7 chapter {* Isabelle/HOL \label{ch:hol} *}
9 section {* Primitive types \label{sec:hol-typedef} *}
12 \begin{matharray}{rcl}
13 @{command_def (HOL) "typedecl"} & : & \isartrans{theory}{theory} \\
14 @{command_def (HOL) "typedef"} & : & \isartrans{theory}{proof(prove)} \\
18 'typedecl' typespec infix?
20 'typedef' altname? abstype '=' repset
23 altname: '(' (name | 'open' | 'open' name) ')'
25 abstype: typespec infix?
27 repset: term ('morphisms' name name)?
33 \item [@{command (HOL) "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n)
34 t"}] is similar to the original @{command "typedecl"} of
35 Isabelle/Pure (see \secref{sec:types-pure}), but also declares type
36 arity @{text "t :: (type, \<dots>, type) type"}, making @{text t} an
37 actual HOL type constructor. %FIXME check, update
39 \item [@{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n)
40 t = A"}] sets up a goal stating non-emptiness of the set @{text A}.
41 After finishing the proof, the theory will be augmented by a
42 Gordon/HOL-style type definition, which establishes a bijection
43 between the representing set @{text A} and the new type @{text t}.
45 Technically, @{command (HOL) "typedef"} defines both a type @{text
46 t} and a set (term constant) of the same name (an alternative base
47 name may be given in parentheses). The injection from type to set
48 is called @{text Rep_t}, its inverse @{text Abs_t} (this may be
49 changed via an explicit @{keyword (HOL) "morphisms"} declaration).
51 Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text
52 Abs_t_inverse} provide the most basic characterization as a
53 corresponding injection/surjection pair (in both directions). Rules
54 @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly
55 more convenient view on the injectivity part, suitable for automated
56 proof tools (e.g.\ in @{method simp} or @{method iff} declarations).
57 Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and @{text
58 Abs_t_cases}/@{text Abs_t_induct} provide alternative views on
59 surjectivity; these are already declared as set or type rules for
60 the generic @{method cases} and @{method induct} methods.
62 An alternative name may be specified in parentheses; the default is
63 to use @{text t} as indicated before. The ``@{text "(open)"}''
64 declaration suppresses a separate constant definition for the
69 Note that raw type declarations are rarely used in practice; the
70 main application is with experimental (or even axiomatic!) theory
71 fragments. Instead of primitive HOL type definitions, user-level
72 theories usually refer to higher-level packages such as @{command
73 (HOL) "record"} (see \secref{sec:hol-record}) or @{command (HOL)
74 "datatype"} (see \secref{sec:hol-datatype}).
78 section {* Adhoc tuples *}
81 \begin{matharray}{rcl}
82 @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & \isaratt \\
86 'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))
92 \item [@{method (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m
93 \<AND> \<dots> \<AND> q\<^sub>1 \<dots> q\<^sub>n"}] puts expressions of
94 low-level tuple types into canonical form as specified by the
95 arguments given; the @{text i}-th collection of arguments refers to
96 occurrences in premise @{text i} of the rule. The ``@{text
97 "(complete)"}'' option causes \emph{all} arguments in function
98 applications to be represented canonically according to their tuple
101 Note that these operations tend to invent funny names for new local
102 parameters to be introduced.
108 section {* Records \label{sec:hol-record} *}
111 In principle, records merely generalize the concept of tuples, where
112 components may be addressed by labels instead of just position. The
113 logical infrastructure of records in Isabelle/HOL is slightly more
114 advanced, though, supporting truly extensible record schemes. This
115 admits operations that are polymorphic with respect to record
116 extension, yielding ``object-oriented'' effects like (single)
117 inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more
118 details on object-oriented verification and record subtyping in HOL.
122 subsection {* Basic concepts *}
125 Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
126 at the level of terms and types. The notation is as follows:
129 \begin{tabular}{l|l|l}
130 & record terms & record types \\ \hline
131 fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
132 schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
133 @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
137 \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
140 A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
141 @{text a} and field @{text y} of value @{text b}. The corresponding
142 type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
143 and @{text "b :: B"}.
145 A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
146 @{text x} and @{text y} as before, but also possibly further fields
147 as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
148 of the syntax). The improper field ``@{text "\<dots>"}'' of a record
149 scheme is called the \emph{more part}. Logically it is just a free
150 variable, which is occasionally referred to as ``row variable'' in
151 the literature. The more part of a record scheme may be
152 instantiated by zero or more further components. For example, the
153 previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
154 c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
155 Fixed records are special instances of record schemes, where
156 ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
157 element. In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
158 for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
160 \medskip Two key observations make extensible records in a simply
161 typed language like HOL work out:
165 \item the more part is internalized, as a free term or type
168 \item field names are externalized, they cannot be accessed within
169 the logic as first-class values.
173 \medskip In Isabelle/HOL record types have to be defined explicitly,
174 fixing their field names and types, and their (optional) parent
175 record. Afterwards, records may be formed using above syntax, while
176 obeying the canonical order of fields as given by their declaration.
177 The record package provides several standard operations like
178 selectors and updates. The common setup for various generic proof
179 tools enable succinct reasoning patterns. See also the Isabelle/HOL
180 tutorial \cite{isabelle-hol-book} for further instructions on using
185 subsection {* Record specifications *}
188 \begin{matharray}{rcl}
189 @{command_def (HOL) "record"} & : & \isartrans{theory}{theory} \\
193 'record' typespec '=' (type '+')? (constdecl +)
199 \item [@{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t
200 = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1 \<dots> c\<^sub>n :: \<sigma>\<^sub>n"}] defines
201 extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
202 derived from the optional parent record @{text "\<tau>"} by adding new
203 field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
205 The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
206 covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
207 \<alpha>\<^sub>m"}. Type constructor @{text t} has to be new, while @{text
208 \<tau>} needs to specify an instance of an existing record type. At
209 least one new field @{text "c\<^sub>i"} has to be specified.
210 Basically, field names need to belong to a unique record. This is
211 not a real restriction in practice, since fields are qualified by
212 the record name internally.
214 The parent record specification @{text \<tau>} is optional; if omitted
215 @{text t} becomes a root record. The hierarchy of all records
216 declared within a theory context forms a forest structure, i.e.\ a
217 set of trees starting with a root record each. There is no way to
218 merge multiple parent records!
220 For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
221 type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
222 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
223 "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
224 @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
231 subsection {* Record operations *}
234 Any record definition of the form presented above produces certain
235 standard operations. Selectors and updates are provided for any
236 field, including the improper one ``@{text more}''. There are also
237 cumulative record constructor functions. To simplify the
238 presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
239 \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
240 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
242 \medskip \textbf{Selectors} and \textbf{updates} are available for
243 any field (including ``@{text more}''):
245 \begin{matharray}{lll}
246 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
247 @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
250 There is special syntax for application of updates: @{text "r\<lparr>x :=
251 a\<rparr>"} abbreviates term @{text "x_update a r"}. Further notation for
252 repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
253 c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}. Note that
254 because of postfix notation the order of fields shown here is
255 reverse than in the actual term. Since repeated updates are just
256 function applications, fields may be freely permuted in @{text "\<lparr>x
257 := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
258 Thus commutativity of independent updates can be proven within the
259 logic for any two fields, but not as a general theorem.
261 \medskip The \textbf{make} operation provides a cumulative record
262 constructor function:
264 \begin{matharray}{lll}
265 @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
268 \medskip We now reconsider the case of non-root records, which are
269 derived of some parent. In general, the latter may depend on
270 another parent as well, resulting in a list of \emph{ancestor
271 records}. Appending the lists of fields of all ancestors results in
272 a certain field prefix. The record package automatically takes care
273 of this by lifting operations over this context of ancestor fields.
274 Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
275 fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
276 the above record operations will get the following types:
280 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
281 @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow>
282 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
283 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
284 @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
285 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
289 \noindent Some further operations address the extension aspect of a
290 derived record scheme specifically: @{text "t.fields"} produces a
291 record fragment consisting of exactly the new fields introduced here
292 (the result may serve as a more part elsewhere); @{text "t.extend"}
293 takes a fixed record and adds a given more part; @{text
294 "t.truncate"} restricts a record scheme to a fixed record.
298 @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
299 @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
300 \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
301 @{text "t.truncate"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
305 \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
310 subsection {* Derived rules and proof tools *}
313 The record package proves several results internally, declaring
314 these facts to appropriate proof tools. This enables users to
315 reason about record structures quite conveniently. Assume that
316 @{text t} is a record type as specified above.
320 \item Standard conversions for selectors or updates applied to
321 record constructor terms are made part of the default Simplifier
322 context; thus proofs by reduction of basic operations merely require
323 the @{method simp} method without further arguments. These rules
324 are available as @{text "t.simps"}, too.
326 \item Selectors applied to updated records are automatically reduced
327 by an internal simplification procedure, which is also part of the
328 standard Simplifier setup.
330 \item Inject equations of a form analogous to @{prop "(x, y) = (x',
331 y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
332 Reasoner as @{attribute iff} rules. These rules are available as
335 \item The introduction rule for record equality analogous to @{text
336 "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
337 and as the basic rule context as ``@{attribute intro}@{text "?"}''.
338 The rule is called @{text "t.equality"}.
340 \item Representations of arbitrary record expressions as canonical
341 constructor terms are provided both in @{method cases} and @{method
342 induct} format (cf.\ the generic proof methods of the same name,
343 \secref{sec:cases-induct}). Several variations are available, for
344 fixed records, record schemes, more parts etc.
346 The generic proof methods are sufficiently smart to pick the most
347 sensible rule according to the type of the indicated record
348 expression: users just need to apply something like ``@{text "(cases
349 r)"}'' to a certain proof problem.
351 \item The derived record operations @{text "t.make"}, @{text
352 "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
353 treated automatically, but usually need to be expanded by hand,
354 using the collective fact @{text "t.defs"}.
360 section {* Datatypes \label{sec:hol-datatype} *}
363 \begin{matharray}{rcl}
364 @{command_def (HOL) "datatype"} & : & \isartrans{theory}{theory} \\
365 @{command_def (HOL) "rep_datatype"} & : & \isartrans{theory}{theory} \\
369 'datatype' (dtspec + 'and')
371 'rep\_datatype' (name *) dtrules
374 dtspec: parname? typespec infix? '=' (cons + '|')
376 cons: name (type *) mixfix?
378 dtrules: 'distinct' thmrefs 'inject' thmrefs 'induction' thmrefs
383 \item [@{command (HOL) "datatype"}] defines inductive datatypes in
386 \item [@{command (HOL) "rep_datatype"}] represents existing types as
387 inductive ones, generating the standard infrastructure of derived
388 concepts (primitive recursion etc.).
392 The induction and exhaustion theorems generated provide case names
393 according to the constructors involved, while parameters are named
394 after the types (see also \secref{sec:cases-induct}).
396 See \cite{isabelle-HOL} for more details on datatypes, but beware of
397 the old-style theory syntax being used there! Apart from proper
398 proof methods for case-analysis and induction, there are also
399 emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
400 induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
401 to refer directly to the internal structure of subgoals (including
402 internally bound parameters).
406 section {* Recursive functions \label{sec:recursion} *}
409 \begin{matharray}{rcl}
410 @{command_def (HOL) "primrec"} & : & \isarkeep{local{\dsh}theory} \\
411 @{command_def (HOL) "fun"} & : & \isarkeep{local{\dsh}theory} \\
412 @{command_def (HOL) "function"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
413 @{command_def (HOL) "termination"} & : & \isartrans{local{\dsh}theory}{proof(prove)} \\
416 \railalias{funopts}{function\_opts} %FIXME ??
419 'primrec' target? fixes 'where' equations
421 equations: (thmdecl? prop + '|')
423 ('fun' | 'function') (funopts)? fixes 'where' clauses
425 clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
427 funopts: '(' (('sequential' | 'in' name | 'domintros' | 'tailrec' |
428 'default' term) + ',') ')'
430 'termination' ( term )?
435 \item [@{command (HOL) "primrec"}] defines primitive recursive
436 functions over datatypes, see also \cite{isabelle-HOL}.
438 \item [@{command (HOL) "function"}] defines functions by general
439 wellfounded recursion. A detailed description with examples can be
440 found in \cite{isabelle-function}. The function is specified by a
441 set of (possibly conditional) recursive equations with arbitrary
442 pattern matching. The command generates proof obligations for the
443 completeness and the compatibility of patterns.
445 The defined function is considered partial, and the resulting
446 simplification rules (named @{text "f.psimps"}) and induction rule
447 (named @{text "f.pinduct"}) are guarded by a generated domain
448 predicate @{text "f_dom"}. The @{command (HOL) "termination"}
449 command can then be used to establish that the function is total.
451 \item [@{command (HOL) "fun"}] is a shorthand notation for
452 ``@{command (HOL) "function"}~@{text "(sequential)"}, followed by
453 automated proof attempts regarding pattern matching and termination.
454 See \cite{isabelle-function} for further details.
456 \item [@{command (HOL) "termination"}~@{text f}] commences a
457 termination proof for the previously defined function @{text f}. If
458 this is omitted, the command refers to the most recent function
459 definition. After the proof is closed, the recursive equations and
460 the induction principle is established.
466 Recursive definitions introduced by both the @{command (HOL)
467 "primrec"} and the @{command (HOL) "function"} command accommodate
468 reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
469 "c.induct"} (where @{text c} is the name of the function definition)
470 refers to a specific induction rule, with parameters named according
471 to the user-specified equations. Case names of @{command (HOL)
472 "primrec"} are that of the datatypes involved, while those of
473 @{command (HOL) "function"} are numbered (starting from 1).
475 The equations provided by these packages may be referred later as
476 theorem list @{text "f.simps"}, where @{text f} is the (collective)
477 name of the functions defined. Individual equations may be named
480 The @{command (HOL) "function"} command accepts the following
485 \item [@{text sequential}] enables a preprocessor which
486 disambiguates overlapping patterns by making them mutually disjoint.
487 Earlier equations take precedence over later ones. This allows to
488 give the specification in a format very similar to functional
489 programming. Note that the resulting simplification and induction
490 rules correspond to the transformed specification, not the one given
491 originally. This usually means that each equation given by the user
492 may result in several theroems. Also note that this automatic
493 transformation only works for ML-style datatype patterns.
495 \item [@{text "\<IN> name"}] gives the target for the definition.
498 \item [@{text domintros}] enables the automated generation of
499 introduction rules for the domain predicate. While mostly not
500 needed, they can be helpful in some proofs about partial functions.
502 \item [@{text tailrec}] generates the unconstrained recursive
503 equations even without a termination proof, provided that the
504 function is tail-recursive. This currently only works
506 \item [@{text "default d"}] allows to specify a default value for a
507 (partial) function, which will ensure that @{text "f x = d x"}
508 whenever @{text "x \<notin> f_dom"}.
514 subsection {* Proof methods related to recursive definitions *}
517 \begin{matharray}{rcl}
518 @{method_def (HOL) pat_completeness} & : & \isarmeth \\
519 @{method_def (HOL) relation} & : & \isarmeth \\
520 @{method_def (HOL) lexicographic_order} & : & \isarmeth \\
526 'lexicographic\_order' (clasimpmod *)
532 \item [@{method (HOL) pat_completeness}] is a specialized method to
533 solve goals regarding the completeness of pattern matching, as
534 required by the @{command (HOL) "function"} package (cf.\
535 \cite{isabelle-function}).
537 \item [@{method (HOL) relation}~@{text R}] introduces a termination
538 proof using the relation @{text R}. The resulting proof state will
539 contain goals expressing that @{text R} is wellfounded, and that the
540 arguments of recursive calls decrease with respect to @{text R}.
541 Usually, this method is used as the initial proof step of manual
544 \item [@{method (HOL) "lexicographic_order"}] attempts a fully
545 automated termination proof by searching for a lexicographic
546 combination of size measures on the arguments of the function. The
547 method accepts the same arguments as the @{method auto} method,
548 which it uses internally to prove local descents. The same context
549 modifiers as for @{method auto} are accepted, see
550 \secref{sec:clasimp}.
552 In case of failure, extensive information is printed, which can help
553 to analyse the situation (cf.\ \cite{isabelle-function}).
559 subsection {* Old-style recursive function definitions (TFL) *}
562 The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
563 "recdef_tc"} for defining recursive are mostly obsolete; @{command
564 (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
566 \begin{matharray}{rcl}
567 @{command_def (HOL) "recdef"} & : & \isartrans{theory}{theory} \\
568 @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & \isartrans{theory}{proof(prove)} \\
572 'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
576 hints: '(' 'hints' (recdefmod *) ')'
578 recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
580 tc: nameref ('(' nat ')')?
586 \item [@{command (HOL) "recdef"}] defines general well-founded
587 recursive functions (using the TFL package), see also
588 \cite{isabelle-HOL}. The ``@{text "(permissive)"}'' option tells
589 TFL to recover from failed proof attempts, returning unfinished
590 results. The @{text recdef_simp}, @{text recdef_cong}, and @{text
591 recdef_wf} hints refer to auxiliary rules to be used in the internal
592 automated proof process of TFL. Additional @{syntax clasimpmod}
593 declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
594 context of the Simplifier (cf.\ \secref{sec:simplifier}) and
595 Classical reasoner (cf.\ \secref{sec:classical}).
597 \item [@{command (HOL) "recdef_tc"}~@{text "c (i)"}] recommences the
598 proof for leftover termination condition number @{text i} (default
599 1) as generated by a @{command (HOL) "recdef"} definition of
602 Note that in most cases, @{command (HOL) "recdef"} is able to finish
603 its internal proofs without manual intervention.
607 \medskip Hints for @{command (HOL) "recdef"} may be also declared
608 globally, using the following attributes.
610 \begin{matharray}{rcl}
611 @{attribute_def (HOL) recdef_simp} & : & \isaratt \\
612 @{attribute_def (HOL) recdef_cong} & : & \isaratt \\
613 @{attribute_def (HOL) recdef_wf} & : & \isaratt \\
617 ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
623 section {* Definition by specification \label{sec:hol-specification} *}
626 \begin{matharray}{rcl}
627 @{command_def (HOL) "specification"} & : & \isartrans{theory}{proof(prove)} \\
628 @{command_def (HOL) "ax_specification"} & : & \isartrans{theory}{proof(prove)} \\
632 ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
634 decl: ((name ':')? term '(' 'overloaded' ')'?)
639 \item [@{command (HOL) "specification"}~@{text "decls \<phi>"}] sets up a
640 goal stating the existence of terms with the properties specified to
641 hold for the constants given in @{text decls}. After finishing the
642 proof, the theory will be augmented with definitions for the given
643 constants, as well as with theorems stating the properties for these
646 \item [@{command (HOL) "ax_specification"}~@{text "decls \<phi>"}] sets
647 up a goal stating the existence of terms with the properties
648 specified to hold for the constants given in @{text decls}. After
649 finishing the proof, the theory will be augmented with axioms
650 expressing the properties given in the first place.
652 \item [@{text decl}] declares a constant to be defined by the
653 specification given. The definition for the constant @{text c} is
654 bound to the name @{text c_def} unless a theorem name is given in
655 the declaration. Overloaded constants should be declared as such.
659 Whether to use @{command (HOL) "specification"} or @{command (HOL)
660 "ax_specification"} is to some extent a matter of style. @{command
661 (HOL) "specification"} introduces no new axioms, and so by
662 construction cannot introduce inconsistencies, whereas @{command
663 (HOL) "ax_specification"} does introduce axioms, but only after the
664 user has explicitly proven it to be safe. A practical issue must be
665 considered, though: After introducing two constants with the same
666 properties using @{command (HOL) "specification"}, one can prove
667 that the two constants are, in fact, equal. If this might be a
668 problem, one should use @{command (HOL) "ax_specification"}.
672 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
675 An \textbf{inductive definition} specifies the least predicate (or
676 set) @{text R} closed under given rules: applying a rule to elements
677 of @{text R} yields a result within @{text R}. For example, a
678 structural operational semantics is an inductive definition of an
681 Dually, a \textbf{coinductive definition} specifies the greatest
682 predicate~/ set @{text R} that is consistent with given rules: every
683 element of @{text R} can be seen as arising by applying a rule to
684 elements of @{text R}. An important example is using bisimulation
685 relations to formalise equivalence of processes and infinite data
688 \medskip The HOL package is related to the ZF one, which is
689 described in a separate paper,\footnote{It appeared in CADE
690 \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
691 which you should refer to in case of difficulties. The package is
692 simpler than that of ZF thanks to implicit type-checking in HOL.
693 The types of the (co)inductive predicates (or sets) determine the
694 domain of the fixedpoint definition, and the package does not have
695 to use inference rules for type-checking.
697 \begin{matharray}{rcl}
698 @{command_def (HOL) "inductive"} & : & \isarkeep{local{\dsh}theory} \\
699 @{command_def (HOL) "inductive_set"} & : & \isarkeep{local{\dsh}theory} \\
700 @{command_def (HOL) "coinductive"} & : & \isarkeep{local{\dsh}theory} \\
701 @{command_def (HOL) "coinductive_set"} & : & \isarkeep{local{\dsh}theory} \\
702 @{attribute_def (HOL) mono} & : & \isaratt \\
706 ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
707 ('where' clauses)? ('monos' thmrefs)?
709 clauses: (thmdecl? prop + '|')
711 'mono' (() | 'add' | 'del')
717 \item [@{command (HOL) "inductive"} and @{command (HOL)
718 "coinductive"}] define (co)inductive predicates from the
719 introduction rules given in the @{keyword "where"} part. The
720 optional @{keyword "for"} part contains a list of parameters of the
721 (co)inductive predicates that remain fixed throughout the
722 definition. The optional @{keyword "monos"} section contains
723 \emph{monotonicity theorems}, which are required for each operator
724 applied to a recursive set in the introduction rules. There
725 \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
726 for each premise @{text "M R\<^sub>i t"} in an introduction rule!
728 \item [@{command (HOL) "inductive_set"} and @{command (HOL)
729 "coinductive_set"}] are wrappers for to the previous commands,
730 allowing the definition of (co)inductive sets.
732 \item [@{attribute (HOL) mono}] declares monotonicity rules. These
733 rule are involved in the automated monotonicity proof of @{command
740 subsection {* Derived rules *}
743 Each (co)inductive definition @{text R} adds definitions to the
744 theory and also proves some theorems:
748 \item [@{text R.intros}] is the list of introduction rules as proven
749 theorems, for the recursive predicates (or sets). The rules are
750 also available individually, using the names given them in the
753 \item [@{text R.cases}] is the case analysis (or elimination) rule;
755 \item [@{text R.induct} or @{text R.coinduct}] is the (co)induction
760 When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
761 defined simultaneously, the list of introduction rules is called
762 @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
763 called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
764 of mutual induction rules is called @{text
765 "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
769 subsection {* Monotonicity theorems *}
772 Each theory contains a default set of theorems that are used in
773 monotonicity proofs. New rules can be added to this set via the
774 @{attribute (HOL) mono} attribute. The HOL theory @{text Inductive}
775 shows how this is done. In general, the following monotonicity
776 theorems may be added:
780 \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
781 monotonicity of inductive definitions whose introduction rules have
782 premises involving terms such as @{text "M R\<^sub>i t"}.
784 \item Monotonicity theorems for logical operators, which are of the
785 general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}. For example, in
786 the case of the operator @{text "\<or>"}, the corresponding theorem is
788 \infer{@{text "P\<^sub>1 \<or> P\<^sub>2 \<longrightarrow> Q\<^sub>1 \<or> Q\<^sub>2"}}{@{text "P\<^sub>1 \<longrightarrow> Q\<^sub>1"} & @{text "P\<^sub>2 \<longrightarrow> Q\<^sub>2"}}
791 \item De Morgan style equations for reasoning about the ``polarity''
794 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
795 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
798 \item Equations for reducing complex operators to more primitive
799 ones whose monotonicity can easily be proved, e.g.
801 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
802 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
807 %FIXME: Example of an inductive definition
811 section {* Arithmetic proof support *}
814 \begin{matharray}{rcl}
815 @{method_def (HOL) arith} & : & \isarmeth \\
816 @{method_def (HOL) arith_split} & : & \isaratt \\
819 The @{method (HOL) arith} method decides linear arithmetic problems
820 (on types @{text nat}, @{text int}, @{text real}). Any current
821 facts are inserted into the goal before running the procedure.
823 The @{method (HOL) arith_split} attribute declares case split rules
824 to be expanded before the arithmetic procedure is invoked.
826 Note that a simpler (but faster) version of arithmetic reasoning is
827 already performed by the Simplifier.
831 section {* Cases and induction: emulating tactic scripts \label{sec:hol-induct-tac} *}
834 The following important tactical tools of Isabelle/HOL have been
835 ported to Isar. These should be never used in proper proof texts!
837 \begin{matharray}{rcl}
838 @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & \isarmeth \\
839 @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & \isarmeth \\
840 @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & \isarmeth \\
841 @{command_def (HOL) "inductive_cases"} & : & \isartrans{theory}{theory} \\
845 'case\_tac' goalspec? term rule?
847 'induct\_tac' goalspec? (insts * 'and') rule?
849 'ind\_cases' (prop +) ('for' (name +)) ?
851 'inductive\_cases' (thmdecl? (prop +) + 'and')
854 rule: ('rule' ':' thmref)
860 \item [@{method (HOL) case_tac} and @{method (HOL) induct_tac}]
861 admit to reason about inductive datatypes only (unless an
862 alternative rule is given explicitly). Furthermore, @{method (HOL)
863 case_tac} does a classical case split on booleans; @{method (HOL)
864 induct_tac} allows only variables to be given as instantiation.
865 These tactic emulations feature both goal addressing and dynamic
866 instantiation. Note that named rule cases are \emph{not} provided
867 as would be by the proper @{method induct} and @{method cases} proof
868 methods (see \secref{sec:cases-induct}).
870 \item [@{method (HOL) ind_cases} and @{command (HOL)
871 "inductive_cases"}] provide an interface to the internal
872 \texttt{mk_cases} operation. Rules are simplified in an
873 unrestricted forward manner.
875 While @{method (HOL) ind_cases} is a proof method to apply the
876 result immediately as elimination rules, @{command (HOL)
877 "inductive_cases"} provides case split theorems at the theory level
878 for later use. The @{keyword "for"} argument of the @{method (HOL)
879 ind_cases} method allows to specify a list of variables that should
880 be generalized before applying the resulting rule.
886 section {* Executable code *}
889 Isabelle/Pure provides two generic frameworks to support code
890 generation from executable specifications. Isabelle/HOL
891 instantiates these mechanisms in a way that is amenable to end-user
894 One framework generates code from both functional and relational
895 programs to SML. See \cite{isabelle-HOL} for further information
896 (this actually covers the new-style theory format as well).
898 \begin{matharray}{rcl}
899 @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
900 @{command_def (HOL) "code_module"} & : & \isartrans{theory}{theory} \\
901 @{command_def (HOL) "code_library"} & : & \isartrans{theory}{theory} \\
902 @{command_def (HOL) "consts_code"} & : & \isartrans{theory}{theory} \\
903 @{command_def (HOL) "types_code"} & : & \isartrans{theory}{theory} \\
904 @{attribute_def (HOL) code} & : & \isaratt \\
911 ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
912 ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
913 'contains' ( ( name '=' term ) + | term + )
916 modespec: '(' ( name * ) ')'
919 'consts\_code' (codespec +)
922 codespec: const template attachment ?
925 'types\_code' (tycodespec +)
928 tycodespec: name template attachment ?
934 template: '(' string ')'
937 attachment: 'attach' modespec ? verblbrace text verbrbrace
946 \item [@{command (HOL) "value"}~@{text t}] evaluates and prints a
947 term using the code generator.
951 \medskip The other framework generates code from functional programs
952 (including overloading using type classes) to SML \cite{SML}, OCaml
953 \cite{OCaml} and Haskell \cite{haskell-revised-report}.
954 Conceptually, code generation is split up in three steps:
955 \emph{selection} of code theorems, \emph{translation} into an
956 abstract executable view and \emph{serialization} to a specific
957 \emph{target language}. See \cite{isabelle-codegen} for an
958 introduction on how to use it.
960 \begin{matharray}{rcl}
961 @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
962 @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
963 @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
964 @{command_def (HOL) "code_datatype"} & : & \isartrans{theory}{theory} \\
965 @{command_def (HOL) "code_const"} & : & \isartrans{theory}{theory} \\
966 @{command_def (HOL) "code_type"} & : & \isartrans{theory}{theory} \\
967 @{command_def (HOL) "code_class"} & : & \isartrans{theory}{theory} \\
968 @{command_def (HOL) "code_instance"} & : & \isartrans{theory}{theory} \\
969 @{command_def (HOL) "code_monad"} & : & \isartrans{theory}{theory} \\
970 @{command_def (HOL) "code_reserved"} & : & \isartrans{theory}{theory} \\
971 @{command_def (HOL) "code_include"} & : & \isartrans{theory}{theory} \\
972 @{command_def (HOL) "code_modulename"} & : & \isartrans{theory}{theory} \\
973 @{command_def (HOL) "code_exception"} & : & \isartrans{theory}{theory} \\
974 @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & \isarkeep{theory~|~proof} \\
975 @{attribute_def (HOL) code} & : & \isaratt \\
979 'export\_code' ( constexpr + ) ? \\
980 ( ( 'in' target ( 'module\_name' string ) ? \\
981 ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
984 'code\_thms' ( constexpr + ) ?
987 'code\_deps' ( constexpr + ) ?
993 constexpr: ( const | 'name.*' | '*' )
996 typeconstructor: nameref
1002 target: 'OCaml' | 'SML' | 'Haskell'
1005 'code\_datatype' const +
1008 'code\_const' (const + 'and') \\
1009 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1012 'code\_type' (typeconstructor + 'and') \\
1013 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1016 'code\_class' (class + 'and') \\
1018 ( ( string ('where' \\
1019 ( const ( '==' | equiv ) string ) + ) ? ) ? + 'and' ) ')' ) + )
1022 'code\_instance' (( typeconstructor '::' class ) + 'and') \\
1023 ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
1026 'code\_monad' const const target
1029 'code\_reserved' target ( string + )
1032 'code\_include' target ( string ( string | '-') )
1035 'code\_modulename' target ( ( string string ) + )
1038 'code\_exception' ( const + )
1041 syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
1044 'code' ('func' | 'inline') ( 'del' )?
1050 \item [@{command (HOL) "export_code"}] is the canonical interface
1051 for generating and serializing code: for a given list of constants,
1052 code is generated for the specified target languages. Abstract code
1053 is cached incrementally. If no constant is given, the currently
1054 cached code is serialized. If no serialization instruction is
1055 given, only abstract code is cached.
1057 Constants may be specified by giving them literally, referring to
1058 all executable contants within a certain theory by giving @{text
1059 "name.*"}, or referring to \emph{all} executable constants currently
1060 available by giving @{text "*"}.
1062 By default, for each involved theory one corresponding name space
1063 module is generated. Alternativly, a module name may be specified
1064 after the @{keyword "module_name"} keyword; then \emph{all} code is
1065 placed in this module.
1067 For \emph{SML} and \emph{OCaml}, the file specification refers to a
1068 single file; for \emph{Haskell}, it refers to a whole directory,
1069 where code is generated in multiple files reflecting the module
1070 hierarchy. The file specification ``@{text "-"}'' denotes standard
1071 output. For \emph{SML}, omitting the file specification compiles
1072 code internally in the context of the current ML session.
1074 Serializers take an optional list of arguments in parentheses. For
1075 \emph{Haskell} a module name prefix may be given using the ``@{text
1076 "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim
1077 "deriving (Read, Show)"}'' clause to each appropriate datatype
1080 \item [@{command (HOL) "code_thms"}] prints a list of theorems
1081 representing the corresponding program containing all given
1082 constants; if no constants are given, the currently cached code
1083 theorems are printed.
1085 \item [@{command (HOL) "code_deps"}] visualizes dependencies of
1086 theorems representing the corresponding program containing all given
1087 constants; if no constants are given, the currently cached code
1088 theorems are visualized.
1090 \item [@{command (HOL) "code_datatype"}] specifies a constructor set
1093 \item [@{command (HOL) "code_const"}] associates a list of constants
1094 with target-specific serializations; omitting a serialization
1095 deletes an existing serialization.
1097 \item [@{command (HOL) "code_type"}] associates a list of type
1098 constructors with target-specific serializations; omitting a
1099 serialization deletes an existing serialization.
1101 \item [@{command (HOL) "code_class"}] associates a list of classes
1102 with target-specific class names; in addition, constants associated
1103 with this class may be given target-specific names used for instance
1104 declarations; omitting a serialization deletes an existing
1105 serialization. This applies only to \emph{Haskell}.
1107 \item [@{command (HOL) "code_instance"}] declares a list of type
1108 constructor / class instance relations as ``already present'' for a
1109 given target. Omitting a ``@{text "-"}'' deletes an existing
1110 ``already present'' declaration. This applies only to
1113 \item [@{command (HOL) "code_monad"}] provides an auxiliary
1114 mechanism to generate monadic code.
1116 \item [@{command (HOL) "code_reserved"}] declares a list of names as
1117 reserved for a given target, preventing it to be shadowed by any
1120 \item [@{command (HOL) "code_include"}] adds arbitrary named content
1121 (``include'') to generated code. A as last argument ``@{text "-"}''
1122 will remove an already added ``include''.
1124 \item [@{command (HOL) "code_modulename"}] declares aliasings from
1125 one module name onto another.
1127 \item [@{command (HOL) "code_exception"}] declares constants which
1128 are not required to have a definition by a defining equations; these
1129 are mapped on exceptions instead.
1131 \item [@{attribute (HOL) code}~@{text func}] explicitly selects (or
1132 with option ``@{text "del:"}'' deselects) a defining equation for
1133 code generation. Usually packages introducing defining equations
1134 provide a resonable default setup for selection.
1136 \item [@{attribute (HOL) code}@{text inline}] declares (or with
1137 option ``@{text "del:"}'' removes) inlining theorems which are
1138 applied as rewrite rules to any defining equation during
1141 \item [@{command (HOL) "print_codesetup"}] gives an overview on
1142 selected defining equations, code generator datatypes and