2 The following sections discuss Isabelle's logical foundations in detail:
3 representing logical syntax in the typed $\lambda$-calculus; expressing
4 inference rules in Isabelle's meta-logic; combining rules by resolution.
6 If you wish to use Isabelle immediately, please turn to
7 page~\pageref{chap:getting}. You can always read about foundations later,
8 either by returning to this point or by looking up particular items in the
13 \neg P & \hbox{abbreviates} & P\imp\bot \\
14 P\bimp Q & \hbox{abbreviates} & (P\imp Q) \conj (Q\imp P)
18 \(\begin{array}{c@{\qquad\qquad}c}
19 \infer[({\conj}I)]{P\conj Q}{P & Q} &
20 \infer[({\conj}E1)]{P}{P\conj Q} \qquad
21 \infer[({\conj}E2)]{Q}{P\conj Q} \\[4ex]
23 \infer[({\disj}I1)]{P\disj Q}{P} \qquad
24 \infer[({\disj}I2)]{P\disj Q}{Q} &
25 \infer[({\disj}E)]{R}{P\disj Q & \infer*{R}{[P]} & \infer*{R}{[Q]}}\\[4ex]
27 \infer[({\imp}I)]{P\imp Q}{\infer*{Q}{[P]}} &
28 \infer[({\imp}E)]{Q}{P\imp Q & P} \\[4ex]
31 \infer[({\bot}E)]{P}{\bot}\\[4ex]
33 \infer[({\forall}I)*]{\forall x.P}{P} &
34 \infer[({\forall}E)]{P[t/x]}{\forall x.P} \\[3ex]
36 \infer[({\exists}I)]{\exists x.P}{P[t/x]} &
37 \infer[({\exists}E)*]{Q}{{\exists x.P} & \infer*{Q}{[P]} } \\[3ex]
39 {t=t} \,(refl) & \vcenter{\infer[(subst)]{P[u/x]}{t=u & P[t/x]}}
43 *{\em Eigenvariable conditions\/}:
45 $\forall I$: provided $x$ is not free in the assumptions
47 $\exists E$: provided $x$ is not free in $Q$ or any assumption except $P$
48 \caption{Intuitionistic first-order logic} \label{fol-fig}
51 \section{Formalizing logical syntax in Isabelle}\label{sec:logical-syntax}
52 \index{first-order logic}
54 Figure~\ref{fol-fig} presents intuitionistic first-order logic,
55 including equality. Let us see how to formalize
56 this logic in Isabelle, illustrating the main features of Isabelle's
57 polymorphic meta-logic.
59 \index{lambda calc@$\lambda$-calculus}
60 Isabelle represents syntax using the simply typed $\lambda$-calculus. We
61 declare a type for each syntactic category of the logic. We declare a
62 constant for each symbol of the logic, giving each $n$-place operation an
63 $n$-argument curried function type. Most importantly,
64 $\lambda$-abstraction represents variable binding in quantifiers.
66 \index{types!syntax of}\index{types!function}\index{*fun type}
67 \index{type constructors}
68 Isabelle has \ML-style polymorphic types such as~$(\alpha)list$, where
69 $list$ is a type constructor and $\alpha$ is a type variable; for example,
70 $(bool)list$ is the type of lists of booleans. Function types have the
71 form $(\sigma,\tau)fun$ or $\sigma\To\tau$, where $\sigma$ and $\tau$ are
72 types. Curried function types may be abbreviated:
73 \[ \sigma@1\To (\cdots \sigma@n\To \tau\cdots) \quad \hbox{as} \quad
74 [\sigma@1, \ldots, \sigma@n] \To \tau \]
76 \index{terms!syntax of} The syntax for terms is summarised below.
77 Note that there are two versions of function application syntax
78 available in Isabelle: either $t\,u$, which is the usual form for
79 higher-order languages, or $t(u)$, trying to look more like
80 first-order. The latter syntax is used throughout the manual.
82 \index{lambda abs@$\lambda$-abstractions}\index{function applications}
84 t :: \tau & \hbox{type constraint, on a term or bound variable} \\
85 \lambda x.t & \hbox{abstraction} \\
86 \lambda x@1\ldots x@n.t
87 & \hbox{curried abstraction, $\lambda x@1. \ldots \lambda x@n.t$} \\
88 t(u) & \hbox{application} \\
89 t (u@1, \ldots, u@n) & \hbox{curried application, $t(u@1)\ldots(u@n)$}
94 \subsection{Simple types and constants}\index{types!simple|bold}
96 The syntactic categories of our logic (Fig.\ts\ref{fol-fig}) are {\bf
97 formulae} and {\bf terms}. Formulae denote truth values, so (following
98 tradition) let us call their type~$o$. To allow~0 and~$Suc(t)$ as terms,
99 let us declare a type~$nat$ of natural numbers. Later, we shall see
100 how to admit terms of other types.
102 \index{constants}\index{*nat type}\index{*o type}
103 After declaring the types~$o$ and~$nat$, we may declare constants for the
104 symbols of our logic. Since $\bot$ denotes a truth value (falsity) and 0
105 denotes a number, we put \begin{eqnarray*}
109 If a symbol requires operands, the corresponding constant must have a
110 function type. In our logic, the successor function
111 ($Suc$) is from natural numbers to natural numbers, negation ($\neg$) is a
112 function from truth values to truth values, and the binary connectives are
113 curried functions taking two truth values as arguments:
115 Suc & :: & nat\To nat \\
116 {\neg} & :: & o\To o \\
117 \conj,\disj,\imp,\bimp & :: & [o,o]\To o
119 The binary connectives can be declared as infixes, with appropriate
120 precedences, so that we write $P\conj Q\disj R$ instead of
121 $\disj(\conj(P,Q), R)$.
123 Section~\ref{sec:defining-theories} below describes the syntax of Isabelle
124 theory files and illustrates it by extending our logic with mathematical
128 \subsection{Polymorphic types and constants} \label{polymorphic}
129 \index{types!polymorphic|bold}
130 \index{equality!polymorphic}
131 \index{constants!polymorphic}
133 Which type should we assign to the equality symbol? If we tried
134 $[nat,nat]\To o$, then equality would be restricted to the natural
135 numbers; we should have to declare different equality symbols for each
136 type. Isabelle's type system is polymorphic, so we could declare
138 {=} & :: & [\alpha,\alpha]\To o,
140 where the type variable~$\alpha$ ranges over all types.
141 But this is also wrong. The declaration is too polymorphic; $\alpha$
142 includes types like~$o$ and $nat\To nat$. Thus, it admits
143 $\bot=\neg(\bot)$ and $Suc=Suc$ as formulae, which is acceptable in
144 higher-order logic but not in first-order logic.
146 Isabelle's {\bf type classes}\index{classes} control
147 polymorphism~\cite{nipkow-prehofer}. Each type variable belongs to a
148 class, which denotes a set of types. Classes are partially ordered by the
149 subclass relation, which is essentially the subset relation on the sets of
150 types. They closely resemble the classes of the functional language
151 Haskell~\cite{haskell-tutorial,haskell-report}.
153 \index{*logic class}\index{*term class}
154 Isabelle provides the built-in class $logic$, which consists of the logical
155 types: the ones we want to reason about. Let us declare a class $term$, to
156 consist of all legal types of terms in our logic. The subclass structure
157 is now $term\le logic$.
160 We put $nat$ in class $term$ by declaring $nat{::}term$. We declare the
163 {=} & :: & [\alpha{::}term,\alpha]\To o
165 where $\alpha{::}term$ constrains the type variable~$\alpha$ to class
166 $term$. Such type variables resemble Standard~\ML's equality type
169 We give~$o$ and function types the class $logic$ rather than~$term$, since
170 they are not legal types for terms. We may introduce new types of class
171 $term$ --- for instance, type $string$ or $real$ --- at any time. We can
172 even declare type constructors such as~$list$, and state that type
173 $(\tau)list$ belongs to class~$term$ provided $\tau$ does; equality
174 applies to lists of natural numbers but not to lists of formulae. We may
175 summarize this paragraph by a set of {\bf arity declarations} for type
176 constructors:\index{arities!declaring}
177 \begin{eqnarray*}\index{*o type}\index{*fun type}
179 fun & :: & (logic,logic)logic \\
180 nat, string, real & :: & term \\
181 list & :: & (term)term
183 (Recall that $fun$ is the type constructor for function types.)
184 In \rmindex{higher-order logic}, equality does apply to truth values and
185 functions; this requires the arity declarations ${o::term}$
186 and ${fun::(term,term)term}$. The class system can also handle
187 overloading.\index{overloading|bold} We could declare $arith$ to be the
188 subclass of $term$ consisting of the `arithmetic' types, such as~$nat$.
189 Then we could declare the operators
191 {+},{-},{\times},{/} & :: & [\alpha{::}arith,\alpha]\To \alpha
193 If we declare new types $real$ and $complex$ of class $arith$, then we
194 in effect have three sets of operators:
196 {+},{-},{\times},{/} & :: & [nat,nat]\To nat \\
197 {+},{-},{\times},{/} & :: & [real,real]\To real \\
198 {+},{-},{\times},{/} & :: & [complex,complex]\To complex
200 Isabelle will regard these as distinct constants, each of which can be defined
201 separately. We could even introduce the type $(\alpha)vector$ and declare
202 its arity as $(arith)arith$. Then we could declare the constant
204 {+} & :: & [(\alpha)vector,(\alpha)vector]\To (\alpha)vector
206 and specify it in terms of ${+} :: [\alpha,\alpha]\To \alpha$.
208 A type variable may belong to any finite number of classes. Suppose that
209 we had declared yet another class $ord \le term$, the class of all
210 `ordered' types, and a constant
212 {\le} & :: & [\alpha{::}ord,\alpha]\To o.
214 In this context the variable $x$ in $x \le (x+x)$ will be assigned type
215 $\alpha{::}\{arith,ord\}$, which means $\alpha$ belongs to both $arith$ and
216 $ord$. Semantically the set $\{arith,ord\}$ should be understood as the
217 intersection of the sets of types represented by $arith$ and $ord$. Such
218 intersections of classes are called \bfindex{sorts}. The empty
219 intersection of classes, $\{\}$, contains all types and is thus the {\bf
222 Even with overloading, each term has a unique, most general type. For this
223 to be possible, the class and type declarations must satisfy certain
224 technical constraints; see
225 \iflabelundefined{sec:ref-defining-theories}%
226 {Sect.\ Defining Theories in the {\em Reference Manual}}%
227 {\S\ref{sec:ref-defining-theories}}.
230 \subsection{Higher types and quantifiers}
231 \index{types!higher|bold}\index{quantifiers}
232 Quantifiers are regarded as operations upon functions. Ignoring polymorphism
233 for the moment, consider the formula $\forall x. P(x)$, where $x$ ranges
234 over type~$nat$. This is true if $P(x)$ is true for all~$x$. Abstracting
235 $P(x)$ into a function, this is the same as saying that $\lambda x.P(x)$
236 returns true for all arguments. Thus, the universal quantifier can be
237 represented by a constant
239 \forall & :: & (nat\To o) \To o,
241 which is essentially an infinitary truth table. The representation of $\forall
242 x. P(x)$ is $\forall(\lambda x. P(x))$.
244 The existential quantifier is treated
245 in the same way. Other binding operators are also easily handled; for
246 instance, the summation operator $\Sigma@{k=i}^j f(k)$ can be represented as
247 $\Sigma(i,j,\lambda k.f(k))$, where
249 \Sigma & :: & [nat,nat, nat\To nat] \To nat.
251 Quantifiers may be polymorphic. We may define $\forall$ and~$\exists$ over
252 all legal types of terms, not just the natural numbers, and
253 allow summations over all arithmetic types:
255 \forall,\exists & :: & (\alpha{::}term\To o) \To o \\
256 \Sigma & :: & [nat,nat, nat\To \alpha{::}arith] \To \alpha
258 Observe that the index variables still have type $nat$, while the values
259 being summed may belong to any arithmetic type.
262 \section{Formalizing logical rules in Isabelle}
263 \index{meta-implication|bold}
264 \index{meta-quantifiers|bold}
265 \index{meta-equality|bold}
267 Object-logics are formalized by extending Isabelle's
268 meta-logic~\cite{paulson-found}, which is intuitionistic higher-order logic.
269 The meta-level connectives are {\bf implication}, the {\bf universal
270 quantifier}, and {\bf equality}.
272 \item The implication \(\phi\Imp \psi\) means `\(\phi\) implies
273 \(\psi\)', and expresses logical {\bf entailment}.
275 \item The quantification \(\Forall x.\phi\) means `\(\phi\) is true for
276 all $x$', and expresses {\bf generality} in rules and axiom schemes.
278 \item The equality \(a\equiv b\) means `$a$ equals $b$', for expressing
279 {\bf definitions} (see~\S\ref{definitions}).\index{definitions}
280 Equalities left over from the unification process, so called {\bf
281 flex-flex constraints},\index{flex-flex constraints} are written $a\qeq
282 b$. The two equality symbols have the same logical meaning.
285 The syntax of the meta-logic is formalized in the same manner
286 as object-logics, using the simply typed $\lambda$-calculus. Analogous to
287 type~$o$ above, there is a built-in type $prop$ of meta-level truth values.
288 Meta-level formulae will have this type. Type $prop$ belongs to
289 class~$logic$; also, $\sigma\To\tau$ belongs to $logic$ provided $\sigma$
290 and $\tau$ do. Here are the types of the built-in connectives:
291 \begin{eqnarray*}\index{*prop type}\index{*logic class}
292 \Imp & :: & [prop,prop]\To prop \\
293 \Forall & :: & (\alpha{::}logic\To prop) \To prop \\
294 {\equiv} & :: & [\alpha{::}\{\},\alpha]\To prop \\
295 \qeq & :: & [\alpha{::}\{\},\alpha]\To prop
297 The polymorphism in $\Forall$ is restricted to class~$logic$ to exclude
298 certain types, those used just for parsing. The type variable
299 $\alpha{::}\{\}$ ranges over the universal sort.
301 In our formalization of first-order logic, we declared a type~$o$ of
302 object-level truth values, rather than using~$prop$ for this purpose. If
303 we declared the object-level connectives to have types such as
304 ${\neg}::prop\To prop$, then these connectives would be applicable to
305 meta-level formulae. Keeping $prop$ and $o$ as separate types maintains
306 the distinction between the meta-level and the object-level. To formalize
307 the inference rules, we shall need to relate the two levels; accordingly,
308 we declare the constant
309 \index{*Trueprop constant}
311 Trueprop & :: & o\To prop.
313 We may regard $Trueprop$ as a meta-level predicate, reading $Trueprop(P)$ as
314 `$P$ is true at the object-level.' Put another way, $Trueprop$ is a coercion
318 \subsection{Expressing propositional rules}
319 \index{rules!propositional}
320 We shall illustrate the use of the meta-logic by formalizing the rules of
321 Fig.\ts\ref{fol-fig}. Each object-level rule is expressed as a meta-level
324 One of the simplest rules is $(\conj E1)$. Making
325 everything explicit, its formalization in the meta-logic is
327 \Forall P\;Q. Trueprop(P\conj Q) \Imp Trueprop(P). \eqno(\conj E1)
329 This may look formidable, but it has an obvious reading: for all object-level
330 truth values $P$ and~$Q$, if $P\conj Q$ is true then so is~$P$. The
331 reading is correct because the meta-logic has simple models, where
332 types denote sets and $\Forall$ really means `for all.'
334 \index{*Trueprop constant}
335 Isabelle adopts notational conventions to ease the writing of rules. We may
336 hide the occurrences of $Trueprop$ by making it an implicit coercion.
337 Outer universal quantifiers may be dropped. Finally, the nested implication
338 \index{meta-implication}
339 \[ \phi@1\Imp(\cdots \phi@n\Imp\psi\cdots) \]
340 may be abbreviated as $\List{\phi@1; \ldots; \phi@n} \Imp \psi$, which
341 formalizes a rule of $n$~premises.
343 Using these conventions, the conjunction rules become the following axioms.
344 These fully specify the properties of~$\conj$:
345 $$ \List{P; Q} \Imp P\conj Q \eqno(\conj I) $$
346 $$ P\conj Q \Imp P \qquad P\conj Q \Imp Q \eqno(\conj E1,2) $$
349 Next, consider the disjunction rules. The discharge of assumption in
350 $(\disj E)$ is expressed using $\Imp$:
351 \index{assumptions!discharge of}%
352 $$ P \Imp P\disj Q \qquad Q \Imp P\disj Q \eqno(\disj I1,2) $$
353 $$ \List{P\disj Q; P\Imp R; Q\Imp R} \Imp R \eqno(\disj E) $$
355 To understand this treatment of assumptions in natural
356 deduction, look at implication. The rule $({\imp}I)$ is the classic
357 example of natural deduction: to prove that $P\imp Q$ is true, assume $P$
358 is true and show that $Q$ must then be true. More concisely, if $P$
359 implies $Q$ (at the meta-level), then $P\imp Q$ is true (at the
360 object-level). Showing the coercion explicitly, this is formalized as
361 \[ (Trueprop(P)\Imp Trueprop(Q)) \Imp Trueprop(P\imp Q). \]
362 The rule $({\imp}E)$ is straightforward; hiding $Trueprop$, the axioms to
364 $$ (P \Imp Q) \Imp P\imp Q \eqno({\imp}I) $$
365 $$ \List{P\imp Q; P} \Imp Q. \eqno({\imp}E) $$
368 Finally, the intuitionistic contradiction rule is formalized as the axiom
369 $$ \bot \Imp P. \eqno(\bot E) $$
372 Earlier versions of Isabelle, and certain
373 papers~\cite{paulson-found,paulson700}, use $\List{P}$ to mean $Trueprop(P)$.
376 \subsection{Quantifier rules and substitution}
377 \index{quantifiers}\index{rules!quantifier}\index{substitution|bold}
378 \index{variables!bound}\index{lambda abs@$\lambda$-abstractions}
379 \index{function applications}
381 Isabelle expresses variable binding using $\lambda$-abstraction; for instance,
382 $\forall x.P$ is formalized as $\forall(\lambda x.P)$. Recall that $F(t)$
383 is Isabelle's syntax for application of the function~$F$ to the argument~$t$;
384 it is not a meta-notation for substitution. On the other hand, a substitution
385 will take place if $F$ has the form $\lambda x.P$; Isabelle transforms
386 $(\lambda x.P)(t)$ to~$P[t/x]$ by $\beta$-conversion. Thus, we can express
387 inference rules that involve substitution for bound variables.
389 \index{parameters|bold}\index{eigenvariables|see{parameters}}
390 A logic may attach provisos to certain of its rules, especially quantifier
391 rules. We cannot hope to formalize arbitrary provisos. Fortunately, those
392 typical of quantifier rules always have the same form, namely `$x$ not free in
393 \ldots {\it (some set of formulae)},' where $x$ is a variable (called a {\bf
394 parameter} or {\bf eigenvariable}) in some premise. Isabelle treats
395 provisos using~$\Forall$, its inbuilt notion of `for all'.
396 \index{meta-quantifiers}
398 The purpose of the proviso `$x$ not free in \ldots' is
399 to ensure that the premise may not make assumptions about the value of~$x$,
400 and therefore holds for all~$x$. We formalize $(\forall I)$ by
401 \[ \left(\Forall x. Trueprop(P(x))\right) \Imp Trueprop(\forall x.P(x)). \]
402 This means, `if $P(x)$ is true for all~$x$, then $\forall x.P(x)$ is true.'
403 The $\forall E$ rule exploits $\beta$-conversion. Hiding $Trueprop$, the
405 $$ \left(\Forall x. P(x)\right) \Imp \forall x.P(x) \eqno(\forall I) $$
406 $$ (\forall x.P(x)) \Imp P(t). \eqno(\forall E) $$
409 We have defined the object-level universal quantifier~($\forall$)
410 using~$\Forall$. But we do not require meta-level counterparts of all the
411 connectives of the object-logic! Consider the existential quantifier:
412 $$ P(t) \Imp \exists x.P(x) \eqno(\exists I) $$
413 $$ \List{\exists x.P(x);\; \Forall x. P(x)\Imp Q} \Imp Q \eqno(\exists E) $$
414 Let us verify $(\exists E)$ semantically. Suppose that the premises
415 hold; since $\exists x.P(x)$ is true, we may choose an~$a$ such that $P(a)$ is
416 true. Instantiating $\Forall x. P(x)\Imp Q$ with $a$ yields $P(a)\Imp Q$, and
417 we obtain the desired conclusion, $Q$.
419 The treatment of substitution deserves mention. The rule
420 \[ \infer{P[u/t]}{t=u & P} \]
421 would be hard to formalize in Isabelle. It calls for replacing~$t$ by $u$
422 throughout~$P$, which cannot be expressed using $\beta$-conversion. Our
423 rule~$(subst)$ uses~$P$ as a template for substitution, inferring $P[u/x]$
424 from~$P[t/x]$. When we formalize this as an axiom, the template becomes a
426 $$ \List{t=u; P(t)} \Imp P(u). \eqno(subst) $$
429 \subsection{Signatures and theories}
430 \index{signatures|bold}
432 A {\bf signature} contains the information necessary for type-checking,
433 parsing and pretty printing a term. It specifies type classes and their
434 relationships, types and their arities, constants and their types, etc. It
435 also contains grammar rules, specified using mixfix declarations.
437 Two signatures can be merged provided their specifications are compatible ---
438 they must not, for example, assign different types to the same constant.
439 Under similar conditions, a signature can be extended. Signatures are
440 managed internally by Isabelle; users seldom encounter them.
442 \index{theories|bold} A {\bf theory} consists of a signature plus a collection
443 of axioms. The Pure theory contains only the meta-logic. Theories can be
444 combined provided their signatures are compatible. A theory definition
445 extends an existing theory with further signature specifications --- classes,
446 types, constants and mixfix declarations --- plus lists of axioms and
447 definitions etc., expressed as strings to be parsed. A theory can formalize a
448 small piece of mathematics, such as lists and their operations, or an entire
449 logic. A mathematical development typically involves many theories in a
450 hierarchy. For example, the Pure theory could be extended to form a theory
451 for Fig.\ts\ref{fol-fig}; this could be extended in two separate ways to form
452 a theory for natural numbers and a theory for lists; the union of these two
453 could be extended into a theory defining the length of a list:
456 \begin{array}{c@{}c@{}c@{}c@{}c}
457 {} & {} &\hbox{Pure}& {} & {} \\
458 {} & {} & \downarrow & {} & {} \\
459 {} & {} &\hbox{FOL} & {} & {} \\
460 {} & \swarrow & {} & \searrow & {} \\
461 \hbox{Nat} & {} & {} & {} & \hbox{List} \\
462 {} & \searrow & {} & \swarrow & {} \\
463 {} & {} &\hbox{Nat}+\hbox{List}& {} & {} \\
464 {} & {} & \downarrow & {} & {} \\
465 {} & {} & \hbox{Length} & {} & {}
469 Each Isabelle proof typically works within a single theory, which is
470 associated with the proof state. However, many different theories may
471 coexist at the same time, and you may work in each of these during a single
474 \begin{warn}\index{constants!clashes with variables}%
475 Confusing problems arise if you work in the wrong theory. Each theory
476 defines its own syntax. An identifier may be regarded in one theory as a
477 constant and in another as a variable, for example.
480 \section{Proof construction in Isabelle}
481 I have elsewhere described the meta-logic and demonstrated it by
482 formalizing first-order logic~\cite{paulson-found}. There is a one-to-one
483 correspondence between meta-level proofs and object-level proofs. To each
484 use of a meta-level axiom, such as $(\forall I)$, there is a use of the
485 corresponding object-level rule. Object-level assumptions and parameters
486 have meta-level counterparts. The meta-level formalization is {\bf
487 faithful}, admitting no incorrect object-level inferences, and {\bf
488 adequate}, admitting all correct object-level inferences. These
489 properties must be demonstrated separately for each object-logic.
491 The meta-logic is defined by a collection of inference rules, including
492 equational rules for the $\lambda$-calculus and logical rules. The rules
493 for~$\Imp$ and~$\Forall$ resemble those for~$\imp$ and~$\forall$ in
494 Fig.\ts\ref{fol-fig}. Proofs performed using the primitive meta-rules
495 would be lengthy; Isabelle proofs normally use certain derived rules.
496 {\bf Resolution}, in particular, is convenient for backward proof.
498 Unification is central to theorem proving. It supports quantifier
499 reasoning by allowing certain `unknown' terms to be instantiated later,
500 possibly in stages. When proving that the time required to sort $n$
501 integers is proportional to~$n^2$, we need not state the constant of
502 proportionality; when proving that a hardware adder will deliver the sum of
503 its inputs, we need not state how many clock ticks will be required. Such
504 quantities often emerge from the proof.
506 Isabelle provides {\bf schematic variables}, or {\bf
507 unknowns},\index{unknowns} for unification. Logically, unknowns are free
508 variables. But while ordinary variables remain fixed, unification may
509 instantiate unknowns. Unknowns are written with a ?\ prefix and are
510 frequently subscripted: $\Var{a}$, $\Var{a@1}$, $\Var{a@2}$, \ldots,
511 $\Var{P}$, $\Var{P@1}$, \ldots.
513 Recall that an inference rule of the form
514 \[ \infer{\phi}{\phi@1 & \ldots & \phi@n} \]
515 is formalized in Isabelle's meta-logic as the axiom
516 $\List{\phi@1; \ldots; \phi@n} \Imp \phi$.\index{resolution}
517 Such axioms resemble Prolog's Horn clauses, and can be combined by
518 resolution --- Isabelle's principal proof method. Resolution yields both
519 forward and backward proof. Backward proof works by unifying a goal with
520 the conclusion of a rule, whose premises become new subgoals. Forward proof
521 works by unifying theorems with the premises of a rule, deriving a new theorem.
523 Isabelle formulae require an extended notion of resolution.
524 They differ from Horn clauses in two major respects:
526 \item They are written in the typed $\lambda$-calculus, and therefore must be
527 resolved using higher-order unification.
529 \item The constituents of a clause need not be atomic formulae. Any
530 formula of the form $Trueprop(\cdots)$ is atomic, but axioms such as
531 ${\imp}I$ and $\forall I$ contain non-atomic formulae.
533 Isabelle has little in common with classical resolution theorem provers
534 such as Otter~\cite{wos-bledsoe}. At the meta-level, Isabelle proves
535 theorems in their positive form, not by refutation. However, an
536 object-logic that includes a contradiction rule may employ a refutation
540 \subsection{Higher-order unification}
541 \index{unification!higher-order|bold}
542 Unification is equation solving. The solution of $f(\Var{x},c) \qeq
543 f(d,\Var{y})$ is $\Var{x}\equiv d$ and $\Var{y}\equiv c$. {\bf
544 Higher-order unification} is equation solving for typed $\lambda$-terms.
545 To handle $\beta$-conversion, it must reduce $(\lambda x.t)u$ to $t[u/x]$.
546 That is easy --- in the typed $\lambda$-calculus, all reduction sequences
547 terminate at a normal form. But it must guess the unknown
548 function~$\Var{f}$ in order to solve the equation
549 \begin{equation} \label{hou-eqn}
550 \Var{f}(t) \qeq g(u@1,\ldots,u@k).
552 Huet's~\cite{huet75} search procedure solves equations by imitation and
553 projection. {\bf Imitation} makes~$\Var{f}$ apply the leading symbol (if a
554 constant) of the right-hand side. To solve equation~(\ref{hou-eqn}), it
556 \[ \Var{f} \equiv \lambda x. g(\Var{h@1}(x),\ldots,\Var{h@k}(x)), \]
557 where $\Var{h@1}$, \ldots, $\Var{h@k}$ are new unknowns. Assuming there are no
558 other occurrences of~$\Var{f}$, equation~(\ref{hou-eqn}) simplifies to the
560 \[ \Var{h@1}(t)\qeq u@1 \quad\ldots\quad \Var{h@k}(t)\qeq u@k. \]
561 If the procedure solves these equations, instantiating $\Var{h@1}$, \ldots,
562 $\Var{h@k}$, then it yields an instantiation for~$\Var{f}$.
564 {\bf Projection} makes $\Var{f}$ apply one of its arguments. To solve
565 equation~(\ref{hou-eqn}), if $t$ expects~$m$ arguments and delivers a
566 result of suitable type, it guesses
567 \[ \Var{f} \equiv \lambda x. x(\Var{h@1}(x),\ldots,\Var{h@m}(x)), \]
568 where $\Var{h@1}$, \ldots, $\Var{h@m}$ are new unknowns. Assuming there are no
569 other occurrences of~$\Var{f}$, equation~(\ref{hou-eqn}) simplifies to the
571 \[ t(\Var{h@1}(t),\ldots,\Var{h@m}(t)) \qeq g(u@1,\ldots,u@k). \]
573 \begin{warn}\index{unification!incompleteness of}%
574 Huet's unification procedure is complete. Isabelle's polymorphic version,
575 which solves for type unknowns as well as for term unknowns, is incomplete.
576 The problem is that projection requires type information. In
577 equation~(\ref{hou-eqn}), if the type of~$t$ is unknown, then projections
578 are possible for all~$m\geq0$, and the types of the $\Var{h@i}$ will be
579 similarly unconstrained. Therefore, Isabelle never attempts such
580 projections, and may fail to find unifiers where a type unknown turns out
581 to be a function type.
584 \index{unknowns!function|bold}
585 Given $\Var{f}(t@1,\ldots,t@n)\qeq u$, Huet's procedure could make up to
586 $n+1$ guesses. The search tree and set of unifiers may be infinite. But
587 higher-order unification can work effectively, provided you are careful
588 with {\bf function unknowns}:
590 \item Equations with no function unknowns are solved using first-order
591 unification, extended to treat bound variables. For example, $\lambda x.x
592 \qeq \lambda x.\Var{y}$ has no solution because $\Var{y}\equiv x$ would
593 capture the free variable~$x$.
595 \item An occurrence of the term $\Var{f}(x,y,z)$, where the arguments are
596 distinct bound variables, causes no difficulties. Its projections can only
597 match the corresponding variables.
599 \item Even an equation such as $\Var{f}(a)\qeq a+a$ is all right. It has
600 four solutions, but Isabelle evaluates them lazily, trying projection before
601 imitation. The first solution is usually the one desired:
602 \[ \Var{f}\equiv \lambda x. x+x \quad
603 \Var{f}\equiv \lambda x. a+x \quad
604 \Var{f}\equiv \lambda x. x+a \quad
605 \Var{f}\equiv \lambda x. a+a \]
606 \item Equations such as $\Var{f}(\Var{x},\Var{y})\qeq t$ and
607 $\Var{f}(\Var{g}(x))\qeq t$ admit vast numbers of unifiers, and must be
610 In problematic cases, you may have to instantiate some unknowns before
611 invoking unification.
614 \subsection{Joining rules by resolution} \label{joining}
615 \index{resolution|bold}
616 Let $\List{\psi@1; \ldots; \psi@m} \Imp \psi$ and $\List{\phi@1; \ldots;
617 \phi@n} \Imp \phi$ be two Isabelle theorems, representing object-level rules.
618 Choosing some~$i$ from~1 to~$n$, suppose that $\psi$ and $\phi@i$ have a
619 higher-order unifier. Writing $Xs$ for the application of substitution~$s$ to
620 expression~$X$, this means there is some~$s$ such that $\psi s\equiv \phi@i s$.
621 By resolution, we may conclude
622 \[ (\List{\phi@1; \ldots; \phi@{i-1}; \psi@1; \ldots; \psi@m;
623 \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s.
625 The substitution~$s$ may instantiate unknowns in both rules. In short,
626 resolution is the following rule:
627 \[ \infer[(\psi s\equiv \phi@i s)]
628 {(\List{\phi@1; \ldots; \phi@{i-1}; \psi@1; \ldots; \psi@m;
629 \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s}
630 {\List{\psi@1; \ldots; \psi@m} \Imp \psi & &
631 \List{\phi@1; \ldots; \phi@n} \Imp \phi}
633 It operates at the meta-level, on Isabelle theorems, and is justified by
634 the properties of $\Imp$ and~$\Forall$. It takes the number~$i$ (for
635 $1\leq i\leq n$) as a parameter and may yield infinitely many conclusions,
636 one for each unifier of $\psi$ with $\phi@i$. Isabelle returns these
637 conclusions as a sequence (lazy list).
639 Resolution expects the rules to have no outer quantifiers~($\Forall$).
640 It may rename or instantiate any schematic variables, but leaves free
641 variables unchanged. When constructing a theory, Isabelle puts the
642 rules into a standard form with all free variables converted into
643 schematic ones; for instance, $({\imp}E)$ becomes
644 \[ \List{\Var{P}\imp \Var{Q}; \Var{P}} \Imp \Var{Q}.
646 When resolving two rules, the unknowns in the first rule are renamed, by
647 subscripting, to make them distinct from the unknowns in the second rule. To
648 resolve $({\imp}E)$ with itself, the first copy of the rule becomes
649 \[ \List{\Var{P@1}\imp \Var{Q@1}; \Var{P@1}} \Imp \Var{Q@1}. \]
650 Resolving this with $({\imp}E)$ in the first premise, unifying $\Var{Q@1}$ with
651 $\Var{P}\imp \Var{Q}$, is the meta-level inference
652 \[ \infer{\List{\Var{P@1}\imp (\Var{P}\imp \Var{Q}); \Var{P@1}; \Var{P}}
654 {\List{\Var{P@1}\imp \Var{Q@1}; \Var{P@1}} \Imp \Var{Q@1} & &
655 \List{\Var{P}\imp \Var{Q}; \Var{P}} \Imp \Var{Q}}
657 Renaming the unknowns in the resolvent, we have derived the
658 object-level rule\index{rules!derived}
659 \[ \infer{Q.}{R\imp (P\imp Q) & R & P} \]
660 Joining rules in this fashion is a simple way of proving theorems. The
661 derived rules are conservative extensions of the object-logic, and may permit
662 simpler proofs. Let us consider another example. Suppose we have the axiom
663 $$ \forall x\,y. Suc(x)=Suc(y)\imp x=y. \eqno (inject) $$
666 The standard form of $(\forall E)$ is
667 $\forall x.\Var{P}(x) \Imp \Var{P}(\Var{t})$.
668 Resolving $(inject)$ with $(\forall E)$ replaces $\Var{P}$ by
669 $\lambda x. \forall y. Suc(x)=Suc(y)\imp x=y$ and leaves $\Var{t}$
671 \[ \forall y. Suc(\Var{t})=Suc(y)\imp \Var{t}=y. \]
672 Resolving this with $(\forall E)$ puts a subscript on~$\Var{t}$
674 \[ Suc(\Var{t@1})=Suc(\Var{t})\imp \Var{t@1}=\Var{t}. \]
675 Resolving this with $({\imp}E)$ increases the subscripts and yields
676 \[ Suc(\Var{t@2})=Suc(\Var{t@1})\Imp \Var{t@2}=\Var{t@1}.
678 We have derived the rule
679 \[ \infer{m=n,}{Suc(m)=Suc(n)} \]
680 which goes directly from $Suc(m)=Suc(n)$ to $m=n$. It is handy for simplifying
681 an equation like $Suc(Suc(Suc(m)))=Suc(Suc(Suc(0)))$.
684 \section{Lifting a rule into a context}
685 The rules $({\imp}I)$ and $(\forall I)$ may seem unsuitable for
686 resolution. They have non-atomic premises, namely $P\Imp Q$ and $\Forall
687 x.P(x)$, while the conclusions of all the rules are atomic (they have the form
688 $Trueprop(\cdots)$). Isabelle gets round the problem through a meta-inference
689 called \bfindex{lifting}. Let us consider how to construct proofs such as
690 \[ \infer[({\imp}I)]{P\imp(Q\imp R)}
691 {\infer[({\imp}I)]{Q\imp R}
694 \infer[(\forall I)]{\forall x\,y.P(x,y)}
695 {\infer[(\forall I)]{\forall y.P(x,y)}{P(x,y)}}
698 \subsection{Lifting over assumptions}
699 \index{assumptions!lifting over}
700 Lifting over $\theta\Imp{}$ is the following meta-inference rule:
701 \[ \infer{\List{\theta\Imp\phi@1; \ldots; \theta\Imp\phi@n} \Imp
703 {\List{\phi@1; \ldots; \phi@n} \Imp \phi} \]
704 This is clearly sound: if $\List{\phi@1; \ldots; \phi@n} \Imp \phi$ is true and
705 $\theta\Imp\phi@1$, \ldots, $\theta\Imp\phi@n$ and $\theta$ are all true then
706 $\phi$ must be true. Iterated lifting over a series of meta-formulae
707 $\theta@k$, \ldots, $\theta@1$ yields an object-rule whose conclusion is
708 $\List{\theta@1; \ldots; \theta@k} \Imp \phi$. Typically the $\theta@i$ are
709 the assumptions in a natural deduction proof; lifting copies them into a rule's
710 premises and conclusion.
712 When resolving two rules, Isabelle lifts the first one if necessary. The
713 standard form of $({\imp}I)$ is
714 \[ (\Var{P} \Imp \Var{Q}) \Imp \Var{P}\imp \Var{Q}. \]
715 To resolve this rule with itself, Isabelle modifies one copy as follows: it
716 renames the unknowns to $\Var{P@1}$ and $\Var{Q@1}$, then lifts the rule over
717 $\Var{P}\Imp{}$ to obtain
718 \[ (\Var{P}\Imp (\Var{P@1} \Imp \Var{Q@1})) \Imp (\Var{P} \Imp
719 (\Var{P@1}\imp \Var{Q@1})). \]
720 Using the $\List{\cdots}$ abbreviation, this can be written as
721 \[ \List{\List{\Var{P}; \Var{P@1}} \Imp \Var{Q@1}; \Var{P}}
722 \Imp \Var{P@1}\imp \Var{Q@1}. \]
723 Unifying $\Var{P}\Imp \Var{P@1}\imp\Var{Q@1}$ with $\Var{P} \Imp
724 \Var{Q}$ instantiates $\Var{Q}$ to ${\Var{P@1}\imp\Var{Q@1}}$.
726 \[ (\List{\Var{P}; \Var{P@1}} \Imp \Var{Q@1}) \Imp
727 \Var{P}\imp(\Var{P@1}\imp\Var{Q@1}). \]
728 This represents the derived rule
729 \[ \infer{P\imp(Q\imp R).}{\infer*{R}{[P,Q]}} \]
731 \subsection{Lifting over parameters}
732 \index{parameters!lifting over}
733 An analogous form of lifting handles premises of the form $\Forall x\ldots\,$.
734 Here, lifting prefixes an object-rule's premises and conclusion with $\Forall
735 x$. At the same time, lifting introduces a dependence upon~$x$. It replaces
736 each unknown $\Var{a}$ in the rule by $\Var{a'}(x)$, where $\Var{a'}$ is a new
737 unknown (by subscripting) of suitable type --- necessarily a function type. In
738 short, lifting is the meta-inference
739 \[ \infer{\List{\Forall x.\phi@1^x; \ldots; \Forall x.\phi@n^x}
740 \Imp \Forall x.\phi^x,}
741 {\List{\phi@1; \ldots; \phi@n} \Imp \phi} \]
743 where $\phi^x$ stands for the result of lifting unknowns over~$x$ in
744 $\phi$. It is not hard to verify that this meta-inference is sound. If
745 $\phi\Imp\psi$ then $\phi^x\Imp\psi^x$ for all~$x$; so if $\phi^x$ is true
746 for all~$x$ then so is $\psi^x$. Thus, from $\phi\Imp\psi$ we conclude
747 $(\Forall x.\phi^x) \Imp (\Forall x.\psi^x)$.
749 For example, $(\disj I)$ might be lifted to
750 \[ (\Forall x.\Var{P@1}(x)) \Imp (\Forall x. \Var{P@1}(x)\disj \Var{Q@1}(x))\]
752 \[ (\Forall x\,y.\Var{P@1}(x,y)) \Imp (\Forall x. \forall y.\Var{P@1}(x,y)). \]
753 Isabelle has renamed a bound variable in $(\forall I)$ from $x$ to~$y$,
754 avoiding a clash. Resolving the above with $(\forall I)$ is the meta-inference
755 \[ \infer{\Forall x\,y.\Var{P@1}(x,y)) \Imp \forall x\,y.\Var{P@1}(x,y)) }
756 {(\Forall x\,y.\Var{P@1}(x,y)) \Imp
757 (\Forall x. \forall y.\Var{P@1}(x,y)) &
758 (\Forall x.\Var{P}(x)) \Imp (\forall x.\Var{P}(x))} \]
759 Here, $\Var{P}$ is replaced by $\lambda x.\forall y.\Var{P@1}(x,y)$; the
760 resolvent expresses the derived rule
761 \[ \vcenter{ \infer{\forall x\,y.Q(x,y)}{Q(x,y)} }
762 \quad\hbox{provided $x$, $y$ not free in the assumptions}
764 I discuss lifting and parameters at length elsewhere~\cite{paulson-found}.
765 Miller goes into even greater detail~\cite{miller-mixed}.
768 \section{Backward proof by resolution}
769 \index{resolution!in backward proof}
771 Resolution is convenient for deriving simple rules and for reasoning
772 forward from facts. It can also support backward proof, where we start
773 with a goal and refine it to progressively simpler subgoals until all have
774 been solved. {\sc lcf} and its descendants {\sc hol} and Nuprl provide
775 tactics and tacticals, which constitute a sophisticated language for
776 expressing proof searches. {\bf Tactics} refine subgoals while {\bf
777 tacticals} combine tactics.
780 Isabelle's tactics and tacticals work differently from {\sc lcf}'s. An
781 Isabelle rule is bidirectional: there is no distinction between
782 inputs and outputs. {\sc lcf} has a separate tactic for each rule;
783 Isabelle performs refinement by any rule in a uniform fashion, using
786 Isabelle works with meta-level theorems of the form
787 \( \List{\phi@1; \ldots; \phi@n} \Imp \phi \).
788 We have viewed this as the {\bf rule} with premises
789 $\phi@1$,~\ldots,~$\phi@n$ and conclusion~$\phi$. It can also be viewed as
790 the {\bf proof state}\index{proof state}
791 with subgoals $\phi@1$,~\ldots,~$\phi@n$ and main
794 To prove the formula~$\phi$, take $\phi\Imp \phi$ as the initial proof
795 state. This assertion is, trivially, a theorem. At a later stage in the
796 backward proof, a typical proof state is $\List{\phi@1; \ldots; \phi@n}
797 \Imp \phi$. This proof state is a theorem, ensuring that the subgoals
798 $\phi@1$,~\ldots,~$\phi@n$ imply~$\phi$. If $n=0$ then we have
799 proved~$\phi$ outright. If $\phi$ contains unknowns, they may become
800 instantiated during the proof; a proof state may be $\List{\phi@1; \ldots;
801 \phi@n} \Imp \phi'$, where $\phi'$ is an instance of~$\phi$.
803 \subsection{Refinement by resolution}
804 To refine subgoal~$i$ of a proof state by a rule, perform the following
806 \[ \infer{\hbox{new proof state}}{\hbox{rule} & & \hbox{proof state}} \]
807 Suppose the rule is $\List{\psi'@1; \ldots; \psi'@m} \Imp \psi'$ after
808 lifting over subgoal~$i$'s assumptions and parameters. If the proof state
809 is $\List{\phi@1; \ldots; \phi@n} \Imp \phi$, then the new proof state is
810 (for~$1\leq i\leq n$)
811 \[ (\List{\phi@1; \ldots; \phi@{i-1}; \psi'@1;
812 \ldots; \psi'@m; \phi@{i+1}; \ldots; \phi@n} \Imp \phi)s. \]
813 Substitution~$s$ unifies $\psi'$ with~$\phi@i$. In the proof state,
814 subgoal~$i$ is replaced by $m$ new subgoals, the rule's instantiated premises.
815 If some of the rule's unknowns are left un-instantiated, they become new
816 unknowns in the proof state. Refinement by~$(\exists I)$, namely
817 \[ \Var{P}(\Var{t}) \Imp \exists x. \Var{P}(x), \]
818 inserts a new unknown derived from~$\Var{t}$ by subscripting and lifting.
819 We do not have to specify an `existential witness' when
820 applying~$(\exists I)$. Further resolutions may instantiate unknowns in
823 \subsection{Proof by assumption}
824 \index{assumptions!use of}
825 In the course of a natural deduction proof, parameters $x@1$, \ldots,~$x@l$ and
826 assumptions $\theta@1$, \ldots, $\theta@k$ accumulate, forming a context for
827 each subgoal. Repeated lifting steps can lift a rule into any context. To
828 aid readability, Isabelle puts contexts into a normal form, gathering the
829 parameters at the front:
830 \begin{equation} \label{context-eqn}
831 \Forall x@1 \ldots x@l. \List{\theta@1; \ldots; \theta@k}\Imp\theta.
833 Under the usual reading of the connectives, this expresses that $\theta$
834 follows from $\theta@1$,~\ldots~$\theta@k$ for arbitrary
835 $x@1$,~\ldots,~$x@l$. It is trivially true if $\theta$ equals any of
836 $\theta@1$,~\ldots~$\theta@k$, or is unifiable with any of them. This
837 models proof by assumption in natural deduction.
839 Isabelle automates the meta-inference for proof by assumption. Its arguments
840 are the meta-theorem $\List{\phi@1; \ldots; \phi@n} \Imp \phi$, and some~$i$
841 from~1 to~$n$, where $\phi@i$ has the form~(\ref{context-eqn}). Its results
842 are meta-theorems of the form
843 \[ (\List{\phi@1; \ldots; \phi@{i-1}; \phi@{i+1}; \phi@n} \Imp \phi)s \]
844 for each $s$ and~$j$ such that $s$ unifies $\lambda x@1 \ldots x@l. \theta@j$
845 with $\lambda x@1 \ldots x@l. \theta$. Isabelle supplies the parameters
846 $x@1$,~\ldots,~$x@l$ to higher-order unification as bound variables, which
847 regards them as unique constants with a limited scope --- this enforces
848 parameter provisos~\cite{paulson-found}.
850 The premise represents a proof state with~$n$ subgoals, of which the~$i$th
851 is to be solved by assumption. Isabelle searches the subgoal's context for
852 an assumption~$\theta@j$ that can solve it. For each unifier, the
853 meta-inference returns an instantiated proof state from which the $i$th
854 subgoal has been removed. Isabelle searches for a unifying assumption; for
855 readability and robustness, proofs do not refer to assumptions by number.
857 Consider the proof state
858 \[ (\List{P(a); P(b)} \Imp P(\Var{x})) \Imp Q(\Var{x}). \]
859 Proof by assumption (with $i=1$, the only possibility) yields two results:
861 \item $Q(a)$, instantiating $\Var{x}\equiv a$
862 \item $Q(b)$, instantiating $\Var{x}\equiv b$
864 Here, proof by assumption affects the main goal. It could also affect
865 other subgoals; if we also had the subgoal ${\List{P(b); P(c)} \Imp
866 P(\Var{x})}$, then $\Var{x}\equiv a$ would transform it to ${\List{P(b);
867 P(c)} \Imp P(a)}$, which might be unprovable.
870 \subsection{A propositional proof} \label{prop-proof}
871 \index{examples!propositional}
872 Our first example avoids quantifiers. Given the main goal $P\disj P\imp
873 P$, Isabelle creates the initial state
874 \[ (P\disj P\imp P)\Imp (P\disj P\imp P). \]
876 Bear in mind that every proof state we derive will be a meta-theorem,
877 expressing that the subgoals imply the main goal. Our aim is to reach the
878 state $P\disj P\imp P$; this meta-theorem is the desired result.
880 The first step is to refine subgoal~1 by (${\imp}I)$, creating a new state
881 where $P\disj P$ is an assumption:
882 \[ (P\disj P\Imp P)\Imp (P\disj P\imp P) \]
883 The next step is $(\disj E)$, which replaces subgoal~1 by three new subgoals.
884 Because of lifting, each subgoal contains a copy of the context --- the
885 assumption $P\disj P$. (In fact, this assumption is now redundant; we shall
886 shortly see how to get rid of it!) The new proof state is the following
887 meta-theorem, laid out for clarity:
888 \[ \begin{array}{l@{}l@{\qquad\qquad}l}
889 \lbrakk\;& P\disj P\Imp \Var{P@1}\disj\Var{Q@1}; & \hbox{(subgoal 1)} \\
890 & \List{P\disj P; \Var{P@1}} \Imp P; & \hbox{(subgoal 2)} \\
891 & \List{P\disj P; \Var{Q@1}} \Imp P & \hbox{(subgoal 3)} \\
892 \rbrakk\;& \Imp (P\disj P\imp P) & \hbox{(main goal)}
895 Notice the unknowns in the proof state. Because we have applied $(\disj E)$,
896 we must prove some disjunction, $\Var{P@1}\disj\Var{Q@1}$. Of course,
897 subgoal~1 is provable by assumption. This instantiates both $\Var{P@1}$ and
898 $\Var{Q@1}$ to~$P$ throughout the proof state:
899 \[ \begin{array}{l@{}l@{\qquad\qquad}l}
900 \lbrakk\;& \List{P\disj P; P} \Imp P; & \hbox{(subgoal 1)} \\
901 & \List{P\disj P; P} \Imp P & \hbox{(subgoal 2)} \\
902 \rbrakk\;& \Imp (P\disj P\imp P) & \hbox{(main goal)}
904 Both of the remaining subgoals can be proved by assumption. After two such
905 steps, the proof state is $P\disj P\imp P$.
908 \subsection{A quantifier proof}
909 \index{examples!with quantifiers}
910 To illustrate quantifiers and $\Forall$-lifting, let us prove
911 $(\exists x.P(f(x)))\imp(\exists x.P(x))$. The initial proof
912 state is the trivial meta-theorem
913 \[ (\exists x.P(f(x)))\imp(\exists x.P(x)) \Imp
914 (\exists x.P(f(x)))\imp(\exists x.P(x)). \]
915 As above, the first step is refinement by (${\imp}I)$:
916 \[ (\exists x.P(f(x))\Imp \exists x.P(x)) \Imp
917 (\exists x.P(f(x)))\imp(\exists x.P(x))
919 The next step is $(\exists E)$, which replaces subgoal~1 by two new subgoals.
920 Both have the assumption $\exists x.P(f(x))$. The new proof
921 state is the meta-theorem
922 \[ \begin{array}{l@{}l@{\qquad\qquad}l}
923 \lbrakk\;& \exists x.P(f(x)) \Imp \exists x.\Var{P@1}(x); & \hbox{(subgoal 1)} \\
924 & \Forall x.\List{\exists x.P(f(x)); \Var{P@1}(x)} \Imp
925 \exists x.P(x) & \hbox{(subgoal 2)} \\
926 \rbrakk\;& \Imp (\exists x.P(f(x)))\imp(\exists x.P(x)) & \hbox{(main goal)}
929 The unknown $\Var{P@1}$ appears in both subgoals. Because we have applied
930 $(\exists E)$, we must prove $\exists x.\Var{P@1}(x)$, where $\Var{P@1}(x)$ may
931 become any formula possibly containing~$x$. Proving subgoal~1 by assumption
932 instantiates $\Var{P@1}$ to~$\lambda x.P(f(x))$:
933 \[ \left(\Forall x.\List{\exists x.P(f(x)); P(f(x))} \Imp
934 \exists x.P(x)\right)
935 \Imp (\exists x.P(f(x)))\imp(\exists x.P(x))
937 The next step is refinement by $(\exists I)$. The rule is lifted into the
938 context of the parameter~$x$ and the assumption $P(f(x))$. This copies
939 the context to the subgoal and allows the existential witness to
941 \[ \left(\Forall x.\List{\exists x.P(f(x)); P(f(x))} \Imp
942 P(\Var{x@2}(x))\right)
943 \Imp (\exists x.P(f(x)))\imp(\exists x.P(x))
945 The existential witness, $\Var{x@2}(x)$, consists of an unknown
946 applied to a parameter. Proof by assumption unifies $\lambda x.P(f(x))$
947 with $\lambda x.P(\Var{x@2}(x))$, instantiating $\Var{x@2}$ to $f$. The final
948 proof state contains no subgoals: $(\exists x.P(f(x)))\imp(\exists x.P(x))$.
951 \subsection{Tactics and tacticals}
952 \index{tactics|bold}\index{tacticals|bold}
953 {\bf Tactics} perform backward proof. Isabelle tactics differ from those
954 of {\sc lcf}, {\sc hol} and Nuprl by operating on entire proof states,
955 rather than on individual subgoals. An Isabelle tactic is a function that
956 takes a proof state and returns a sequence (lazy list) of possible
957 successor states. Lazy lists are coded in ML as functions, a standard
958 technique~\cite{paulson-ml2}. Isabelle represents proof states by theorems.
960 Basic tactics execute the meta-rules described above, operating on a
961 given subgoal. The {\bf resolution tactics} take a list of rules and
962 return next states for each combination of rule and unifier. The {\bf
963 assumption tactic} examines the subgoal's assumptions and returns next
964 states for each combination of assumption and unifier. Lazy lists are
965 essential because higher-order resolution may return infinitely many
966 unifiers. If there are no matching rules or assumptions then no next
967 states are generated; a tactic application that returns an empty list is
970 Sequences realize their full potential with {\bf tacticals} --- operators
971 for combining tactics. Depth-first search, breadth-first search and
972 best-first search (where a heuristic function selects the best state to
973 explore) return their outcomes as a sequence. Isabelle provides such
974 procedures in the form of tacticals. Simpler procedures can be expressed
975 directly using the basic tacticals {\tt THEN}, {\tt ORELSE} and {\tt REPEAT}:
976 \begin{ttdescription}
977 \item[$tac1$ THEN $tac2$] is a tactic for sequential composition. Applied
978 to a proof state, it returns all states reachable in two steps by applying
979 $tac1$ followed by~$tac2$.
981 \item[$tac1$ ORELSE $tac2$] is a choice tactic. Applied to a state, it
982 tries~$tac1$ and returns the result if non-empty; otherwise, it uses~$tac2$.
984 \item[REPEAT $tac$] is a repetition tactic. Applied to a state, it
985 returns all states reachable by applying~$tac$ as long as possible --- until
988 For instance, this tactic repeatedly applies $tac1$ and~$tac2$, giving
991 REPEAT($tac1$ ORELSE $tac2$)
995 \section{Variations on resolution}
996 In principle, resolution and proof by assumption suffice to prove all
997 theorems. However, specialized forms of resolution are helpful for working
998 with elimination rules. Elim-resolution applies an elimination rule to an
999 assumption; destruct-resolution is similar, but applies a rule in a forward
1002 The last part of the section shows how the techniques for proving theorems
1003 can also serve to derive rules.
1005 \subsection{Elim-resolution}
1006 \index{elim-resolution|bold}\index{assumptions!deleting}
1008 Consider proving the theorem $((R\disj R)\disj R)\disj R\imp R$. By
1009 $({\imp}I)$, we prove~$R$ from the assumption $((R\disj R)\disj R)\disj R$.
1010 Applying $(\disj E)$ to this assumption yields two subgoals, one that
1011 assumes~$R$ (and is therefore trivial) and one that assumes $(R\disj
1012 R)\disj R$. This subgoal admits another application of $(\disj E)$. Since
1013 natural deduction never discards assumptions, we eventually generate a
1014 subgoal containing much that is redundant:
1015 \[ \List{((R\disj R)\disj R)\disj R; (R\disj R)\disj R; R\disj R; R} \Imp R. \]
1016 In general, using $(\disj E)$ on the assumption $P\disj Q$ creates two new
1017 subgoals with the additional assumption $P$ or~$Q$. In these subgoals,
1018 $P\disj Q$ is redundant. Other elimination rules behave
1019 similarly. In first-order logic, only universally quantified
1020 assumptions are sometimes needed more than once --- say, to prove
1021 $P(f(f(a)))$ from the assumptions $\forall x.P(x)\imp P(f(x))$ and~$P(a)$.
1023 Many logics can be formulated as sequent calculi that delete redundant
1024 assumptions after use. The rule $(\disj E)$ might become
1025 \[ \infer[\disj\hbox{-left}]
1026 {\Gamma,P\disj Q,\Delta \turn \Theta}
1027 {\Gamma,P,\Delta \turn \Theta && \Gamma,Q,\Delta \turn \Theta} \]
1028 In backward proof, a goal containing $P\disj Q$ on the left of the~$\turn$
1029 (that is, as an assumption) splits into two subgoals, replacing $P\disj Q$
1030 by $P$ or~$Q$. But the sequent calculus, with its explicit handling of
1031 assumptions, can be tiresome to use.
1033 Elim-resolution is Isabelle's way of getting sequent calculus behaviour
1034 from natural deduction rules. It lets an elimination rule consume an
1035 assumption. Elim-resolution combines two meta-theorems:
1037 \item a rule $\List{\psi@1; \ldots; \psi@m} \Imp \psi$
1038 \item a proof state $\List{\phi@1; \ldots; \phi@n} \Imp \phi$
1040 The rule must have at least one premise, thus $m>0$. Write the rule's
1041 lifted form as $\List{\psi'@1; \ldots; \psi'@m} \Imp \psi'$. Suppose we
1042 wish to change subgoal number~$i$.
1044 Ordinary resolution would attempt to reduce~$\phi@i$,
1045 replacing subgoal~$i$ by $m$ new ones. Elim-resolution tries
1046 simultaneously to reduce~$\phi@i$ and to solve~$\psi'@1$ by assumption; it
1047 returns a sequence of next states. Each of these replaces subgoal~$i$ by
1048 instances of $\psi'@2$, \ldots, $\psi'@m$ from which the selected
1049 assumption has been deleted. Suppose $\phi@i$ has the parameter~$x$ and
1050 assumptions $\theta@1$,~\ldots,~$\theta@k$. Then $\psi'@1$, the rule's first
1051 premise after lifting, will be
1052 \( \Forall x. \List{\theta@1; \ldots; \theta@k}\Imp \psi^{x}@1 \).
1053 Elim-resolution tries to unify $\psi'\qeq\phi@i$ and
1054 $\lambda x. \theta@j \qeq \lambda x. \psi^{x}@1$ simultaneously, for
1057 Let us redo the example from~\S\ref{prop-proof}. The elimination rule
1059 \[ \List{\Var{P}\disj \Var{Q};\; \Var{P}\Imp \Var{R};\; \Var{Q}\Imp \Var{R}}
1061 and the proof state is $(P\disj P\Imp P)\Imp (P\disj P\imp P)$. The
1063 \[ \begin{array}{l@{}l}
1064 \lbrakk\;& P\disj P \Imp \Var{P@1}\disj\Var{Q@1}; \\
1065 & \List{P\disj P ;\; \Var{P@1}} \Imp \Var{R@1}; \\
1066 & \List{P\disj P ;\; \Var{Q@1}} \Imp \Var{R@1} \\
1067 \rbrakk\;& \Imp (P\disj P \Imp \Var{R@1})
1070 Unification takes the simultaneous equations
1071 $P\disj P \qeq \Var{P@1}\disj\Var{Q@1}$ and $\Var{R@1} \qeq P$, yielding
1072 $\Var{P@1}\equiv\Var{Q@1}\equiv\Var{R@1} \equiv P$. The new proof state
1074 \[ \List{P \Imp P;\; P \Imp P} \Imp (P\disj P\imp P).
1076 Elim-resolution's simultaneous unification gives better control
1077 than ordinary resolution. Recall the substitution rule:
1078 $$ \List{\Var{t}=\Var{u}; \Var{P}(\Var{t})} \Imp \Var{P}(\Var{u})
1080 Unsuitable for ordinary resolution because $\Var{P}(\Var{u})$ admits many
1081 unifiers, $(subst)$ works well with elim-resolution. It deletes some
1082 assumption of the form $x=y$ and replaces every~$y$ by~$x$ in the subgoal
1083 formula. The simultaneous unification instantiates $\Var{u}$ to~$y$; if
1084 $y$ is not an unknown, then $\Var{P}(y)$ can easily be unified with another
1087 In logical parlance, the premise containing the connective to be eliminated
1088 is called the \bfindex{major premise}. Elim-resolution expects the major
1089 premise to come first. The order of the premises is significant in
1092 \subsection{Destruction rules} \label{destruct}
1093 \index{rules!destruction}\index{rules!elimination}
1094 \index{forward proof}
1096 Looking back to Fig.\ts\ref{fol-fig}, notice that there are two kinds of
1097 elimination rule. The rules $({\conj}E1)$, $({\conj}E2)$, $({\imp}E)$ and
1098 $({\forall}E)$ extract the conclusion from the major premise. In Isabelle
1099 parlance, such rules are called {\bf destruction rules}; they are readable
1100 and easy to use in forward proof. The rules $({\disj}E)$, $({\bot}E)$ and
1101 $({\exists}E)$ work by discharging assumptions; they support backward proof
1102 in a style reminiscent of the sequent calculus.
1104 The latter style is the most general form of elimination rule. In natural
1105 deduction, there is no way to recast $({\disj}E)$, $({\bot}E)$ or
1106 $({\exists}E)$ as destruction rules. But we can write general elimination
1107 rules for $\conj$, $\imp$ and~$\forall$:
1109 \infer{R}{P\conj Q & \infer*{R}{[P,Q]}} \qquad
1110 \infer{R}{P\imp Q & P & \infer*{R}{[Q]}} \qquad
1111 \infer{Q}{\forall x.P & \infer*{Q}{[P[t/x]]}}
1113 Because they are concise, destruction rules are simpler to derive than the
1114 corresponding elimination rules. To facilitate their use in backward
1115 proof, Isabelle provides a means of transforming a destruction rule such as
1116 \[ \infer[\quad\hbox{to the elimination rule}\quad]{Q}{P@1 & \ldots & P@m}
1117 \infer{R.}{P@1 & \ldots & P@m & \infer*{R}{[Q]}}
1119 {\bf Destruct-resolution}\index{destruct-resolution} combines this
1120 transformation with elim-resolution. It applies a destruction rule to some
1121 assumption of a subgoal. Given the rule above, it replaces the
1122 assumption~$P@1$ by~$Q$, with new subgoals of showing instances of $P@2$,
1123 \ldots,~$P@m$. Destruct-resolution works forward from a subgoal's
1124 assumptions. Ordinary resolution performs forward reasoning from theorems,
1125 as illustrated in~\S\ref{joining}.
1128 \subsection{Deriving rules by resolution} \label{deriving}
1129 \index{rules!derived|bold}\index{meta-assumptions!syntax of}
1130 The meta-logic, itself a form of the predicate calculus, is defined by a
1131 system of natural deduction rules. Each theorem may depend upon
1132 meta-assumptions. The theorem that~$\phi$ follows from the assumptions
1133 $\phi@1$, \ldots, $\phi@n$ is written
1134 \[ \phi \quad [\phi@1,\ldots,\phi@n]. \]
1135 A more conventional notation might be $\phi@1,\ldots,\phi@n \turn \phi$,
1136 but Isabelle's notation is more readable with large formulae.
1138 Meta-level natural deduction provides a convenient mechanism for deriving
1139 new object-level rules. To derive the rule
1140 \[ \infer{\phi,}{\theta@1 & \ldots & \theta@k} \]
1141 assume the premises $\theta@1$,~\ldots,~$\theta@k$ at the
1142 meta-level. Then prove $\phi$, possibly using these assumptions.
1143 Starting with a proof state $\phi\Imp\phi$, assumptions may accumulate,
1144 reaching a final state such as
1145 \[ \phi \quad [\theta@1,\ldots,\theta@k]. \]
1146 The meta-rule for $\Imp$ introduction discharges an assumption.
1147 Discharging them in the order $\theta@k,\ldots,\theta@1$ yields the
1148 meta-theorem $\List{\theta@1; \ldots; \theta@k} \Imp \phi$, with no
1149 assumptions. This represents the desired rule.
1150 Let us derive the general $\conj$ elimination rule:
1151 $$ \infer{R}{P\conj Q & \infer*{R}{[P,Q]}} \eqno(\conj E) $$
1152 We assume $P\conj Q$ and $\List{P;Q}\Imp R$, and commence backward proof in
1153 the state $R\Imp R$. Resolving this with the second assumption yields the
1155 \[ \phantom{\List{P\conj Q;\; P\conj Q}}
1156 \llap{$\List{P;Q}$}\Imp R \quad [\,\List{P;Q}\Imp R\,]. \]
1157 Resolving subgoals~1 and~2 with~$({\conj}E1)$ and~$({\conj}E2)$,
1158 respectively, yields the state
1159 \[ \List{P\conj \Var{Q@1};\; \Var{P@2}\conj Q}\Imp R
1160 \quad [\,\List{P;Q}\Imp R\,].
1162 The unknowns $\Var{Q@1}$ and~$\Var{P@2}$ arise from unconstrained
1163 subformulae in the premises of~$({\conj}E1)$ and~$({\conj}E2)$. Resolving
1164 both subgoals with the assumption $P\conj Q$ instantiates the unknowns to yield
1165 \[ R \quad [\, \List{P;Q}\Imp R, P\conj Q \,]. \]
1166 The proof may use the meta-assumptions in any order, and as often as
1167 necessary; when finished, we discharge them in the correct order to
1168 obtain the desired form:
1169 \[ \List{P\conj Q;\; \List{P;Q}\Imp R} \Imp R \]
1170 We have derived the rule using free variables, which prevents their
1171 premature instantiation during the proof; we may now replace them by
1172 schematic variables.
1175 Schematic variables are not allowed in meta-assumptions, for a variety of
1176 reasons. Meta-assumptions remain fixed throughout a proof.