doc-src/Intro/foundations.tex
 author blanchet Fri Apr 30 09:36:45 2010 +0200 (2010-04-30) changeset 36569 3a29eb7606c3 parent 9695 ec7d7f877712 child 42637 381fdcab0f36 permissions -rw-r--r--
added "no_atp" for theorems that are automatically used or included by Sledgehammer when appropriate (about combinators and fequal)
     1 %% $Id$

     2

     3 \part{Foundations}

     4 The following sections discuss Isabelle's logical foundations in detail:

     5 representing logical syntax in the typed $\lambda$-calculus; expressing

     6 inference rules in Isabelle's meta-logic; combining rules by resolution.

     7

     8 If you wish to use Isabelle immediately, please turn to

     9 page~\pageref{chap:getting}.  You can always read about foundations later,

    10 either by returning to this point or by looking up particular items in the

    11 index.

    12

    13 \begin{figure}

    14 \begin{eqnarray*}

    15   \neg P   & \hbox{abbreviates} & P\imp\bot \\

    16   P\bimp Q & \hbox{abbreviates} & (P\imp Q) \conj (Q\imp P)

    17 \end{eqnarray*}

    18 \vskip 4ex

    19

    20 $$\begin{array}{c@{\qquad\qquad}c}   21 \infer[({\conj}I)]{P\conj Q}{P & Q} &   22 \infer[({\conj}E1)]{P}{P\conj Q} \qquad   23 \infer[({\conj}E2)]{Q}{P\conj Q} \$4ex]   24   25 \infer[({\disj}I1)]{P\disj Q}{P} \qquad   26 \infer[({\disj}I2)]{P\disj Q}{Q} &   27 \infer[({\disj}E)]{R}{P\disj Q & \infer*{R}{[P]} & \infer*{R}{[Q]}}\\[4ex]   28   29 \infer[({\imp}I)]{P\imp Q}{\infer*{Q}{[P]}} &   30 \infer[({\imp}E)]{Q}{P\imp Q & P} \\[4ex]   31   32 &   33 \infer[({\bot}E)]{P}{\bot}\\[4ex]   34   35 \infer[({\forall}I)*]{\forall x.P}{P} &   36 \infer[({\forall}E)]{P[t/x]}{\forall x.P} \\[3ex]   37   38 \infer[({\exists}I)]{\exists x.P}{P[t/x]} &   39 \infer[({\exists}E)*]{Q}{{\exists x.P} & \infer*{Q}{[P]} } \\[3ex]   40   41 {t=t} \,(refl) & \vcenter{\infer[(subst)]{P[u/x]}{t=u & P[t/x]}}   42 \end{array}$$   43   44 \bigskip\bigskip   45 *{\em Eigenvariable conditions\/}:   46   47 \forall I: provided x is not free in the assumptions   48   49 \exists E: provided x is not free in Q or any assumption except P   50 \caption{Intuitionistic first-order logic} \label{fol-fig}   51 \end{figure}   52   53 \section{Formalizing logical syntax in Isabelle}\label{sec:logical-syntax}   54 \index{first-order logic}   55   56 Figure~\ref{fol-fig} presents intuitionistic first-order logic,   57 including equality. Let us see how to formalize   58 this logic in Isabelle, illustrating the main features of Isabelle's   59 polymorphic meta-logic.   60   61 \index{lambda calc@\lambda-calculus}   62 Isabelle represents syntax using the simply typed \lambda-calculus. We   63 declare a type for each syntactic category of the logic. We declare a   64 constant for each symbol of the logic, giving each n-place operation an   65 n-argument curried function type. Most importantly,   66 \lambda-abstraction represents variable binding in quantifiers.   67   68 \index{types!syntax of}\index{types!function}\index{*fun type}   69 \index{type constructors}   70 Isabelle has \ML-style polymorphic types such as~(\alpha)list, where   71 list is a type constructor and \alpha is a type variable; for example,   72 (bool)list is the type of lists of booleans. Function types have the   73 form (\sigma,\tau)fun or \sigma\To\tau, where \sigma and \tau are   74 types. Curried function types may be abbreviated:   75 \[ \sigma@1\To (\cdots \sigma@n\To \tau\cdots) \quad \hbox{as} \quad   76 [\sigma@1, \ldots, \sigma@n] \To \tau$

    77

    78 \index{terms!syntax of} The syntax for terms is summarised below.

    79 Note that there are two versions of function application syntax

    80 available in Isabelle: either $t\,u$, which is the usual form for

    81 higher-order languages, or $t(u)$, trying to look more like

    82 first-order.  The latter syntax is used throughout the manual.

    83 $  84 \index{lambda abs@\lambda-abstractions}\index{function applications}   85 \begin{array}{ll}   86 t :: \tau & \hbox{type constraint, on a term or bound variable} \\   87 \lambda x.t & \hbox{abstraction} \\   88 \lambda x@1\ldots x@n.t   89 & \hbox{curried abstraction, \lambda x@1. \ldots \lambda x@n.t} \\   90 t(u) & \hbox{application} \\   91 t (u@1, \ldots, u@n) & \hbox{curried application, t(u@1)\ldots(u@n)}   92 \end{array}   93$

    94

    95

    96 \subsection{Simple types and constants}\index{types!simple|bold}

    97

    98 The syntactic categories of our logic (Fig.\ts\ref{fol-fig}) are {\bf

    99   formulae} and {\bf terms}.  Formulae denote truth values, so (following

   100 tradition) let us call their type~$o$.  To allow~0 and~$Suc(t)$ as terms,

   101 let us declare a type~$nat$ of natural numbers.  Later, we shall see

   102 how to admit terms of other types.

   103

   104 \index{constants}\index{*nat type}\index{*o type}

   105 After declaring the types~$o$ and~$nat$, we may declare constants for the

   106 symbols of our logic.  Since $\bot$ denotes a truth value (falsity) and 0

   107 denotes a number, we put \begin{eqnarray*}

   108   \bot  & :: & o \\

   109   0     & :: & nat.

   110 \end{eqnarray*}

   111 If a symbol requires operands, the corresponding constant must have a

   112 function type.  In our logic, the successor function

   113 ($Suc$) is from natural numbers to natural numbers, negation ($\neg$) is a

   114 function from truth values to truth values, and the binary connectives are

   115 curried functions taking two truth values as arguments:

   116 \begin{eqnarray*}

   117   Suc    & :: & nat\To nat  \\

   118   {\neg} & :: & o\To o      \\

   119   \conj,\disj,\imp,\bimp  & :: & [o,o]\To o

   120 \end{eqnarray*}

   121 The binary connectives can be declared as infixes, with appropriate

   122 precedences, so that we write $P\conj Q\disj R$ instead of

   123 $\disj(\conj(P,Q), R)$.

   124

   125 Section~\ref{sec:defining-theories} below describes the syntax of Isabelle

   126 theory files and illustrates it by extending our logic with mathematical

   127 induction.

   128

   129

   130 \subsection{Polymorphic types and constants} \label{polymorphic}

   131 \index{types!polymorphic|bold}

   132 \index{equality!polymorphic}

   133 \index{constants!polymorphic}

   134

   135 Which type should we assign to the equality symbol?  If we tried

   136 $[nat,nat]\To o$, then equality would be restricted to the natural

   137 numbers; we should have to declare different equality symbols for each

   138 type.  Isabelle's type system is polymorphic, so we could declare

   139 \begin{eqnarray*}

   140   {=}  & :: & [\alpha,\alpha]\To o,

   141 \end{eqnarray*}

   142 where the type variable~$\alpha$ ranges over all types.

   143 But this is also wrong.  The declaration is too polymorphic; $\alpha$

   144 includes types like~$o$ and $nat\To nat$.  Thus, it admits

   145 $\bot=\neg(\bot)$ and $Suc=Suc$ as formulae, which is acceptable in

   146 higher-order logic but not in first-order logic.

   147

   148 Isabelle's {\bf type classes}\index{classes} control

   149 polymorphism~\cite{nipkow-prehofer}.  Each type variable belongs to a

   150 class, which denotes a set of types.  Classes are partially ordered by the

   151 subclass relation, which is essentially the subset relation on the sets of

   152 types.  They closely resemble the classes of the functional language

   153 Haskell~\cite{haskell-tutorial,haskell-report}.

   154

   155 \index{*logic class}\index{*term class}

   156 Isabelle provides the built-in class $logic$, which consists of the logical

   157 types: the ones we want to reason about.  Let us declare a class $term$, to

   158 consist of all legal types of terms in our logic.  The subclass structure

   159 is now $term\le logic$.

   160

   161 \index{*nat type}

   162 We put $nat$ in class $term$ by declaring $nat{::}term$.  We declare the

   163 equality constant by

   164 \begin{eqnarray*}

   165   {=}  & :: & [\alpha{::}term,\alpha]\To o

   166 \end{eqnarray*}

   167 where $\alpha{::}term$ constrains the type variable~$\alpha$ to class

   168 $term$.  Such type variables resemble Standard~\ML's equality type

   169 variables.

   170

   171 We give~$o$ and function types the class $logic$ rather than~$term$, since

   172 they are not legal types for terms.  We may introduce new types of class

   173 $term$ --- for instance, type $string$ or $real$ --- at any time.  We can

   174 even declare type constructors such as~$list$, and state that type

   175 $(\tau)list$ belongs to class~$term$ provided $\tau$ does; equality

   176 applies to lists of natural numbers but not to lists of formulae.  We may

   177 summarize this paragraph by a set of {\bf arity declarations} for type

   178 constructors:\index{arities!declaring}

   179 \begin{eqnarray*}\index{*o type}\index{*fun type}

   180   o             & :: & logic \\

   181   fun           & :: & (logic,logic)logic \\

   182   nat, string, real     & :: & term \\

   183   list          & :: & (term)term

   184 \end{eqnarray*}

   185 (Recall that $fun$ is the type constructor for function types.)

   186 In \rmindex{higher-order logic}, equality does apply to truth values and

   187 functions;  this requires the arity declarations ${o::term}$

   188 and ${fun::(term,term)term}$.  The class system can also handle

   189 overloading.\index{overloading|bold} We could declare $arith$ to be the

   190 subclass of $term$ consisting of the arithmetic' types, such as~$nat$.

   191 Then we could declare the operators

   192 \begin{eqnarray*}

   193   {+},{-},{\times},{/}  & :: & [\alpha{::}arith,\alpha]\To \alpha

   194 \end{eqnarray*}

   195 If we declare new types $real$ and $complex$ of class $arith$, then we

   196 in effect have three sets of operators:

   197 \begin{eqnarray*}

   198   {+},{-},{\times},{/}  & :: & [nat,nat]\To nat \\

   199   {+},{-},{\times},{/}  & :: & [real,real]\To real \\

   200   {+},{-},{\times},{/}  & :: & [complex,complex]\To complex

   201 \end{eqnarray*}

   202 Isabelle will regard these as distinct constants, each of which can be defined

   203 separately.  We could even introduce the type $(\alpha)vector$ and declare

   204 its arity as $(arith)arith$.  Then we could declare the constant

   205 \begin{eqnarray*}

   206   {+}  & :: & [(\alpha)vector,(\alpha)vector]\To (\alpha)vector

   207 \end{eqnarray*}

   208 and specify it in terms of ${+} :: [\alpha,\alpha]\To \alpha$.

   209

   210 A type variable may belong to any finite number of classes.  Suppose that

   211 we had declared yet another class $ord \le term$, the class of all

   212 ordered' types, and a constant

   213 \begin{eqnarray*}

   214   {\le}  & :: & [\alpha{::}ord,\alpha]\To o.

   215 \end{eqnarray*}

   216 In this context the variable $x$ in $x \le (x+x)$ will be assigned type

   217 $\alpha{::}\{arith,ord\}$, which means $\alpha$ belongs to both $arith$ and

   218 $ord$.  Semantically the set $\{arith,ord\}$ should be understood as the

   219 intersection of the sets of types represented by $arith$ and $ord$.  Such

   220 intersections of classes are called \bfindex{sorts}.  The empty

   221 intersection of classes, $\{\}$, contains all types and is thus the {\bf

   222   universal sort}.

   223

   224 Even with overloading, each term has a unique, most general type.  For this

   225 to be possible, the class and type declarations must satisfy certain

   226 technical constraints; see

   227 \iflabelundefined{sec:ref-defining-theories}%

   228            {Sect.\ Defining Theories in the {\em Reference Manual}}%

   229            {\S\ref{sec:ref-defining-theories}}.

   230

   231

   232 \subsection{Higher types and quantifiers}

   233 \index{types!higher|bold}\index{quantifiers}

   234 Quantifiers are regarded as operations upon functions.  Ignoring polymorphism

   235 for the moment, consider the formula $\forall x. P(x)$, where $x$ ranges

   236 over type~$nat$.  This is true if $P(x)$ is true for all~$x$.  Abstracting

   237 $P(x)$ into a function, this is the same as saying that $\lambda x.P(x)$

   238 returns true for all arguments.  Thus, the universal quantifier can be

   239 represented by a constant

   240 \begin{eqnarray*}

   241   \forall  & :: & (nat\To o) \To o,

   242 \end{eqnarray*}

   243 which is essentially an infinitary truth table.  The representation of $\forall   244 x. P(x)$ is $\forall(\lambda x. P(x))$.

   245

   246 The existential quantifier is treated

   247 in the same way.  Other binding operators are also easily handled; for

   248 instance, the summation operator $\Sigma@{k=i}^j f(k)$ can be represented as

   249 $\Sigma(i,j,\lambda k.f(k))$, where

   250 \begin{eqnarray*}

   251   \Sigma  & :: & [nat,nat, nat\To nat] \To nat.

   252 \end{eqnarray*}

   253 Quantifiers may be polymorphic.  We may define $\forall$ and~$\exists$ over

   254 all legal types of terms, not just the natural numbers, and

   255 allow summations over all arithmetic types:

   256 \begin{eqnarray*}

   257    \forall,\exists      & :: & (\alpha{::}term\To o) \To o \\

   258    \Sigma               & :: & [nat,nat, nat\To \alpha{::}arith] \To \alpha

   259 \end{eqnarray*}

   260 Observe that the index variables still have type $nat$, while the values

   261 being summed may belong to any arithmetic type.

   262

   263

   264 \section{Formalizing logical rules in Isabelle}

   265 \index{meta-implication|bold}

   266 \index{meta-quantifiers|bold}

   267 \index{meta-equality|bold}

   268

   269 Object-logics are formalized by extending Isabelle's

   270 meta-logic~\cite{paulson-found}, which is intuitionistic higher-order logic.

   271 The meta-level connectives are {\bf implication}, the {\bf universal

   272   quantifier}, and {\bf equality}.

   273 \begin{itemize}

   274   \item The implication $$\phi\Imp \psi$$ means $$\phi$$ implies

   275 $$\psi$$', and expresses logical {\bf entailment}.

   276

   277   \item The quantification $$\Forall x.\phi$$ means $$\phi$$ is true for

   278 all $x$', and expresses {\bf generality} in rules and axiom schemes.

   279

   280 \item The equality $$a\equiv b$$ means $a$ equals $b$', for expressing

   281   {\bf definitions} (see~\S\ref{definitions}).\index{definitions}

   282   Equalities left over from the unification process, so called {\bf

   283     flex-flex constraints},\index{flex-flex constraints} are written $a\qeq   284 b$.  The two equality symbols have the same logical meaning.

   285

   286 \end{itemize}

   287 The syntax of the meta-logic is formalized in the same manner

   288 as object-logics, using the simply typed $\lambda$-calculus.  Analogous to

   289 type~$o$ above, there is a built-in type $prop$ of meta-level truth values.

   290 Meta-level formulae will have this type.  Type $prop$ belongs to

   291 class~$logic$; also, $\sigma\To\tau$ belongs to $logic$ provided $\sigma$

   292 and $\tau$ do.  Here are the types of the built-in connectives:

   293 \begin{eqnarray*}\index{*prop type}\index{*logic class}

   294   \Imp     & :: & [prop,prop]\To prop \\

   295   \Forall  & :: & (\alpha{::}logic\To prop) \To prop \\

   296   {\equiv} & :: & [\alpha{::}\{\},\alpha]\To prop \\

   297   \qeq & :: & [\alpha{::}\{\},\alpha]\To prop

   298 \end{eqnarray*}

   299 The polymorphism in $\Forall$ is restricted to class~$logic$ to exclude

   300 certain types, those used just for parsing.  The type variable

   301 $\alpha{::}\{\}$ ranges over the universal sort.

   302

   303 In our formalization of first-order logic, we declared a type~$o$ of

   304 object-level truth values, rather than using~$prop$ for this purpose.  If

   305 we declared the object-level connectives to have types such as

   306 ${\neg}::prop\To prop$, then these connectives would be applicable to

   307 meta-level formulae.  Keeping $prop$ and $o$ as separate types maintains

   308 the distinction between the meta-level and the object-level.  To formalize

   309 the inference rules, we shall need to relate the two levels; accordingly,

   310 we declare the constant

   311 \index{*Trueprop constant}

   312 \begin{eqnarray*}

   313   Trueprop & :: & o\To prop.

   314 \end{eqnarray*}

   315 We may regard $Trueprop$ as a meta-level predicate, reading $Trueprop(P)$ as

   316 $P$ is true at the object-level.'  Put another way, $Trueprop$ is a coercion

   317 from $o$ to $prop$.

   318

   319

   320 \subsection{Expressing propositional rules}

   321 \index{rules!propositional}

   322 We shall illustrate the use of the meta-logic by formalizing the rules of

   323 Fig.\ts\ref{fol-fig}.  Each object-level rule is expressed as a meta-level

   324 axiom.

   325

   326 One of the simplest rules is $(\conj E1)$.  Making

   327 everything explicit, its formalization in the meta-logic is

   328 $$  329 \Forall P\;Q. Trueprop(P\conj Q) \Imp Trueprop(P). \eqno(\conj E1)   330$$

   331 This may look formidable, but it has an obvious reading: for all object-level

   332 truth values $P$ and~$Q$, if $P\conj Q$ is true then so is~$P$.  The

   333 reading is correct because the meta-logic has simple models, where

   334 types denote sets and $\Forall$ really means for all.'

   335

   336 \index{*Trueprop constant}

   337 Isabelle adopts notational conventions to ease the writing of rules.  We may

   338 hide the occurrences of $Trueprop$ by making it an implicit coercion.

   339 Outer universal quantifiers may be dropped.  Finally, the nested implication

   340 \index{meta-implication}

   341 $\phi@1\Imp(\cdots \phi@n\Imp\psi\cdots)$

   342 may be abbreviated as $\List{\phi@1; \ldots; \phi@n} \Imp \psi$, which

   343 formalizes a rule of $n$~premises.

   344

   345 Using these conventions, the conjunction rules become the following axioms.

   346 These fully specify the properties of~$\conj$:

   347 $$\List{P; Q} \Imp P\conj Q \eqno(\conj I)$$

   348 $$P\conj Q \Imp P \qquad P\conj Q \Imp Q \eqno(\conj E1,2)$$

   349

   350 \noindent

   351 Next, consider the disjunction rules.  The discharge of assumption in

   352 $(\disj E)$ is expressed  using $\Imp$:

   353 \index{assumptions!discharge of}%

   354 $$P \Imp P\disj Q \qquad Q \Imp P\disj Q \eqno(\disj I1,2)$$

   355 $$\List{P\disj Q; P\Imp R; Q\Imp R} \Imp R \eqno(\disj E)$$

   356 %

   357 To understand this treatment of assumptions in natural

   358 deduction, look at implication.  The rule $({\imp}I)$ is the classic

   359 example of natural deduction: to prove that $P\imp Q$ is true, assume $P$

   360 is true and show that $Q$ must then be true.  More concisely, if $P$

   361 implies $Q$ (at the meta-level), then $P\imp Q$ is true (at the

   362 object-level).  Showing the coercion explicitly, this is formalized as

   363 $(Trueprop(P)\Imp Trueprop(Q)) \Imp Trueprop(P\imp Q).$

   364 The rule $({\imp}E)$ is straightforward; hiding $Trueprop$, the axioms to

   365 specify $\imp$ are

   366 $$(P \Imp Q) \Imp P\imp Q \eqno({\imp}I)$$

   367 $$\List{P\imp Q; P} \Imp Q. \eqno({\imp}E)$$

   368

   369 \noindent

   370 Finally, the intuitionistic contradiction rule is formalized as the axiom

   371 $$\bot \Imp P. \eqno(\bot E)$$

   372

   373 \begin{warn}

   374 Earlier versions of Isabelle, and certain

   375 papers~\cite{paulson-found,paulson700}, use $\List{P}$ to mean $Trueprop(P)$.

   376 \end{warn}

   377

   378 \subsection{Quantifier rules and substitution}

   379 \index{quantifiers}\index{rules!quantifier}\index{substitution|bold}

   380 \index{variables!bound}\index{lambda abs@$\lambda$-abstractions}

   381 \index{function applications}

   382

   383 Isabelle expresses variable binding using $\lambda$-abstraction; for instance,

   384 $\forall x.P$ is formalized as $\forall(\lambda x.P)$.  Recall that $F(t)$

   385 is Isabelle's syntax for application of the function~$F$ to the argument~$t$;

   386 it is not a meta-notation for substitution.  On the other hand, a substitution

   387 will take place if $F$ has the form $\lambda x.P$;  Isabelle transforms

   388 $(\lambda x.P)(t)$ to~$P[t/x]$ by $\beta$-conversion.  Thus, we can express

   389 inference rules that involve substitution for bound variables.

   390

   391 \index{parameters|bold}\index{eigenvariables|see{parameters}}

   392 A logic may attach provisos to certain of its rules, especially quantifier

   393 rules.  We cannot hope to formalize arbitrary provisos.  Fortunately, those

   394 typical of quantifier rules always have the same form, namely $x$ not free in

   395 \ldots {\it (some set of formulae)},' where $x$ is a variable (called a {\bf

   396 parameter} or {\bf eigenvariable}) in some premise.  Isabelle treats

   397 provisos using~$\Forall$, its inbuilt notion of for all'.

   398 \index{meta-quantifiers}

   399

   400 The purpose of the proviso $x$ not free in \ldots' is

   401 to ensure that the premise may not make assumptions about the value of~$x$,

   402 and therefore holds for all~$x$.  We formalize $(\forall I)$ by

   403 $\left(\Forall x. Trueprop(P(x))\right) \Imp Trueprop(\forall x.P(x)).$

   404 This means, if $P(x)$ is true for all~$x$, then $\forall x.P(x)$ is true.'

   405 The $\forall E$ rule exploits $\beta$-conversion.  Hiding $Trueprop$, the

   406 $\forall$ axioms are

   407 $$\left(\Forall x. P(x)\right) \Imp \forall x.P(x) \eqno(\forall I)$$

   408 $$(\forall x.P(x)) \Imp P(t). \eqno(\forall E)$$

   409

   410 \noindent

   411 We have defined the object-level universal quantifier~($\forall$)

   412 using~$\Forall$.  But we do not require meta-level counterparts of all the

   413 connectives of the object-logic!  Consider the existential quantifier:

   414 $$P(t) \Imp \exists x.P(x) \eqno(\exists I)$$

   415 $$\List{\exists x.P(x);\; \Forall x. P(x)\Imp Q} \Imp Q \eqno(\exists E)$$

   416 Let us verify $(\exists E)$ semantically.  Suppose that the premises

   417 hold; since $\exists x.P(x)$ is true, we may choose an~$a$ such that $P(a)$ is

   418 true.  Instantiating $\Forall x. P(x)\Imp Q$ with $a$ yields $P(a)\Imp Q$, and

   419 we obtain the desired conclusion, $Q$.

   420

   421 The treatment of substitution deserves mention.  The rule

   422 $\infer{P[u/t]}{t=u & P}$

   423 would be hard to formalize in Isabelle.  It calls for replacing~$t$ by $u$

   424 throughout~$P$, which cannot be expressed using $\beta$-conversion.  Our

   425 rule~$(subst)$ uses~$P$ as a template for substitution, inferring $P[u/x]$

   426 from~$P[t/x]$.  When we formalize this as an axiom, the template becomes a

   427 function variable:

   428 $$\List{t=u; P(t)} \Imp P(u). \eqno(subst)$$

   429

   430

   431 \subsection{Signatures and theories}

   432 \index{signatures|bold}

   433

   434 A {\bf signature} contains the information necessary for type-checking,

   435 parsing and pretty printing a term.  It specifies type classes and their

   436 relationships, types and their arities, constants and their types, etc.  It

   437 also contains grammar rules, specified using mixfix declarations.

   438

   439 Two signatures can be merged provided their specifications are compatible ---

   440 they must not, for example, assign different types to the same constant.

   441 Under similar conditions, a signature can be extended.  Signatures are

   442 managed internally by Isabelle; users seldom encounter them.

   443

   444 \index{theories|bold} A {\bf theory} consists of a signature plus a collection

   445 of axioms.  The Pure theory contains only the meta-logic.  Theories can be

   446 combined provided their signatures are compatible.  A theory definition

   447 extends an existing theory with further signature specifications --- classes,

   448 types, constants and mixfix declarations --- plus lists of axioms and

   449 definitions etc., expressed as strings to be parsed.  A theory can formalize a

   450 small piece of mathematics, such as lists and their operations, or an entire

   451 logic.  A mathematical development typically involves many theories in a

   452 hierarchy.  For example, the Pure theory could be extended to form a theory

   453 for Fig.\ts\ref{fol-fig}; this could be extended in two separate ways to form

   454 a theory for natural numbers and a theory for lists; the union of these two

   455 could be extended into a theory defining the length of a list:

   456 \begin{tt}

   457 $  458 \begin{array}{c@{}c@{}c@{}c@{}c}   459 {} & {} &\hbox{Pure}& {} & {} \\   460 {} & {} & \downarrow & {} & {} \\   461 {} & {} &\hbox{FOL} & {} & {} \\   462 {} & \swarrow & {} & \searrow & {} \\   463 \hbox{Nat} & {} & {} & {} & \hbox{List} \\   464 {} & \searrow & {} & \swarrow & {} \\   465 {} & {} &\hbox{Nat}+\hbox{List}& {} & {} \\   466 {} & {} & \downarrow & {} & {} \\   467 {} & {} & \hbox{Length} & {} & {}   468 \end{array}   469$

   470 \end{tt}%

   471 Each Isabelle proof typically works within a single theory, which is

   472 associated with the proof state.  However, many different theories may

   473 coexist at the same time, and you may work in each of these during a single

   474 session.

   475

   476 \begin{warn}\index{constants!clashes with variables}%

   477   Confusing problems arise if you work in the wrong theory.  Each theory

   478   defines its own syntax.  An identifier may be regarded in one theory as a

   479   constant and in another as a variable, for example.

   480 \end{warn}

   481

   482 \section{Proof construction in Isabelle}

   483 I have elsewhere described the meta-logic and demonstrated it by

   484 formalizing first-order logic~\cite{paulson-found}.  There is a one-to-one

   485 correspondence between meta-level proofs and object-level proofs.  To each

   486 use of a meta-level axiom, such as $(\forall I)$, there is a use of the

   487 corresponding object-level rule.  Object-level assumptions and parameters

   488 have meta-level counterparts.  The meta-level formalization is {\bf

   489   faithful}, admitting no incorrect object-level inferences, and {\bf

   490   adequate}, admitting all correct object-level inferences.  These

   491 properties must be demonstrated separately for each object-logic.

   492

   493 The meta-logic is defined by a collection of inference rules, including

   494 equational rules for the $\lambda$-calculus and logical rules.  The rules

   495 for~$\Imp$ and~$\Forall$ resemble those for~$\imp$ and~$\forall$ in

   496 Fig.\ts\ref{fol-fig}.  Proofs performed using the primitive meta-rules

   497 would be lengthy; Isabelle proofs normally use certain derived rules.

   498 {\bf Resolution}, in particular, is convenient for backward proof.

   499

   500 Unification is central to theorem proving.  It supports quantifier

   501 reasoning by allowing certain unknown' terms to be instantiated later,

   502 possibly in stages.  When proving that the time required to sort $n$

   503 integers is proportional to~$n^2$, we need not state the constant of

   504 proportionality; when proving that a hardware adder will deliver the sum of

   505 its inputs, we need not state how many clock ticks will be required.  Such

   506 quantities often emerge from the proof.

   507

   508 Isabelle provides {\bf schematic variables}, or {\bf

   509   unknowns},\index{unknowns} for unification.  Logically, unknowns are free

   510 variables.  But while ordinary variables remain fixed, unification may

   511 instantiate unknowns.  Unknowns are written with a ?\ prefix and are

   512 frequently subscripted: $\Var{a}$, $\Var{a@1}$, $\Var{a@2}$, \ldots,

   513 $\Var{P}$, $\Var{P@1}$, \ldots.

   514

   515 Recall that an inference rule of the form

   516 $\infer{\phi}{\phi@1 & \ldots & \phi@n}$

   517 is formalized in Isabelle's meta-logic as the axiom

   518 $\List{\phi@1; \ldots; \phi@n} \Imp \phi$.\index{resolution}

   519 Such axioms resemble Prolog's Horn clauses, and can be combined by

   520 resolution --- Isabelle's principal proof method.  Resolution yields both

   521 forward and backward proof.  Backward proof works by unifying a goal with

   522 the conclusion of a rule, whose premises become new subgoals.  Forward proof

   523 works by unifying theorems with the premises of a rule, deriving a new theorem.

   524

   525 Isabelle formulae require an extended notion of resolution.

   526 They differ from Horn clauses in two major respects:

   527 \begin{itemize}

   528   \item They are written in the typed $\lambda$-calculus, and therefore must be

   529 resolved using higher-order unification.

   530

   531 \item The constituents of a clause need not be atomic formulae.  Any

   532   formula of the form $Trueprop(\cdots)$ is atomic, but axioms such as

   533   ${\imp}I$ and $\forall I$ contain non-atomic formulae.

   534 \end{itemize}

   535 Isabelle has little in common with classical resolution theorem provers

   536 such as Otter~\cite{wos-bledsoe}.  At the meta-level, Isabelle proves

   537 theorems in their positive form, not by refutation.  However, an

   538 object-logic that includes a contradiction rule may employ a refutation

   539 proof procedure.

   540

   541

   542 \subsection{Higher-order unification}

   543 \index{unification!higher-order|bold}

   544 Unification is equation solving.  The solution of $f(\Var{x},c) \qeq   545 f(d,\Var{y})$ is $\Var{x}\equiv d$ and $\Var{y}\equiv c$.  {\bf

   546 Higher-order unification} is equation solving for typed $\lambda$-terms.

   547 To handle $\beta$-conversion, it must reduce $(\lambda x.t)u$ to $t[u/x]$.

   548 That is easy --- in the typed $\lambda$-calculus, all reduction sequences

   549 terminate at a normal form.  But it must guess the unknown

   550 function~$\Var{f}$ in order to solve the equation

   551  \label{hou-eqn}

   552  \Var{f}(t) \qeq g(u@1,\ldots,u@k).

   553

   554 Huet's~\cite{huet75} search procedure solves equations by imitation and

   555 projection.  {\bf Imitation} makes~$\Var{f}$ apply the leading symbol (if a

   556 constant) of the right-hand side.  To solve equation~(\ref{hou-eqn}), it

   557 guesses

   558 $\Var{f} \equiv \lambda x. g(\Var{h@1}(x),\ldots,\Var{h@k}(x)),$

   559 where $\Var{h@1}$, \ldots, $\Var{h@k}$ are new unknowns.  Assuming there are no

   560 other occurrences of~$\Var{f}$, equation~(\ref{hou-eqn}) simplifies to the

   561 set of equations

   562 $\Var{h@1}(t)\qeq u@1 \quad\ldots\quad \Var{h@k}(t)\qeq u@k.$

   563 If the procedure solves these equations, instantiating $\Var{h@1}$, \ldots,

   564 $\Var{h@k}$, then it yields an instantiation for~$\Var{f}$.

   565

   566 {\bf Projection} makes $\Var{f}$ apply one of its arguments.  To solve

   567 equation~(\ref{hou-eqn}), if $t$ expects~$m$ arguments and delivers a

   568 result of suitable type, it guesses

   569 $\Var{f} \equiv \lambda x. x(\Var{h@1}(x),\ldots,\Var{h@m}(x)),$

   570 where $\Var{h@1}$, \ldots, $\Var{h@m}$ are new unknowns.  Assuming there are no

   571 other occurrences of~$\Var{f}$, equation~(\ref{hou-eqn}) simplifies to the

   572 equation

   573 $t(\Var{h@1}(t),\ldots,\Var{h@m}(t)) \qeq g(u@1,\ldots,u@k).$

   574

   575 \begin{warn}\index{unification!incompleteness of}%

   576 Huet's unification procedure is complete.  Isabelle's polymorphic version,

   577 which solves for type unknowns as well as for term unknowns, is incomplete.

   578 The problem is that projection requires type information.  In

   579 equation~(\ref{hou-eqn}), if the type of~$t$ is unknown, then projections

   580 are possible for all~$m\geq0$, and the types of the $\Var{h@i}$ will be

   581 similarly unconstrained.  Therefore, Isabelle never attempts such

   582 projections, and may fail to find unifiers where a type unknown turns out

   583 to be a function type.

   584 \end{warn}

   585

   586 \index{unknowns!function|bold}

   587 Given $\Var{f}(t@1,\ldots,t@n)\qeq u$, Huet's procedure could make up to

   588 $n+1$ guesses.  The search tree and set of unifiers may be infinite.  But

   589 higher-order unification can work effectively, provided you are careful

   590 with {\bf function unknowns}:

   591 \begin{itemize}

   592   \item Equations with no function unknowns are solved using first-order

   593 unification, extended to treat bound variables.  For example, $\lambda x.x   594 \qeq \lambda x.\Var{y}$ has no solution because $\Var{y}\equiv x$ would

   595 capture the free variable~$x$.

   596

   597   \item An occurrence of the term $\Var{f}(x,y,z)$, where the arguments are

   598 distinct bound variables, causes no difficulties.  Its projections can only

   599 match the corresponding variables.

   600

   601   \item Even an equation such as $\Var{f}(a)\qeq a+a$ is all right.  It has

   602 four solutions, but Isabelle evaluates them lazily, trying projection before

   603 imitation.  The first solution is usually the one desired:

   604 $\Var{f}\equiv \lambda x. x+x \quad   605 \Var{f}\equiv \lambda x. a+x \quad   606 \Var{f}\equiv \lambda x. x+a \quad   607 \Var{f}\equiv \lambda x. a+a$

   608   \item  Equations such as $\Var{f}(\Var{x},\Var{y})\qeq t$ and

   609 $\Var{f}(\Var{g}(x))\qeq t$ admit vast numbers of unifiers, and must be

   610 avoided.

   611 \end{itemize}

   612 In problematic cases, you may have to instantiate some unknowns before

   613 invoking unification.

   614

   615

   616 \subsection{Joining rules by resolution} \label{joining}

   617 \index{resolution|bold}

   618 Let $\List{\psi@1; \ldots; \psi@m} \Imp \psi$ and $\List{\phi@1; \ldots;   619 \phi@n} \Imp \phi$ be two Isabelle theorems, representing object-level rules.

   620 Choosing some~$i$ from~1 to~$n$, suppose that $\psi$ and $\phi@i$ have a

   621 higher-order unifier.  Writing $Xs$ for the application of substitution~$s$ to

   622 expression~$X$, this means there is some~$s$ such that $\psi s\equiv \phi@i s$.

   623 By resolution, we may conclude

   624 $(\List{\phi@1; \ldots; \phi@{i-1}; \psi@1; \ldots; \psi@m;   625 \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s.   626$

   627 The substitution~$s$ may instantiate unknowns in both rules.  In short,

   628 resolution is the following rule:

   629 $\infer[(\psi s\equiv \phi@i s)]   630 {(\List{\phi@1; \ldots; \phi@{i-1}; \psi@1; \ldots; \psi@m;   631 \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s}   632 {\List{\psi@1; \ldots; \psi@m} \Imp \psi & &   633 \List{\phi@1; \ldots; \phi@n} \Imp \phi}   634$

   635 It operates at the meta-level, on Isabelle theorems, and is justified by

   636 the properties of $\Imp$ and~$\Forall$.  It takes the number~$i$ (for

   637 $1\leq i\leq n$) as a parameter and may yield infinitely many conclusions,

   638 one for each unifier of $\psi$ with $\phi@i$.  Isabelle returns these

   639 conclusions as a sequence (lazy list).

   640

   641 Resolution expects the rules to have no outer quantifiers~($\Forall$).

   642 It may rename or instantiate any schematic variables, but leaves free

   643 variables unchanged.  When constructing a theory, Isabelle puts the

   644 rules into a standard form with all free variables converted into

   645 schematic ones; for instance, $({\imp}E)$ becomes

   646 $\List{\Var{P}\imp \Var{Q}; \Var{P}} \Imp \Var{Q}.   647$

   648 When resolving two rules, the unknowns in the first rule are renamed, by

   649 subscripting, to make them distinct from the unknowns in the second rule.  To

   650 resolve $({\imp}E)$ with itself, the first copy of the rule becomes

   651 $\List{\Var{P@1}\imp \Var{Q@1}; \Var{P@1}} \Imp \Var{Q@1}.$

   652 Resolving this with $({\imp}E)$ in the first premise, unifying $\Var{Q@1}$ with

   653 $\Var{P}\imp \Var{Q}$, is the meta-level inference

   654 $\infer{\List{\Var{P@1}\imp (\Var{P}\imp \Var{Q}); \Var{P@1}; \Var{P}}   655 \Imp\Var{Q}.}   656 {\List{\Var{P@1}\imp \Var{Q@1}; \Var{P@1}} \Imp \Var{Q@1} & &   657 \List{\Var{P}\imp \Var{Q}; \Var{P}} \Imp \Var{Q}}   658$

   659 Renaming the unknowns in the resolvent, we have derived the

   660 object-level rule\index{rules!derived}

   661 $\infer{Q.}{R\imp (P\imp Q) & R & P}$

   662 Joining rules in this fashion is a simple way of proving theorems.  The

   663 derived rules are conservative extensions of the object-logic, and may permit

   664 simpler proofs.  Let us consider another example.  Suppose we have the axiom

   665 $$\forall x\,y. Suc(x)=Suc(y)\imp x=y. \eqno (inject)$$

   666

   667 \noindent

   668 The standard form of $(\forall E)$ is

   669 $\forall x.\Var{P}(x) \Imp \Var{P}(\Var{t})$.

   670 Resolving $(inject)$ with $(\forall E)$ replaces $\Var{P}$ by

   671 $\lambda x. \forall y. Suc(x)=Suc(y)\imp x=y$ and leaves $\Var{t}$

   672 unchanged, yielding

   673 $\forall y. Suc(\Var{t})=Suc(y)\imp \Var{t}=y.$

   674 Resolving this with $(\forall E)$ puts a subscript on~$\Var{t}$

   675 and yields

   676 $Suc(\Var{t@1})=Suc(\Var{t})\imp \Var{t@1}=\Var{t}.$

   677 Resolving this with $({\imp}E)$ increases the subscripts and yields

   678 $Suc(\Var{t@2})=Suc(\Var{t@1})\Imp \Var{t@2}=\Var{t@1}.   679$

   680 We have derived the rule

   681 $\infer{m=n,}{Suc(m)=Suc(n)}$

   682 which goes directly from $Suc(m)=Suc(n)$ to $m=n$.  It is handy for simplifying

   683 an equation like $Suc(Suc(Suc(m)))=Suc(Suc(Suc(0)))$.

   684

   685

   686 \section{Lifting a rule into a context}

   687 The rules $({\imp}I)$ and $(\forall I)$ may seem unsuitable for

   688 resolution.  They have non-atomic premises, namely $P\Imp Q$ and $\Forall   689 x.P(x)$, while the conclusions of all the rules are atomic (they have the form

   690 $Trueprop(\cdots)$).  Isabelle gets round the problem through a meta-inference

   691 called \bfindex{lifting}.  Let us consider how to construct proofs such as

   692 $\infer[({\imp}I)]{P\imp(Q\imp R)}   693 {\infer[({\imp}I)]{Q\imp R}   694 {\infer*{R}{[P,Q]}}}   695 \qquad   696 \infer[(\forall I)]{\forall x\,y.P(x,y)}   697 {\infer[(\forall I)]{\forall y.P(x,y)}{P(x,y)}}   698$

   699

   700 \subsection{Lifting over assumptions}

   701 \index{assumptions!lifting over}

   702 Lifting over $\theta\Imp{}$ is the following meta-inference rule:

   703 $\infer{\List{\theta\Imp\phi@1; \ldots; \theta\Imp\phi@n} \Imp   704 (\theta \Imp \phi)}   705 {\List{\phi@1; \ldots; \phi@n} \Imp \phi}$

   706 This is clearly sound: if $\List{\phi@1; \ldots; \phi@n} \Imp \phi$ is true and

   707 $\theta\Imp\phi@1$, \ldots, $\theta\Imp\phi@n$ and $\theta$ are all true then

   708 $\phi$ must be true.  Iterated lifting over a series of meta-formulae

   709 $\theta@k$, \ldots, $\theta@1$ yields an object-rule whose conclusion is

   710 $\List{\theta@1; \ldots; \theta@k} \Imp \phi$.  Typically the $\theta@i$ are

   711 the assumptions in a natural deduction proof; lifting copies them into a rule's

   712 premises and conclusion.

   713

   714 When resolving two rules, Isabelle lifts the first one if necessary.  The

   715 standard form of $({\imp}I)$ is

   716 $(\Var{P} \Imp \Var{Q}) \Imp \Var{P}\imp \Var{Q}.$

   717 To resolve this rule with itself, Isabelle modifies one copy as follows: it

   718 renames the unknowns to $\Var{P@1}$ and $\Var{Q@1}$, then lifts the rule over

   719 $\Var{P}\Imp{}$ to obtain

   720 $(\Var{P}\Imp (\Var{P@1} \Imp \Var{Q@1})) \Imp (\Var{P} \Imp   721 (\Var{P@1}\imp \Var{Q@1})).$

   722 Using the $\List{\cdots}$ abbreviation, this can be written as

   723 $\List{\List{\Var{P}; \Var{P@1}} \Imp \Var{Q@1}; \Var{P}}   724 \Imp \Var{P@1}\imp \Var{Q@1}.$

   725 Unifying $\Var{P}\Imp \Var{P@1}\imp\Var{Q@1}$ with $\Var{P} \Imp   726 \Var{Q}$ instantiates $\Var{Q}$ to ${\Var{P@1}\imp\Var{Q@1}}$.

   727 Resolution yields

   728 $(\List{\Var{P}; \Var{P@1}} \Imp \Var{Q@1}) \Imp   729 \Var{P}\imp(\Var{P@1}\imp\Var{Q@1}).$

   730 This represents the derived rule

   731 $\infer{P\imp(Q\imp R).}{\infer*{R}{[P,Q]}}$

   732

   733 \subsection{Lifting over parameters}

   734 \index{parameters!lifting over}

   735 An analogous form of lifting handles premises of the form $\Forall x\ldots\,$.

   736 Here, lifting prefixes an object-rule's premises and conclusion with $\Forall   737 x$.  At the same time, lifting introduces a dependence upon~$x$.  It replaces

   738 each unknown $\Var{a}$ in the rule by $\Var{a'}(x)$, where $\Var{a'}$ is a new

   739 unknown (by subscripting) of suitable type --- necessarily a function type.  In

   740 short, lifting is the meta-inference

   741 $\infer{\List{\Forall x.\phi@1^x; \ldots; \Forall x.\phi@n^x}   742 \Imp \Forall x.\phi^x,}   743 {\List{\phi@1; \ldots; \phi@n} \Imp \phi}$

   744 %

   745 where $\phi^x$ stands for the result of lifting unknowns over~$x$ in

   746 $\phi$.  It is not hard to verify that this meta-inference is sound.  If

   747 $\phi\Imp\psi$ then $\phi^x\Imp\psi^x$ for all~$x$; so if $\phi^x$ is true

   748 for all~$x$ then so is $\psi^x$.  Thus, from $\phi\Imp\psi$ we conclude

   749 $(\Forall x.\phi^x) \Imp (\Forall x.\psi^x)$.

   750

   751 For example, $(\disj I)$ might be lifted to

   752 $(\Forall x.\Var{P@1}(x)) \Imp (\Forall x. \Var{P@1}(x)\disj \Var{Q@1}(x))$

   753 and $(\forall I)$ to

   754 $(\Forall x\,y.\Var{P@1}(x,y)) \Imp (\Forall x. \forall y.\Var{P@1}(x,y)).$

   755 Isabelle has renamed a bound variable in $(\forall I)$ from $x$ to~$y$,

   756 avoiding a clash.  Resolving the above with $(\forall I)$ is the meta-inference

   757 $\infer{\Forall x\,y.\Var{P@1}(x,y)) \Imp \forall x\,y.\Var{P@1}(x,y)) }   758 {(\Forall x\,y.\Var{P@1}(x,y)) \Imp   759 (\Forall x. \forall y.\Var{P@1}(x,y)) &   760 (\Forall x.\Var{P}(x)) \Imp (\forall x.\Var{P}(x))}$

   761 Here, $\Var{P}$ is replaced by $\lambda x.\forall y.\Var{P@1}(x,y)$; the

   762 resolvent expresses the derived rule

   763 $\vcenter{ \infer{\forall x\,y.Q(x,y)}{Q(x,y)} }   764 \quad\hbox{provided x, y not free in the assumptions}   765$

   766 I discuss lifting and parameters at length elsewhere~\cite{paulson-found}.

   767 Miller goes into even greater detail~\cite{miller-mixed}.

   768

   769

   770 \section{Backward proof by resolution}

   771 \index{resolution!in backward proof}

   772

   773 Resolution is convenient for deriving simple rules and for reasoning

   774 forward from facts.  It can also support backward proof, where we start

   775 with a goal and refine it to progressively simpler subgoals until all have

   776 been solved.  {\sc lcf} and its descendants {\sc hol} and Nuprl provide

   777 tactics and tacticals, which constitute a sophisticated language for

   778 expressing proof searches.  {\bf Tactics} refine subgoals while {\bf

   779   tacticals} combine tactics.

   780

   781 \index{LCF system}

   782 Isabelle's tactics and tacticals work differently from {\sc lcf}'s.  An

   783 Isabelle rule is bidirectional: there is no distinction between

   784 inputs and outputs.  {\sc lcf} has a separate tactic for each rule;

   785 Isabelle performs refinement by any rule in a uniform fashion, using

   786 resolution.

   787

   788 Isabelle works with meta-level theorems of the form

   789 $$\List{\phi@1; \ldots; \phi@n} \Imp \phi$$.

   790 We have viewed this as the {\bf rule} with premises

   791 $\phi@1$,~\ldots,~$\phi@n$ and conclusion~$\phi$.  It can also be viewed as

   792 the {\bf proof state}\index{proof state}

   793 with subgoals $\phi@1$,~\ldots,~$\phi@n$ and main

   794 goal~$\phi$.

   795

   796 To prove the formula~$\phi$, take $\phi\Imp \phi$ as the initial proof

   797 state.  This assertion is, trivially, a theorem.  At a later stage in the

   798 backward proof, a typical proof state is $\List{\phi@1; \ldots; \phi@n}   799 \Imp \phi$.  This proof state is a theorem, ensuring that the subgoals

   800 $\phi@1$,~\ldots,~$\phi@n$ imply~$\phi$.  If $n=0$ then we have

   801 proved~$\phi$ outright.  If $\phi$ contains unknowns, they may become

   802 instantiated during the proof; a proof state may be $\List{\phi@1; \ldots;   803 \phi@n} \Imp \phi'$, where $\phi'$ is an instance of~$\phi$.

   804

   805 \subsection{Refinement by resolution}

   806 To refine subgoal~$i$ of a proof state by a rule, perform the following

   807 resolution:

   808 $\infer{\hbox{new proof state}}{\hbox{rule} & & \hbox{proof state}}$

   809 Suppose the rule is $\List{\psi'@1; \ldots; \psi'@m} \Imp \psi'$ after

   810 lifting over subgoal~$i$'s assumptions and parameters.  If the proof state

   811 is $\List{\phi@1; \ldots; \phi@n} \Imp \phi$, then the new proof state is

   812 (for~$1\leq i\leq n$)

   813 $(\List{\phi@1; \ldots; \phi@{i-1}; \psi'@1;   814 \ldots; \psi'@m; \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s.$

   815 Substitution~$s$ unifies $\psi'$ with~$\phi@i$.  In the proof state,

   816 subgoal~$i$ is replaced by $m$ new subgoals, the rule's instantiated premises.

   817 If some of the rule's unknowns are left un-instantiated, they become new

   818 unknowns in the proof state.  Refinement by~$(\exists I)$, namely

   819 $\Var{P}(\Var{t}) \Imp \exists x. \Var{P}(x),$

   820 inserts a new unknown derived from~$\Var{t}$ by subscripting and lifting.

   821 We do not have to specify an existential witness' when

   822 applying~$(\exists I)$.  Further resolutions may instantiate unknowns in

   823 the proof state.

   824

   825 \subsection{Proof by assumption}

   826 \index{assumptions!use of}

   827 In the course of a natural deduction proof, parameters $x@1$, \ldots,~$x@l$ and

   828 assumptions $\theta@1$, \ldots, $\theta@k$ accumulate, forming a context for

   829 each subgoal.  Repeated lifting steps can lift a rule into any context.  To

   830 aid readability, Isabelle puts contexts into a normal form, gathering the

   831 parameters at the front:

   832  \label{context-eqn}

   833 \Forall x@1 \ldots x@l. \List{\theta@1; \ldots; \theta@k}\Imp\theta.

   834

   835 Under the usual reading of the connectives, this expresses that $\theta$

   836 follows from $\theta@1$,~\ldots~$\theta@k$ for arbitrary

   837 $x@1$,~\ldots,~$x@l$.  It is trivially true if $\theta$ equals any of

   838 $\theta@1$,~\ldots~$\theta@k$, or is unifiable with any of them.  This

   839 models proof by assumption in natural deduction.

   840

   841 Isabelle automates the meta-inference for proof by assumption.  Its arguments

   842 are the meta-theorem $\List{\phi@1; \ldots; \phi@n} \Imp \phi$, and some~$i$

   843 from~1 to~$n$, where $\phi@i$ has the form~(\ref{context-eqn}).  Its results

   844 are meta-theorems of the form

   845 $(\List{\phi@1; \ldots; \phi@{i-1}; \phi@{i+1}; \phi@n} \Imp \phi)s$

   846 for each $s$ and~$j$ such that $s$ unifies $\lambda x@1 \ldots x@l. \theta@j$

   847 with $\lambda x@1 \ldots x@l. \theta$.  Isabelle supplies the parameters

   848 $x@1$,~\ldots,~$x@l$ to higher-order unification as bound variables, which

   849 regards them as unique constants with a limited scope --- this enforces

   850 parameter provisos~\cite{paulson-found}.

   851

   852 The premise represents a proof state with~$n$ subgoals, of which the~$i$th

   853 is to be solved by assumption.  Isabelle searches the subgoal's context for

   854 an assumption~$\theta@j$ that can solve it.  For each unifier, the

   855 meta-inference returns an instantiated proof state from which the $i$th

   856 subgoal has been removed.  Isabelle searches for a unifying assumption; for

   857 readability and robustness, proofs do not refer to assumptions by number.

   858

   859 Consider the proof state

   860 $(\List{P(a); P(b)} \Imp P(\Var{x})) \Imp Q(\Var{x}).$

   861 Proof by assumption (with $i=1$, the only possibility) yields two results:

   862 \begin{itemize}

   863   \item $Q(a)$, instantiating $\Var{x}\equiv a$

   864   \item $Q(b)$, instantiating $\Var{x}\equiv b$

   865 \end{itemize}

   866 Here, proof by assumption affects the main goal.  It could also affect

   867 other subgoals; if we also had the subgoal ${\List{P(b); P(c)} \Imp   868 P(\Var{x})}$, then $\Var{x}\equiv a$ would transform it to ${\List{P(b);   869 P(c)} \Imp P(a)}$, which might be unprovable.

   870

   871

   872 \subsection{A propositional proof} \label{prop-proof}

   873 \index{examples!propositional}

   874 Our first example avoids quantifiers.  Given the main goal $P\disj P\imp   875 P$, Isabelle creates the initial state

   876 $(P\disj P\imp P)\Imp (P\disj P\imp P).$

   877 %

   878 Bear in mind that every proof state we derive will be a meta-theorem,

   879 expressing that the subgoals imply the main goal.  Our aim is to reach the

   880 state $P\disj P\imp P$; this meta-theorem is the desired result.

   881

   882 The first step is to refine subgoal~1 by (${\imp}I)$, creating a new state

   883 where $P\disj P$ is an assumption:

   884 $(P\disj P\Imp P)\Imp (P\disj P\imp P)$

   885 The next step is $(\disj E)$, which replaces subgoal~1 by three new subgoals.

   886 Because of lifting, each subgoal contains a copy of the context --- the

   887 assumption $P\disj P$.  (In fact, this assumption is now redundant; we shall

   888 shortly see how to get rid of it!)  The new proof state is the following

   889 meta-theorem, laid out for clarity:

   890 $\begin{array}{l@{}l@{\qquad\qquad}l}   891 \lbrakk\;& P\disj P\Imp \Var{P@1}\disj\Var{Q@1}; & \hbox{(subgoal 1)} \\   892 & \List{P\disj P; \Var{P@1}} \Imp P; & \hbox{(subgoal 2)} \\   893 & \List{P\disj P; \Var{Q@1}} \Imp P & \hbox{(subgoal 3)} \\   894 \rbrakk\;& \Imp (P\disj P\imp P) & \hbox{(main goal)}   895 \end{array}   896$

   897 Notice the unknowns in the proof state.  Because we have applied $(\disj E)$,

   898 we must prove some disjunction, $\Var{P@1}\disj\Var{Q@1}$.  Of course,

   899 subgoal~1 is provable by assumption.  This instantiates both $\Var{P@1}$ and

   900 $\Var{Q@1}$ to~$P$ throughout the proof state:

   901 $\begin{array}{l@{}l@{\qquad\qquad}l}   902 \lbrakk\;& \List{P\disj P; P} \Imp P; & \hbox{(subgoal 1)} \\   903 & \List{P\disj P; P} \Imp P & \hbox{(subgoal 2)} \\   904 \rbrakk\;& \Imp (P\disj P\imp P) & \hbox{(main goal)}   905 \end{array}$

   906 Both of the remaining subgoals can be proved by assumption.  After two such

   907 steps, the proof state is $P\disj P\imp P$.

   908

   909

   910 \subsection{A quantifier proof}

   911 \index{examples!with quantifiers}

   912 To illustrate quantifiers and $\Forall$-lifting, let us prove

   913 $(\exists x.P(f(x)))\imp(\exists x.P(x))$.  The initial proof

   914 state is the trivial meta-theorem

   915 $(\exists x.P(f(x)))\imp(\exists x.P(x)) \Imp   916 (\exists x.P(f(x)))\imp(\exists x.P(x)).$

   917 As above, the first step is refinement by (${\imp}I)$:

   918 $(\exists x.P(f(x))\Imp \exists x.P(x)) \Imp   919 (\exists x.P(f(x)))\imp(\exists x.P(x))   920$

   921 The next step is $(\exists E)$, which replaces subgoal~1 by two new subgoals.

   922 Both have the assumption $\exists x.P(f(x))$.  The new proof

   923 state is the meta-theorem

   924 $\begin{array}{l@{}l@{\qquad\qquad}l}   925 \lbrakk\;& \exists x.P(f(x)) \Imp \exists x.\Var{P@1}(x); & \hbox{(subgoal 1)} \\   926 & \Forall x.\List{\exists x.P(f(x)); \Var{P@1}(x)} \Imp   927 \exists x.P(x) & \hbox{(subgoal 2)} \\   928 \rbrakk\;& \Imp (\exists x.P(f(x)))\imp(\exists x.P(x)) & \hbox{(main goal)}   929 \end{array}   930$

   931 The unknown $\Var{P@1}$ appears in both subgoals.  Because we have applied

   932 $(\exists E)$, we must prove $\exists x.\Var{P@1}(x)$, where $\Var{P@1}(x)$ may

   933 become any formula possibly containing~$x$.  Proving subgoal~1 by assumption

   934 instantiates $\Var{P@1}$ to~$\lambda x.P(f(x))$:

   935 $\left(\Forall x.\List{\exists x.P(f(x)); P(f(x))} \Imp   936 \exists x.P(x)\right)   937 \Imp (\exists x.P(f(x)))\imp(\exists x.P(x))   938$

   939 The next step is refinement by $(\exists I)$.  The rule is lifted into the

   940 context of the parameter~$x$ and the assumption $P(f(x))$.  This copies

   941 the context to the subgoal and allows the existential witness to

   942 depend upon~$x$:

   943 $\left(\Forall x.\List{\exists x.P(f(x)); P(f(x))} \Imp   944 P(\Var{x@2}(x))\right)   945 \Imp (\exists x.P(f(x)))\imp(\exists x.P(x))   946$

   947 The existential witness, $\Var{x@2}(x)$, consists of an unknown

   948 applied to a parameter.  Proof by assumption unifies $\lambda x.P(f(x))$

   949 with $\lambda x.P(\Var{x@2}(x))$, instantiating $\Var{x@2}$ to $f$.  The final

   950 proof state contains no subgoals: $(\exists x.P(f(x)))\imp(\exists x.P(x))$.

   951

   952

   953 \subsection{Tactics and tacticals}

   954 \index{tactics|bold}\index{tacticals|bold}

   955 {\bf Tactics} perform backward proof.  Isabelle tactics differ from those

   956 of {\sc lcf}, {\sc hol} and Nuprl by operating on entire proof states,

   957 rather than on individual subgoals.  An Isabelle tactic is a function that

   958 takes a proof state and returns a sequence (lazy list) of possible

   959 successor states.  Lazy lists are coded in ML as functions, a standard

   960 technique~\cite{paulson-ml2}.  Isabelle represents proof states by theorems.

   961

   962 Basic tactics execute the meta-rules described above, operating on a

   963 given subgoal.  The {\bf resolution tactics} take a list of rules and

   964 return next states for each combination of rule and unifier.  The {\bf

   965 assumption tactic} examines the subgoal's assumptions and returns next

   966 states for each combination of assumption and unifier.  Lazy lists are

   967 essential because higher-order resolution may return infinitely many

   968 unifiers.  If there are no matching rules or assumptions then no next

   969 states are generated; a tactic application that returns an empty list is

   970 said to {\bf fail}.

   971

   972 Sequences realize their full potential with {\bf tacticals} --- operators

   973 for combining tactics.  Depth-first search, breadth-first search and

   974 best-first search (where a heuristic function selects the best state to

   975 explore) return their outcomes as a sequence.  Isabelle provides such

   976 procedures in the form of tacticals.  Simpler procedures can be expressed

   977 directly using the basic tacticals {\tt THEN}, {\tt ORELSE} and {\tt REPEAT}:

   978 \begin{ttdescription}

   979 \item[$tac1$ THEN $tac2$] is a tactic for sequential composition.  Applied

   980 to a proof state, it returns all states reachable in two steps by applying

   981 $tac1$ followed by~$tac2$.

   982

   983 \item[$tac1$ ORELSE $tac2$] is a choice tactic.  Applied to a state, it

   984 tries~$tac1$ and returns the result if non-empty; otherwise, it uses~$tac2$.

   985

   986 \item[REPEAT $tac$] is a repetition tactic.  Applied to a state, it

   987 returns all states reachable by applying~$tac$ as long as possible --- until

   988 it would fail.

   989 \end{ttdescription}

   990 For instance, this tactic repeatedly applies $tac1$ and~$tac2$, giving

   991 $tac1$ priority:

   992 \begin{center} \tt

   993 REPEAT($tac1$ ORELSE $tac2$)

   994 \end{center}

   995

   996

   997 \section{Variations on resolution}

   998 In principle, resolution and proof by assumption suffice to prove all

   999 theorems.  However, specialized forms of resolution are helpful for working

  1000 with elimination rules.  Elim-resolution applies an elimination rule to an

  1001 assumption; destruct-resolution is similar, but applies a rule in a forward

  1002 style.

  1003

  1004 The last part of the section shows how the techniques for proving theorems

  1005 can also serve to derive rules.

  1006

  1007 \subsection{Elim-resolution}

  1008 \index{elim-resolution|bold}\index{assumptions!deleting}

  1009

  1010 Consider proving the theorem $((R\disj R)\disj R)\disj R\imp R$.  By

  1011 $({\imp}I)$, we prove~$R$ from the assumption $((R\disj R)\disj R)\disj R$.

  1012 Applying $(\disj E)$ to this assumption yields two subgoals, one that

  1013 assumes~$R$ (and is therefore trivial) and one that assumes $(R\disj   1014 R)\disj R$.  This subgoal admits another application of $(\disj E)$.  Since

  1015 natural deduction never discards assumptions, we eventually generate a

  1016 subgoal containing much that is redundant:

  1017 $\List{((R\disj R)\disj R)\disj R; (R\disj R)\disj R; R\disj R; R} \Imp R.$

  1018 In general, using $(\disj E)$ on the assumption $P\disj Q$ creates two new

  1019 subgoals with the additional assumption $P$ or~$Q$.  In these subgoals,

  1020 $P\disj Q$ is redundant.  Other elimination rules behave

  1021 similarly.  In first-order logic, only universally quantified

  1022 assumptions are sometimes needed more than once --- say, to prove

  1023 $P(f(f(a)))$ from the assumptions $\forall x.P(x)\imp P(f(x))$ and~$P(a)$.

  1024

  1025 Many logics can be formulated as sequent calculi that delete redundant

  1026 assumptions after use.  The rule $(\disj E)$ might become

  1027 $\infer[\disj\hbox{-left}]   1028 {\Gamma,P\disj Q,\Delta \turn \Theta}   1029 {\Gamma,P,\Delta \turn \Theta && \Gamma,Q,\Delta \turn \Theta}$

  1030 In backward proof, a goal containing $P\disj Q$ on the left of the~$\turn$

  1031 (that is, as an assumption) splits into two subgoals, replacing $P\disj Q$

  1032 by $P$ or~$Q$.  But the sequent calculus, with its explicit handling of

  1033 assumptions, can be tiresome to use.

  1034

  1035 Elim-resolution is Isabelle's way of getting sequent calculus behaviour

  1036 from natural deduction rules.  It lets an elimination rule consume an

  1037 assumption.  Elim-resolution combines two meta-theorems:

  1038 \begin{itemize}

  1039   \item a rule $\List{\psi@1; \ldots; \psi@m} \Imp \psi$

  1040   \item a proof state $\List{\phi@1; \ldots; \phi@n} \Imp \phi$

  1041 \end{itemize}

  1042 The rule must have at least one premise, thus $m>0$.  Write the rule's

  1043 lifted form as $\List{\psi'@1; \ldots; \psi'@m} \Imp \psi'$.  Suppose we

  1044 wish to change subgoal number~$i$.

  1045

  1046 Ordinary resolution would attempt to reduce~$\phi@i$,

  1047 replacing subgoal~$i$ by $m$ new ones.  Elim-resolution tries

  1048 simultaneously to reduce~$\phi@i$ and to solve~$\psi'@1$ by assumption; it

  1049 returns a sequence of next states.  Each of these replaces subgoal~$i$ by

  1050 instances of $\psi'@2$, \ldots, $\psi'@m$ from which the selected

  1051 assumption has been deleted.  Suppose $\phi@i$ has the parameter~$x$ and

  1052 assumptions $\theta@1$,~\ldots,~$\theta@k$.  Then $\psi'@1$, the rule's first

  1053 premise after lifting, will be

  1054 $$\Forall x. \List{\theta@1; \ldots; \theta@k}\Imp \psi^{x}@1$$.

  1055 Elim-resolution tries to unify $\psi'\qeq\phi@i$ and

  1056 $\lambda x. \theta@j \qeq \lambda x. \psi^{x}@1$ simultaneously, for

  1057 $j=1$,~\ldots,~$k$.

  1058

  1059 Let us redo the example from~\S\ref{prop-proof}.  The elimination rule

  1060 is~$(\disj E)$,

  1061 $\List{\Var{P}\disj \Var{Q};\; \Var{P}\Imp \Var{R};\; \Var{Q}\Imp \Var{R}}   1062 \Imp \Var{R}$

  1063 and the proof state is $(P\disj P\Imp P)\Imp (P\disj P\imp P)$.  The

  1064 lifted rule is

  1065 $\begin{array}{l@{}l}   1066 \lbrakk\;& P\disj P \Imp \Var{P@1}\disj\Var{Q@1}; \\   1067 & \List{P\disj P ;\; \Var{P@1}} \Imp \Var{R@1}; \\   1068 & \List{P\disj P ;\; \Var{Q@1}} \Imp \Var{R@1} \\   1069 \rbrakk\;& \Imp (P\disj P \Imp \Var{R@1})   1070 \end{array}   1071$

  1072 Unification takes the simultaneous equations

  1073 $P\disj P \qeq \Var{P@1}\disj\Var{Q@1}$ and $\Var{R@1} \qeq P$, yielding

  1074 $\Var{P@1}\equiv\Var{Q@1}\equiv\Var{R@1} \equiv P$.  The new proof state

  1075 is simply

  1076 $\List{P \Imp P;\; P \Imp P} \Imp (P\disj P\imp P).   1077$

  1078 Elim-resolution's simultaneous unification gives better control

  1079 than ordinary resolution.  Recall the substitution rule:

  1080 $$\List{\Var{t}=\Var{u}; \Var{P}(\Var{t})} \Imp \Var{P}(\Var{u})   1081 \eqno(subst)$$

  1082 Unsuitable for ordinary resolution because $\Var{P}(\Var{u})$ admits many

  1083 unifiers, $(subst)$ works well with elim-resolution.  It deletes some

  1084 assumption of the form $x=y$ and replaces every~$y$ by~$x$ in the subgoal

  1085 formula.  The simultaneous unification instantiates $\Var{u}$ to~$y$; if

  1086 $y$ is not an unknown, then $\Var{P}(y)$ can easily be unified with another

  1087 formula.

  1088

  1089 In logical parlance, the premise containing the connective to be eliminated

  1090 is called the \bfindex{major premise}.  Elim-resolution expects the major

  1091 premise to come first.  The order of the premises is significant in

  1092 Isabelle.

  1093

  1094 \subsection{Destruction rules} \label{destruct}

  1095 \index{rules!destruction}\index{rules!elimination}

  1096 \index{forward proof}

  1097

  1098 Looking back to Fig.\ts\ref{fol-fig}, notice that there are two kinds of

  1099 elimination rule.  The rules $({\conj}E1)$, $({\conj}E2)$, $({\imp}E)$ and

  1100 $({\forall}E)$ extract the conclusion from the major premise.  In Isabelle

  1101 parlance, such rules are called {\bf destruction rules}; they are readable

  1102 and easy to use in forward proof.  The rules $({\disj}E)$, $({\bot}E)$ and

  1103 $({\exists}E)$ work by discharging assumptions; they support backward proof

  1104 in a style reminiscent of the sequent calculus.

  1105

  1106 The latter style is the most general form of elimination rule.  In natural

  1107 deduction, there is no way to recast $({\disj}E)$, $({\bot}E)$ or

  1108 $({\exists}E)$ as destruction rules.  But we can write general elimination

  1109 rules for $\conj$, $\imp$ and~$\forall$:

  1110 $  1111 \infer{R}{P\conj Q & \infer*{R}{[P,Q]}} \qquad   1112 \infer{R}{P\imp Q & P & \infer*{R}{[Q]}} \qquad   1113 \infer{Q}{\forall x.P & \infer*{Q}{[P[t/x]]}}   1114$

  1115 Because they are concise, destruction rules are simpler to derive than the

  1116 corresponding elimination rules.  To facilitate their use in backward

  1117 proof, Isabelle provides a means of transforming a destruction rule such as

  1118 $\infer[\quad\hbox{to the elimination rule}\quad]{Q}{P@1 & \ldots & P@m}   1119 \infer{R.}{P@1 & \ldots & P@m & \infer*{R}{[Q]}}   1120$

  1121 {\bf Destruct-resolution}\index{destruct-resolution} combines this

  1122 transformation with elim-resolution.  It applies a destruction rule to some

  1123 assumption of a subgoal.  Given the rule above, it replaces the

  1124 assumption~$P@1$ by~$Q$, with new subgoals of showing instances of $P@2$,

  1125 \ldots,~$P@m$.  Destruct-resolution works forward from a subgoal's

  1126 assumptions.  Ordinary resolution performs forward reasoning from theorems,

  1127 as illustrated in~\S\ref{joining}.

  1128

  1129

  1130 \subsection{Deriving rules by resolution}  \label{deriving}

  1131 \index{rules!derived|bold}\index{meta-assumptions!syntax of}

  1132 The meta-logic, itself a form of the predicate calculus, is defined by a

  1133 system of natural deduction rules.  Each theorem may depend upon

  1134 meta-assumptions.  The theorem that~$\phi$ follows from the assumptions

  1135 $\phi@1$, \ldots, $\phi@n$ is written

  1136 $\phi \quad [\phi@1,\ldots,\phi@n].$

  1137 A more conventional notation might be $\phi@1,\ldots,\phi@n \turn \phi$,

  1138 but Isabelle's notation is more readable with large formulae.

  1139

  1140 Meta-level natural deduction provides a convenient mechanism for deriving

  1141 new object-level rules.  To derive the rule

  1142 $\infer{\phi,}{\theta@1 & \ldots & \theta@k}$

  1143 assume the premises $\theta@1$,~\ldots,~$\theta@k$ at the

  1144 meta-level.  Then prove $\phi$, possibly using these assumptions.

  1145 Starting with a proof state $\phi\Imp\phi$, assumptions may accumulate,

  1146 reaching a final state such as

  1147 $\phi \quad [\theta@1,\ldots,\theta@k].$

  1148 The meta-rule for $\Imp$ introduction discharges an assumption.

  1149 Discharging them in the order $\theta@k,\ldots,\theta@1$ yields the

  1150 meta-theorem $\List{\theta@1; \ldots; \theta@k} \Imp \phi$, with no

  1151 assumptions.  This represents the desired rule.

  1152 Let us derive the general $\conj$ elimination rule:

  1153 $$\infer{R}{P\conj Q & \infer*{R}{[P,Q]}} \eqno(\conj E)$$

  1154 We assume $P\conj Q$ and $\List{P;Q}\Imp R$, and commence backward proof in

  1155 the state $R\Imp R$.  Resolving this with the second assumption yields the

  1156 state

  1157 $\phantom{\List{P\conj Q;\; P\conj Q}}   1158 \llap{\List{P;Q}}\Imp R \quad [\,\List{P;Q}\Imp R\,].$

  1159 Resolving subgoals~1 and~2 with~$({\conj}E1)$ and~$({\conj}E2)$,

  1160 respectively, yields the state

  1161 $\List{P\conj \Var{Q@1};\; \Var{P@2}\conj Q}\Imp R   1162 \quad [\,\List{P;Q}\Imp R\,].   1163$

  1164 The unknowns $\Var{Q@1}$ and~$\Var{P@2}$ arise from unconstrained

  1165 subformulae in the premises of~$({\conj}E1)$ and~$({\conj}E2)$.  Resolving

  1166 both subgoals with the assumption $P\conj Q$ instantiates the unknowns to yield

  1167 $R \quad [\, \List{P;Q}\Imp R, P\conj Q \,].$

  1168 The proof may use the meta-assumptions in any order, and as often as

  1169 necessary; when finished, we discharge them in the correct order to

  1170 obtain the desired form:

  1171 $\List{P\conj Q;\; \List{P;Q}\Imp R} \Imp R$

  1172 We have derived the rule using free variables, which prevents their

  1173 premature instantiation during the proof; we may now replace them by

  1174 schematic variables.

  1175

  1176 \begin{warn}

  1177   Schematic variables are not allowed in meta-assumptions, for a variety of

  1178   reasons.  Meta-assumptions remain fixed throughout a proof.

  1179 \end{warn}

  1180
`