begin{eqnarray*}
lcp@105
  1037
    Nil         & :: & \alpha list \\
lcp@105
  1038
    {:}         & :: & [\alpha,\alpha list] \To \alpha list \\
lcp@105
  1039
    app & :: & [\alpha list,\alpha list,\alpha list] \To o \\
lcp@105
  1040
    rev & :: & [\alpha list,\alpha list] \To o 
lcp@105
  1041
\end{eqnarray*}
lcp@284
  1042
The predicate $app$ should satisfy the Prolog-style rules
lcp@105
  1043
\[ {app(Nil,ys,ys)} \qquad
lcp@105
  1044
   {app(xs,ys,zs) \over app(x:xs, ys, x:zs)} \]
lcp@105
  1045
We define the naive version of $rev$, which calls~$app$:
lcp@105
  1046
\[ {rev(Nil,Nil)} \qquad
lcp@105
  1047
   {rev(xs,ys)\quad  app(ys, x:Nil, zs) \over
lcp@105
  1048
    rev(x:xs, zs)} 
lcp@105
  1049
\]
lcp@105
  1050
lcp@105
  1051
\index{examples!of theories}
lcp@310
  1052
Theory \thydx{Prolog} extends first-order logic in order to make use
lcp@105
  1053
of the class~$term$ and the type~$o$.  The interpreter does not use the
paulson@5205
  1054
rules of~\texttt{FOL}.
lcp@105
  1055
\begin{ttbox}
lcp@105
  1056
Prolog = FOL +
lcp@296
  1057
types   'a list
lcp@105
  1058
arities list    :: (term)term
clasohm@1387
  1059
consts  Nil     :: 'a list
clasohm@1387
  1060
        ":"     :: ['a, 'a list]=> 'a list            (infixr 60)
clasohm@1387
  1061
        app     :: ['a list, 'a list, 'a list] => o
clasohm@1387
  1062
        rev     :: ['a list, 'a list] => o
lcp@105
  1063
rules   appNil  "app(Nil,ys,ys)"
lcp@105
  1064
        appCons "app(xs,ys,zs) ==> app(x:xs, ys, x:zs)"
lcp@105
  1065
        revNil  "rev(Nil,Nil)"
lcp@105
  1066
        revCons "[| rev(xs,ys); app(ys,x:Nil,zs) |] ==> rev(x:xs,zs)"
lcp@105
  1067
end
lcp@105
  1068
\end{ttbox}
lcp@105
  1069
\subsection{Simple executions}
lcp@284
  1070
Repeated application of the rules solves Prolog goals.  Let us
lcp@105
  1071
append the lists $[a,b,c]$ and~$[d,e]$.  As the rules are applied, the
paulson@5205
  1072
answer builds up in~\texttt{?x}.
lcp@105
  1073
\begin{ttbox}
paulson@5205
  1074
Goal "app(a:b:c:Nil, d:e:Nil, ?x)";
lcp@105
  1075
{\out Level 0}
lcp@105
  1076
{\out app(a : b : c : Nil, d : e : Nil, ?x)}
lcp@105
  1077
{\out  1. app(a : b : c : Nil, d : e : Nil, ?x)}
lcp@105
  1078
\ttbreak
lcp@105
  1079
by (resolve_tac [appNil,appCons] 1);
lcp@105
  1080
{\out Level 1}
lcp@105
  1081
{\out app(a : b : c : Nil, d : e : Nil, a : ?zs1)}
lcp@105
  1082
{\out  1. app(b : c : Nil, d : e : Nil, ?zs1)}
lcp@105
  1083
\ttbreak
lcp@105
  1084
by (resolve_tac [appNil,appCons] 1);
lcp@105
  1085
{\out Level 2}
lcp@105
  1086
{\out app(a : b : c : Nil, d : e : Nil, a : b : ?zs2)}
lcp@105
  1087
{\out  1. app(c : Nil, d : e : Nil, ?zs2)}
lcp@105
  1088
\end{ttbox}
lcp@105
  1089
At this point, the first two elements of the result are~$a$ and~$b$.
lcp@105
  1090
\begin{ttbox}
lcp@105
  1091
by (resolve_tac [appNil,appCons] 1);
lcp@105
  1092
{\out Level 3}
lcp@105
  1093
{\out app(a : b : c : Nil, d : e : Nil, a : b : c : ?zs3)}
lcp@105
  1094
{\out  1. app(Nil, d : e : Nil, ?zs3)}
lcp@105
  1095
\ttbreak
lcp@105
  1096
by (resolve_tac [appNil,appCons] 1);
lcp@105
  1097
{\out Level 4}
lcp@105
  1098
{\out app(a : b : c : Nil, d : e : Nil, a : b : c : d : e : Nil)}
lcp@105
  1099
{\out No subgoals!}
lcp@105
  1100
\end{ttbox}
lcp@105
  1101
lcp@284
  1102
Prolog can run functions backwards.  Which list can be appended
lcp@105
  1103
with $[c,d]$ to produce $[a,b,c,d]$?
lcp@105
  1104
Using \ttindex{REPEAT}, we find the answer at once, $[a,b]$:
lcp@105
  1105
\begin{ttbox}
paulson@5205
  1106
Goal "app(?x, c:d:Nil, a:b:c:d:Nil)";
lcp@105
  1107
{\out Level 0}
lcp@105
  1108
{\out app(?x, c : d : Nil, a : b : c : d : Nil)}
lcp@105
  1109
{\out  1. app(?x, c : d : Nil, a : b : c : d : Nil)}
lcp@105
  1110
\ttbreak
lcp@105
  1111
by (REPEAT (resolve_tac [appNil,appCons] 1));
lcp@105
  1112
{\out Level 1}
lcp@105
  1113
{\out app(a : b : Nil, c : d : Nil, a : b : c : d : Nil)}
lcp@105
  1114
{\out No subgoals!}
lcp@105
  1115
\end{ttbox}
lcp@105
  1116
lcp@105
  1117
lcp@310
  1118
\subsection{Backtracking}\index{backtracking!Prolog style}
lcp@296
  1119
Prolog backtracking can answer questions that have multiple solutions.
lcp@296
  1120
Which lists $x$ and $y$ can be appended to form the list $[a,b,c,d]$?  This
lcp@296
  1121
question has five solutions.  Using \ttindex{REPEAT} to apply the rules, we
lcp@296
  1122
quickly find the first solution, namely $x=[]$ and $y=[a,b,c,d]$:
lcp@105
  1123
\begin{ttbox}
paulson@5205
  1124
Goal "app(?x, ?y, a:b:c:d:Nil)";
lcp@105
  1125
{\out Level 0}
lcp@105
  1126
{\out app(?x, ?y, a : b : c : d : Nil)}
lcp@105
  1127
{\out  1. app(?x, ?y, a : b : c : d : Nil)}
lcp@105
  1128
\ttbreak
lcp@105
  1129
by (REPEAT (resolve_tac [appNil,appCons] 1));
lcp@105
  1130
{\out Level 1}
lcp@105
  1131
{\out app(Nil, a : b : c : d : Nil, a : b : c : d : Nil)}
lcp@105
  1132
{\out No subgoals!}
lcp@105
  1133
\end{ttbox}
lcp@284
  1134
Isabelle can lazily generate all the possibilities.  The \ttindex{back}
lcp@284
  1135
command returns the tactic's next outcome, namely $x=[a]$ and $y=[b,c,d]$:
lcp@105
  1136
\begin{ttbox}
lcp@105
  1137
back();
lcp@105
  1138
{\out Level 1}
lcp@105
  1139
{\out app(a : Nil, b : c : d : Nil, a : b : c : d : Nil)}
lcp@105
  1140
{\out No subgoals!}
lcp@105
  1141
\end{ttbox}
lcp@105
  1142
The other solutions are generated similarly.
lcp@105
  1143
\begin{ttbox}
lcp@105
  1144
back();
lcp@105
  1145
{\out Level 1}
lcp@105
  1146
{\out app(a : b : Nil, c : d : Nil, a : b : c : d : Nil)}
lcp@105
  1147
{\out No subgoals!}
lcp@105
  1148
\ttbreak
lcp@105
  1149
back();
lcp@105
  1150
{\out Level 1}
lcp@105
  1151
{\out app(a : b : c : Nil, d : Nil, a : b : c : d : Nil)}
lcp@105
  1152
{\out No subgoals!}
lcp@105
  1153
\ttbreak
lcp@105
  1154
back();
lcp@105
  1155
{\out Level 1}
lcp@105
  1156
{\out app(a : b : c : d : Nil, Nil, a : b : c : d : Nil)}
lcp@105
  1157
{\out No subgoals!}
lcp@105
  1158
\end{ttbox}
lcp@105
  1159
lcp@105
  1160
lcp@105
  1161
\subsection{Depth-first search}
lcp@105
  1162
\index{search!depth-first}
lcp@105
  1163
Now let us try $rev$, reversing a list.
paulson@5205
  1164
Bundle the rules together as the \ML{} identifier \texttt{rules}.  Naive
lcp@105
  1165
reverse requires 120 inferences for this 14-element list, but the tactic
lcp@105
  1166
terminates in a few seconds.
lcp@105
  1167
\begin{ttbox}
paulson@5205
  1168
Goal "rev(a:b:c:d:e:f:g:h:i:j:k:l:m:n:Nil, ?w)";
lcp@105
  1169
{\out Level 0}
lcp@105
  1170
{\out rev(a : b : c : d : e : f : g : h : i : j : k : l : m : n : Nil, ?w)}
lcp@284
  1171
{\out  1. rev(a : b : c : d : e : f : g : h : i : j : k : l : m : n : Nil,}
lcp@284
  1172
{\out         ?w)}
lcp@284
  1173
\ttbreak
lcp@105
  1174
val rules = [appNil,appCons,revNil,revCons];
lcp@105
  1175
\ttbreak
lcp@105
  1176
by (REPEAT (resolve_tac rules 1));
lcp@105
  1177
{\out Level 1}
lcp@105
  1178
{\out rev(a : b : c : d : e : f : g : h : i : j : k : l : m : n : Nil,}
lcp@105
  1179
{\out     n : m : l : k : j : i : h : g : f : e : d : c : b : a : Nil)}
lcp@105
  1180
{\out No subgoals!}
lcp@105
  1181
\end{ttbox}
lcp@105
  1182
We may execute $rev$ backwards.  This, too, should reverse a list.  What
lcp@105
  1183
is the reverse of $[a,b,c]$?
lcp@105
  1184
\begin{ttbox}
paulson@5205
  1185
Goal "rev(?x, a:b:c:Nil)";
lcp@105
  1186
{\out Level 0}
lcp@105
  1187
{\out rev(?x, a : b : c : Nil)}
lcp@105
  1188
{\out  1. rev(?x, a : b : c : Nil)}
lcp@105
  1189
\ttbreak
lcp@105
  1190
by (REPEAT (resolve_tac rules 1));
lcp@105
  1191
{\out Level 1}
lcp@105
  1192
{\out rev(?x1 : Nil, a : b : c : Nil)}
lcp@105
  1193
{\out  1. app(Nil, ?x1 : Nil, a : b : c : Nil)}
lcp@105
  1194
\end{ttbox}
lcp@105
  1195
The tactic has failed to find a solution!  It reached a dead end at
lcp@331
  1196
subgoal~1: there is no~$\Var{x@1}$ such that [] appended with~$[\Var{x@1}]$
lcp@105
  1197
equals~$[a,b,c]$.  Backtracking explores other outcomes.
lcp@105
  1198
\begin{ttbox}
lcp@105
  1199
back();
lcp@105
  1200
{\out Level 1}
lcp@105
  1201
{\out rev(?x1 : a : Nil, a : b : c : Nil)}
lcp@105
  1202
{\out  1. app(Nil, ?x1 : Nil, b : c : Nil)}
lcp@105
  1203
\end{ttbox}
lcp@105
  1204
This too is a dead end, but the next outcome is successful.
lcp@105
  1205
\begin{ttbox}
lcp@105
  1206
back();
lcp@105
  1207
{\out Level 1}
lcp@105
  1208
{\out rev(c : b : a : Nil, a : b : c : Nil)}
lcp@105
  1209
{\out No subgoals!}
lcp@105
  1210
\end{ttbox}
lcp@310
  1211
\ttindex{REPEAT} goes wrong because it is only a repetition tactical, not a
paulson@5205
  1212
search tactical.  \texttt{REPEAT} stops when it cannot continue, regardless of
lcp@310
  1213
which state is reached.  The tactical \ttindex{DEPTH_FIRST} searches for a
lcp@310
  1214
satisfactory state, as specified by an \ML{} predicate.  Below,
lcp@105
  1215
\ttindex{has_fewer_prems} specifies that the proof state should have no
lcp@310
  1216
subgoals.
lcp@105
  1217
\begin{ttbox}
lcp@105
  1218
val prolog_tac = DEPTH_FIRST (has_fewer_prems 1) 
lcp@105
  1219
                             (resolve_tac rules 1);
lcp@105
  1220
\end{ttbox}
lcp@284
  1221
Since Prolog uses depth-first search, this tactic is a (slow!) 
lcp@296
  1222
Prolog interpreter.  We return to the start of the proof using
paulson@5205
  1223
\ttindex{choplev}, and apply \texttt{prolog_tac}:
lcp@105
  1224
\begin{ttbox}
lcp@105
  1225
choplev 0;
lcp@105
  1226
{\out Level 0}
lcp@105
  1227
{\out rev(?x, a : b : c : Nil)}
lcp@105
  1228
{\out  1. rev(?x, a : b : c : Nil)}
lcp@105
  1229
\ttbreak
paulson@14148
  1230
by prolog_tac;
lcp@105
  1231
{\out Level 1}
lcp@105
  1232
{\out rev(c : b : a : Nil, a : b : c : Nil)}
lcp@105
  1233
{\out No subgoals!}
lcp@105
  1234
\end{ttbox}
paulson@5205
  1235
Let us try \texttt{prolog_tac} on one more example, containing four unknowns:
lcp@105
  1236
\begin{ttbox}
paulson@5205
  1237
Goal "rev(a:?x:c:?y:Nil, d:?z:b:?u)";
lcp@105
  1238
{\out Level 0}
lcp@105
  1239
{\out rev(a : ?x : c : ?y : Nil, d : ?z : b : ?u)}
lcp@105
  1240
{\out  1. rev(a : ?x : c : ?y : Nil, d : ?z : b : ?u)}
lcp@105
  1241
\ttbreak
lcp@105
  1242
by prolog_tac;
lcp@105
  1243
{\out Level 1}
lcp@105
  1244
{\out rev(a : b : c : d : Nil, d : c : b : a : Nil)}
lcp@105
  1245
{\out No subgoals!}
lcp@105
  1246
\end{ttbox}
lcp@284
  1247
Although Isabelle is much slower than a Prolog system, Isabelle
lcp@156
  1248
tactics can exploit logic programming techniques.  
lcp@156
  1249