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.
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
15 \neg P & \hbox{abbreviates} & P\imp\bot \\
16 P\bimp Q & \hbox{abbreviates} & (P\imp Q) \conj (Q\imp P)
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]
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]
29 \infer[({\imp}I)]{P\imp Q}{\infer*{Q}{[P]}} &
30 \infer[({\imp}E)]{Q}{P\imp Q & P} \\[4ex]
33 \infer[({\bot}E)]{P}{\bot}\\[4ex]
35 \infer[({\forall}I)*]{\forall x.P}{P} &
36 \infer[({\forall}E)]{P[t/x]}{\forall x.P} \\[3ex]
38 \infer[({\exists}I)]{\exists x.P}{P[t/x]} &
39 \infer[({\exists}E)*]{Q}{{\exists x.P} & \infer*{Q}{[P]} } \\[3ex]
41 {t=t} \,(refl) & \vcenter{\infer[(subst)]{P[u/x]}{t=u & P[t/x]}}
45 *{\em Eigenvariable conditions\/}:
47 $\forall I$: provided $x$ is not free in the assumptions
49 $\exists E$: provided $x$ is not free in $Q$ or any assumption except $P$
50 \caption{Intuitionistic first-order logic} \label{fol-fig}
53 \section{Formalizing logical syntax in Isabelle}\label{sec:logical-syntax}
54 \index{first-order logic}
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.
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.
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 \[ \sigma@1\To (\cdots \sigma@n\To \tau\cdots) \quad \hbox{as} \quad
76 [\sigma@1, \ldots, \sigma@n] \To \tau \]
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.
84 \index{lambda abs@$\lambda$-abstractions}\index{function applications}
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, $t(u@1)\ldots(u@n)$}
96 \subsection{Simple types and constants}\index{types!simple|bold}
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.
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*}
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:
117 Suc & :: & nat\To nat \\
118 {\neg} & :: & o\To o \\
119 \conj,\disj,\imp,\bimp & :: & [o,o]\To o
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 $\disj(\conj(P,Q), R)$.
125 Section~\ref{sec:defining-theories} below describes the syntax of Isabelle
126 theory files and illustrates it by extending our logic with mathematical
130 \subsection{Polymorphic types and constants} \label{polymorphic}
131 \index{types!polymorphic|bold}
132 \index{equality!polymorphic}
133 \index{constants!polymorphic}
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
140 {=} & :: & [\alpha,\alpha]\To o,
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.
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}.
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 $term\le logic$.
162 We put $nat$ in class $term$ by declaring $nat{::}term$. We declare the
165 {=} & :: & [\alpha{::}term,\alpha]\To o
167 where $\alpha{::}term$ constrains the type variable~$\alpha$ to class
168 $term$. Such type variables resemble Standard~\ML's equality type
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}
181 fun & :: & (logic,logic)logic \\
182 nat, string, real & :: & term \\
183 list & :: & (term)term
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
193 {+},{-},{\times},{/} & :: & [\alpha{::}arith,\alpha]\To \alpha
195 If we declare new types $real$ and $complex$ of class $arith$, then we
196 in effect have three sets of operators:
198 {+},{-},{\times},{/} & :: & [nat,nat]\To nat \\
199 {+},{-},{\times},{/} & :: & [real,real]\To real \\
200 {+},{-},{\times},{/} & :: & [complex,complex]\To complex
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
206 {+} & :: & [(\alpha)vector,(\alpha)vector]\To (\alpha)vector
208 and specify it in terms of ${+} :: [\alpha,\alpha]\To \alpha$.
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
214 {\le} & :: & [\alpha{::}ord,\alpha]\To o.
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
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}}.
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
241 \forall & :: & (nat\To o) \To o,
243 which is essentially an infinitary truth table. The representation of $\forall
244 x. P(x)$ is $\forall(\lambda x. P(x))$.
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
251 \Sigma & :: & [nat,nat, nat\To nat] \To nat.
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:
257 \forall,\exists & :: & (\alpha{::}term\To o) \To o \\
258 \Sigma & :: & [nat,nat, nat\To \alpha{::}arith] \To \alpha
260 Observe that the index variables still have type $nat$, while the values
261 being summed may belong to any arithmetic type.
264 \section{Formalizing logical rules in Isabelle}
265 \index{meta-implication|bold}
266 \index{meta-quantifiers|bold}
267 \index{meta-equality|bold}
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}.
274 \item The implication \(\phi\Imp \psi\) means `\(\phi\) implies
275 \(\psi\)', and expresses logical {\bf entailment}.
277 \item The quantification \(\Forall x.\phi\) means `\(\phi\) is true for
278 all $x$', and expresses {\bf generality} in rules and axiom schemes.
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.
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
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.
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}
313 Trueprop & :: & o\To prop.
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
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
326 One of the simplest rules is $(\conj E1)$. Making
327 everything explicit, its formalization in the meta-logic is
329 \Forall P\;Q. Trueprop(P\conj Q) \Imp Trueprop(P). \eqno(\conj E1)
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.'
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.
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 $$ P\conj Q \Imp P \qquad P\conj Q \Imp Q \eqno(\conj E1,2) $$
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 $$ \List{P\disj Q; P\Imp R; Q\Imp R} \Imp R \eqno(\disj E) $$
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
366 $$ (P \Imp Q) \Imp P\imp Q \eqno({\imp}I) $$
367 $$ \List{P\imp Q; P} \Imp Q. \eqno({\imp}E) $$
370 Finally, the intuitionistic contradiction rule is formalized as the axiom
371 $$ \bot \Imp P. \eqno(\bot E) $$
374 Earlier versions of Isabelle, and certain
375 papers~\cite{paulson-found,paulson700}, use $\List{P}$ to mean $Trueprop(P)$.
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}
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.
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}
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
407 $$ \left(\Forall x. P(x)\right) \Imp \forall x.P(x) \eqno(\forall I) $$
408 $$ (\forall x.P(x)) \Imp P(t). \eqno(\forall E) $$
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, $Q$.
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
428 $$ \List{t=u; P(t)} \Imp P(u). \eqno(subst) $$
431 \subsection{Signatures and theories}
432 \index{signatures|bold}
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.
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.
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:
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} & {} & {}
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
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.
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.
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.
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.
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.
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.
525 Isabelle formulae require an extended notion of resolution.
526 They differ from Horn clauses in two major respects:
528 \item They are written in the typed $\lambda$-calculus, and therefore must be
529 resolved using higher-order unification.
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.
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
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 \begin{equation} \label{hou-eqn}
552 \Var{f}(t) \qeq g(u@1,\ldots,u@k).
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
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
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~$\Var{f}$.
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
573 \[ t(\Var{h@1}(t),\ldots,\Var{h@m}(t)) \qeq g(u@1,\ldots,u@k). \]
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.
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}:
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~$x$.
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.
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
612 In problematic cases, you may have to instantiate some unknowns before
613 invoking unification.
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.
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}
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).
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}.
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}}
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}}
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 $$ \forall x\,y. Suc(x)=Suc(y)\imp x=y. \eqno (inject) $$
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 $\Var{t}$
673 \[ \forall y. Suc(\Var{t})=Suc(y)\imp \Var{t}=y. \]
674 Resolving this with $(\forall E)$ puts a subscript on~$\Var{t}$
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}.
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 $Suc(Suc(Suc(m)))=Suc(Suc(Suc(0)))$.
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}
696 \infer[(\forall I)]{\forall x\,y.P(x,y)}
697 {\infer[(\forall I)]{\forall y.P(x,y)}{P(x,y)}}
700 \subsection{Lifting over assumptions}
701 \index{assumptions!lifting over}
702 Lifting over $\theta\Imp{}$ is the following meta-inference rule:
703 \[ \infer{\List{\theta\Imp\phi@1; \ldots; \theta\Imp\phi@n} \Imp
705 {\List{\phi@1; \ldots; \phi@n} \Imp \phi} \]
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.
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 ${\Var{P@1}\imp\Var{Q@1}}$.
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 \[ \infer{P\imp(Q\imp R).}{\infer*{R}{[P,Q]}} \]
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 \[ \infer{\List{\Forall x.\phi@1^x; \ldots; \Forall x.\phi@n^x}
742 \Imp \Forall x.\phi^x,}
743 {\List{\phi@1; \ldots; \phi@n} \Imp \phi} \]
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 $(\Forall x.\phi^x) \Imp (\Forall x.\psi^x)$.
751 For example, $(\disj I)$ might be lifted to
752 \[ (\Forall x.\Var{P@1}(x)) \Imp (\Forall x. \Var{P@1}(x)\disj \Var{Q@1}(x))\]
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}
766 I discuss lifting and parameters at length elsewhere~\cite{paulson-found}.
767 Miller goes into even greater detail~\cite{miller-mixed}.
770 \section{Backward proof by resolution}
771 \index{resolution!in backward proof}
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.
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
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
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~$\phi$.
805 \subsection{Refinement by resolution}
806 To refine subgoal~$i$ of a proof state by a rule, perform the following
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
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 \begin{equation} \label{context-eqn}
833 \Forall x@1 \ldots x@l. \List{\theta@1; \ldots; \theta@k}\Imp\theta.
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.
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}.
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.
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:
863 \item $Q(a)$, instantiating $\Var{x}\equiv a$
864 \item $Q(b)$, instantiating $\Var{x}\equiv b$
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.
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 \[ (P\disj P\imp P)\Imp (P\disj P\imp P). \]
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.
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)}
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)}
906 Both of the remaining subgoals can be proved by assumption. After two such
907 steps, the proof state is $P\disj P\imp P$.
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))
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)}
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))
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
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))
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: $(\exists x.P(f(x)))\imp(\exists x.P(x))$.
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.
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
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~$tac2$.
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~$tac2$.
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
990 For instance, this tactic repeatedly applies $tac1$ and~$tac2$, giving
993 REPEAT($tac1$ ORELSE $tac2$)
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
1004 The last part of the section shows how the techniques for proving theorems
1005 can also serve to derive rules.
1007 \subsection{Elim-resolution}
1008 \index{elim-resolution|bold}\index{assumptions!deleting}
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~$P(a)$.
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.
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:
1039 \item a rule $\List{\psi@1; \ldots; \psi@m} \Imp \psi$
1040 \item a proof state $\List{\phi@1; \ldots; \phi@n} \Imp \phi$
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~$i$.
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
1059 Let us redo the example from~\S\ref{prop-proof}. The elimination rule
1061 \[ \List{\Var{P}\disj \Var{Q};\; \Var{P}\Imp \Var{R};\; \Var{Q}\Imp \Var{R}}
1063 and the proof state is $(P\disj P\Imp P)\Imp (P\disj P\imp P)$. The
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})
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
1076 \[ \List{P \Imp P;\; P \Imp P} \Imp (P\disj P\imp P).
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})
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
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
1094 \subsection{Destruction rules} \label{destruct}
1095 \index{rules!destruction}\index{rules!elimination}
1096 \index{forward proof}
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.
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~$\forall$:
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]]}}
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]}}
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}.
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.
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
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\,].
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.
1177 Schematic variables are not allowed in meta-assumptions, for a variety of
1178 reasons. Meta-assumptions remain fixed throughout a proof.