5 chapter {* Isabelle/HOL \label{ch:hol} *}
7 section {* Typedef axiomatization \label{sec:hol-typedef} *}
10 \begin{matharray}{rcl}
11 @{command_def (HOL) "typedef"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
15 'typedef' altname? abstype '=' repset
18 altname: '(' (name | 'open' | 'open' name) ')'
20 abstype: typespecsorts mixfix?
22 repset: term ('morphisms' name name)?
28 \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"}
29 axiomatizes a Gordon/HOL-style type definition in the background
30 theory of the current context, depending on a non-emptiness result
31 of the set @{text A} (which needs to be proven interactively).
33 The raw type may not depend on parameters or assumptions of the
34 context --- this is logically impossible in Isabelle/HOL --- but the
35 non-emptiness property can be local, potentially resulting in
36 multiple interpretations in target contexts. Thus the established
37 bijection between the representing set @{text A} and the new type
38 @{text t} may semantically depend on local assumptions.
40 By default, @{command (HOL) "typedef"} defines both a type @{text t}
41 and a set (term constant) of the same name, unless an alternative
42 base name is given in parentheses, or the ``@{text "(open)"}''
43 declaration is used to suppress a separate constant definition
44 altogether. The injection from type to set is called @{text Rep_t},
45 its inverse @{text Abs_t} --- this may be changed via an explicit
46 @{keyword (HOL) "morphisms"} declaration.
48 Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text
49 Abs_t_inverse} provide the most basic characterization as a
50 corresponding injection/surjection pair (in both directions). Rules
51 @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly
52 more convenient view on the injectivity part, suitable for automated
53 proof tools (e.g.\ in @{attribute simp} or @{attribute iff}
54 declarations). Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and
55 @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views
56 on surjectivity; these are already declared as set or type rules for
57 the generic @{method cases} and @{method induct} methods.
59 An alternative name for the set definition (and other derived
60 entities) may be specified in parentheses; the default is to use
61 @{text t} as indicated before.
67 section {* Adhoc tuples *}
70 \begin{matharray}{rcl}
71 @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\
75 'split\_format' ((( name * ) + 'and') | ('(' 'complete' ')'))
81 \item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots>
82 \<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into
83 canonical form as specified by the arguments given; the @{text i}-th
84 collection of arguments refers to occurrences in premise @{text i}
85 of the rule. The ``@{text "(complete)"}'' option causes \emph{all}
86 arguments in function applications to be represented canonically
87 according to their tuple type structure.
89 Note that these operations tend to invent funny names for new local
90 parameters to be introduced.
96 section {* Records \label{sec:hol-record} *}
99 In principle, records merely generalize the concept of tuples, where
100 components may be addressed by labels instead of just position. The
101 logical infrastructure of records in Isabelle/HOL is slightly more
102 advanced, though, supporting truly extensible record schemes. This
103 admits operations that are polymorphic with respect to record
104 extension, yielding ``object-oriented'' effects like (single)
105 inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more
106 details on object-oriented verification and record subtyping in HOL.
110 subsection {* Basic concepts *}
113 Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
114 at the level of terms and types. The notation is as follows:
117 \begin{tabular}{l|l|l}
118 & record terms & record types \\ \hline
119 fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
120 schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
121 @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
125 \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
128 A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
129 @{text a} and field @{text y} of value @{text b}. The corresponding
130 type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
131 and @{text "b :: B"}.
133 A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
134 @{text x} and @{text y} as before, but also possibly further fields
135 as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
136 of the syntax). The improper field ``@{text "\<dots>"}'' of a record
137 scheme is called the \emph{more part}. Logically it is just a free
138 variable, which is occasionally referred to as ``row variable'' in
139 the literature. The more part of a record scheme may be
140 instantiated by zero or more further components. For example, the
141 previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
142 c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
143 Fixed records are special instances of record schemes, where
144 ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
145 element. In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
146 for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
148 \medskip Two key observations make extensible records in a simply
149 typed language like HOL work out:
153 \item the more part is internalized, as a free term or type
156 \item field names are externalized, they cannot be accessed within
157 the logic as first-class values.
161 \medskip In Isabelle/HOL record types have to be defined explicitly,
162 fixing their field names and types, and their (optional) parent
163 record. Afterwards, records may be formed using above syntax, while
164 obeying the canonical order of fields as given by their declaration.
165 The record package provides several standard operations like
166 selectors and updates. The common setup for various generic proof
167 tools enable succinct reasoning patterns. See also the Isabelle/HOL
168 tutorial \cite{isabelle-hol-book} for further instructions on using
173 subsection {* Record specifications *}
176 \begin{matharray}{rcl}
177 @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\
181 'record' typespecsorts '=' (type '+')? (constdecl +)
187 \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1
188 \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
189 derived from the optional parent record @{text "\<tau>"} by adding new
190 field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
192 The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
193 covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
194 \<alpha>\<^sub>m"}. Type constructor @{text t} has to be new, while @{text
195 \<tau>} needs to specify an instance of an existing record type. At
196 least one new field @{text "c\<^sub>i"} has to be specified.
197 Basically, field names need to belong to a unique record. This is
198 not a real restriction in practice, since fields are qualified by
199 the record name internally.
201 The parent record specification @{text \<tau>} is optional; if omitted
202 @{text t} becomes a root record. The hierarchy of all records
203 declared within a theory context forms a forest structure, i.e.\ a
204 set of trees starting with a root record each. There is no way to
205 merge multiple parent records!
207 For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
208 type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
209 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
210 "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
211 @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
218 subsection {* Record operations *}
221 Any record definition of the form presented above produces certain
222 standard operations. Selectors and updates are provided for any
223 field, including the improper one ``@{text more}''. There are also
224 cumulative record constructor functions. To simplify the
225 presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
226 \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
227 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
229 \medskip \textbf{Selectors} and \textbf{updates} are available for
230 any field (including ``@{text more}''):
232 \begin{matharray}{lll}
233 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
234 @{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>"} \\
237 There is special syntax for application of updates: @{text "r\<lparr>x :=
238 a\<rparr>"} abbreviates term @{text "x_update a r"}. Further notation for
239 repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
240 c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}. Note that
241 because of postfix notation the order of fields shown here is
242 reverse than in the actual term. Since repeated updates are just
243 function applications, fields may be freely permuted in @{text "\<lparr>x
244 := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
245 Thus commutativity of independent updates can be proven within the
246 logic for any two fields, but not as a general theorem.
248 \medskip The \textbf{make} operation provides a cumulative record
249 constructor function:
251 \begin{matharray}{lll}
252 @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
255 \medskip We now reconsider the case of non-root records, which are
256 derived of some parent. In general, the latter may depend on
257 another parent as well, resulting in a list of \emph{ancestor
258 records}. Appending the lists of fields of all ancestors results in
259 a certain field prefix. The record package automatically takes care
260 of this by lifting operations over this context of ancestor fields.
261 Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
262 fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
263 the above record operations will get the following types:
267 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
268 @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow>
269 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
270 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
271 @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
272 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
276 \noindent Some further operations address the extension aspect of a
277 derived record scheme specifically: @{text "t.fields"} produces a
278 record fragment consisting of exactly the new fields introduced here
279 (the result may serve as a more part elsewhere); @{text "t.extend"}
280 takes a fixed record and adds a given more part; @{text
281 "t.truncate"} restricts a record scheme to a fixed record.
285 @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
286 @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
287 \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
288 @{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>"} \\
292 \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
297 subsection {* Derived rules and proof tools *}
300 The record package proves several results internally, declaring
301 these facts to appropriate proof tools. This enables users to
302 reason about record structures quite conveniently. Assume that
303 @{text t} is a record type as specified above.
307 \item Standard conversions for selectors or updates applied to
308 record constructor terms are made part of the default Simplifier
309 context; thus proofs by reduction of basic operations merely require
310 the @{method simp} method without further arguments. These rules
311 are available as @{text "t.simps"}, too.
313 \item Selectors applied to updated records are automatically reduced
314 by an internal simplification procedure, which is also part of the
315 standard Simplifier setup.
317 \item Inject equations of a form analogous to @{prop "(x, y) = (x',
318 y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
319 Reasoner as @{attribute iff} rules. These rules are available as
322 \item The introduction rule for record equality analogous to @{text
323 "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
324 and as the basic rule context as ``@{attribute intro}@{text "?"}''.
325 The rule is called @{text "t.equality"}.
327 \item Representations of arbitrary record expressions as canonical
328 constructor terms are provided both in @{method cases} and @{method
329 induct} format (cf.\ the generic proof methods of the same name,
330 \secref{sec:cases-induct}). Several variations are available, for
331 fixed records, record schemes, more parts etc.
333 The generic proof methods are sufficiently smart to pick the most
334 sensible rule according to the type of the indicated record
335 expression: users just need to apply something like ``@{text "(cases
336 r)"}'' to a certain proof problem.
338 \item The derived record operations @{text "t.make"}, @{text
339 "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
340 treated automatically, but usually need to be expanded by hand,
341 using the collective fact @{text "t.defs"}.
347 section {* Datatypes \label{sec:hol-datatype} *}
350 \begin{matharray}{rcl}
351 @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\
352 @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
356 'datatype' (dtspec + 'and')
358 'rep\_datatype' ('(' (name +) ')')? (term +)
361 dtspec: parname? typespec mixfix? '=' (cons + '|')
363 cons: name ( type * ) mixfix?
368 \item @{command (HOL) "datatype"} defines inductive datatypes in
371 \item @{command (HOL) "rep_datatype"} represents existing types as
372 inductive ones, generating the standard infrastructure of derived
373 concepts (primitive recursion etc.).
377 The induction and exhaustion theorems generated provide case names
378 according to the constructors involved, while parameters are named
379 after the types (see also \secref{sec:cases-induct}).
381 See \cite{isabelle-HOL} for more details on datatypes, but beware of
382 the old-style theory syntax being used there! Apart from proper
383 proof methods for case-analysis and induction, there are also
384 emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
385 induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
386 to refer directly to the internal structure of subgoals (including
387 internally bound parameters).
391 section {* Recursive functions \label{sec:recursion} *}
394 \begin{matharray}{rcl}
395 @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
396 @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
397 @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
398 @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
402 'primrec' target? fixes 'where' equations
404 ('fun' | 'function') target? functionopts? fixes \\ 'where' equations
406 equations: (thmdecl? prop + '|')
408 functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
410 'termination' ( term )?
415 \item @{command (HOL) "primrec"} defines primitive recursive
416 functions over datatypes, see also \cite{isabelle-HOL}.
418 \item @{command (HOL) "function"} defines functions by general
419 wellfounded recursion. A detailed description with examples can be
420 found in \cite{isabelle-function}. The function is specified by a
421 set of (possibly conditional) recursive equations with arbitrary
422 pattern matching. The command generates proof obligations for the
423 completeness and the compatibility of patterns.
425 The defined function is considered partial, and the resulting
426 simplification rules (named @{text "f.psimps"}) and induction rule
427 (named @{text "f.pinduct"}) are guarded by a generated domain
428 predicate @{text "f_dom"}. The @{command (HOL) "termination"}
429 command can then be used to establish that the function is total.
431 \item @{command (HOL) "fun"} is a shorthand notation for ``@{command
432 (HOL) "function"}~@{text "(sequential)"}, followed by automated
433 proof attempts regarding pattern matching and termination. See
434 \cite{isabelle-function} for further details.
436 \item @{command (HOL) "termination"}~@{text f} commences a
437 termination proof for the previously defined function @{text f}. If
438 this is omitted, the command refers to the most recent function
439 definition. After the proof is closed, the recursive equations and
440 the induction principle is established.
444 Recursive definitions introduced by the @{command (HOL) "function"}
446 reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
447 "c.induct"} (where @{text c} is the name of the function definition)
448 refers to a specific induction rule, with parameters named according
449 to the user-specified equations. Cases are numbered (starting from 1).
451 For @{command (HOL) "primrec"}, the induction principle coincides
452 with structural recursion on the datatype the recursion is carried
455 The equations provided by these packages may be referred later as
456 theorem list @{text "f.simps"}, where @{text f} is the (collective)
457 name of the functions defined. Individual equations may be named
460 The @{command (HOL) "function"} command accepts the following
465 \item @{text sequential} enables a preprocessor which disambiguates
466 overlapping patterns by making them mutually disjoint. Earlier
467 equations take precedence over later ones. This allows to give the
468 specification in a format very similar to functional programming.
469 Note that the resulting simplification and induction rules
470 correspond to the transformed specification, not the one given
471 originally. This usually means that each equation given by the user
472 may result in several theorems. Also note that this automatic
473 transformation only works for ML-style datatype patterns.
475 \item @{text domintros} enables the automated generation of
476 introduction rules for the domain predicate. While mostly not
477 needed, they can be helpful in some proofs about partial functions.
479 \item @{text tailrec} generates the unconstrained recursive
480 equations even without a termination proof, provided that the
481 function is tail-recursive. This currently only works
483 \item @{text "default d"} allows to specify a default value for a
484 (partial) function, which will ensure that @{text "f x = d x"}
485 whenever @{text "x \<notin> f_dom"}.
491 subsection {* Proof methods related to recursive definitions *}
494 \begin{matharray}{rcl}
495 @{method_def (HOL) pat_completeness} & : & @{text method} \\
496 @{method_def (HOL) relation} & : & @{text method} \\
497 @{method_def (HOL) lexicographic_order} & : & @{text method} \\
498 @{method_def (HOL) size_change} & : & @{text method} \\
504 'lexicographic\_order' ( clasimpmod * )
506 'size\_change' ( orders ( clasimpmod * ) )
508 orders: ( 'max' | 'min' | 'ms' ) *
513 \item @{method (HOL) pat_completeness} is a specialized method to
514 solve goals regarding the completeness of pattern matching, as
515 required by the @{command (HOL) "function"} package (cf.\
516 \cite{isabelle-function}).
518 \item @{method (HOL) relation}~@{text R} introduces a termination
519 proof using the relation @{text R}. The resulting proof state will
520 contain goals expressing that @{text R} is wellfounded, and that the
521 arguments of recursive calls decrease with respect to @{text R}.
522 Usually, this method is used as the initial proof step of manual
525 \item @{method (HOL) "lexicographic_order"} attempts a fully
526 automated termination proof by searching for a lexicographic
527 combination of size measures on the arguments of the function. The
528 method accepts the same arguments as the @{method auto} method,
529 which it uses internally to prove local descents. The same context
530 modifiers as for @{method auto} are accepted, see
531 \secref{sec:clasimp}.
533 In case of failure, extensive information is printed, which can help
534 to analyse the situation (cf.\ \cite{isabelle-function}).
536 \item @{method (HOL) "size_change"} also works on termination goals,
537 using a variation of the size-change principle, together with a
538 graph decomposition technique (see \cite{krauss_phd} for details).
539 Three kinds of orders are used internally: @{text max}, @{text min},
540 and @{text ms} (multiset), which is only available when the theory
541 @{text Multiset} is loaded. When no order kinds are given, they are
542 tried in order. The search for a termination proof uses SAT solving
545 For local descent proofs, the same context modifiers as for @{method
546 auto} are accepted, see \secref{sec:clasimp}.
552 subsection {* Old-style recursive function definitions (TFL) *}
555 The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
556 "recdef_tc"} for defining recursive are mostly obsolete; @{command
557 (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
559 \begin{matharray}{rcl}
560 @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
561 @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
565 'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
569 hints: '(' 'hints' ( recdefmod * ) ')'
571 recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
573 tc: nameref ('(' nat ')')?
579 \item @{command (HOL) "recdef"} defines general well-founded
580 recursive functions (using the TFL package), see also
581 \cite{isabelle-HOL}. The ``@{text "(permissive)"}'' option tells
582 TFL to recover from failed proof attempts, returning unfinished
583 results. The @{text recdef_simp}, @{text recdef_cong}, and @{text
584 recdef_wf} hints refer to auxiliary rules to be used in the internal
585 automated proof process of TFL. Additional @{syntax clasimpmod}
586 declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
587 context of the Simplifier (cf.\ \secref{sec:simplifier}) and
588 Classical reasoner (cf.\ \secref{sec:classical}).
590 \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
591 proof for leftover termination condition number @{text i} (default
592 1) as generated by a @{command (HOL) "recdef"} definition of
595 Note that in most cases, @{command (HOL) "recdef"} is able to finish
596 its internal proofs without manual intervention.
600 \medskip Hints for @{command (HOL) "recdef"} may be also declared
601 globally, using the following attributes.
603 \begin{matharray}{rcl}
604 @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
605 @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
606 @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
610 ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
616 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
619 An \textbf{inductive definition} specifies the least predicate (or
620 set) @{text R} closed under given rules: applying a rule to elements
621 of @{text R} yields a result within @{text R}. For example, a
622 structural operational semantics is an inductive definition of an
625 Dually, a \textbf{coinductive definition} specifies the greatest
626 predicate~/ set @{text R} that is consistent with given rules: every
627 element of @{text R} can be seen as arising by applying a rule to
628 elements of @{text R}. An important example is using bisimulation
629 relations to formalise equivalence of processes and infinite data
632 \medskip The HOL package is related to the ZF one, which is
633 described in a separate paper,\footnote{It appeared in CADE
634 \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
635 which you should refer to in case of difficulties. The package is
636 simpler than that of ZF thanks to implicit type-checking in HOL.
637 The types of the (co)inductive predicates (or sets) determine the
638 domain of the fixedpoint definition, and the package does not have
639 to use inference rules for type-checking.
641 \begin{matharray}{rcl}
642 @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
643 @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
644 @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
645 @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
646 @{attribute_def (HOL) mono} & : & @{text attribute} \\
650 ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
651 ('where' clauses)? ('monos' thmrefs)?
653 clauses: (thmdecl? prop + '|')
655 'mono' (() | 'add' | 'del')
661 \item @{command (HOL) "inductive"} and @{command (HOL)
662 "coinductive"} define (co)inductive predicates from the
663 introduction rules given in the @{keyword "where"} part. The
664 optional @{keyword "for"} part contains a list of parameters of the
665 (co)inductive predicates that remain fixed throughout the
666 definition. The optional @{keyword "monos"} section contains
667 \emph{monotonicity theorems}, which are required for each operator
668 applied to a recursive set in the introduction rules. There
669 \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
670 for each premise @{text "M R\<^sub>i t"} in an introduction rule!
672 \item @{command (HOL) "inductive_set"} and @{command (HOL)
673 "coinductive_set"} are wrappers for to the previous commands,
674 allowing the definition of (co)inductive sets.
676 \item @{attribute (HOL) mono} declares monotonicity rules. These
677 rule are involved in the automated monotonicity proof of @{command
684 subsection {* Derived rules *}
687 Each (co)inductive definition @{text R} adds definitions to the
688 theory and also proves some theorems:
692 \item @{text R.intros} is the list of introduction rules as proven
693 theorems, for the recursive predicates (or sets). The rules are
694 also available individually, using the names given them in the
697 \item @{text R.cases} is the case analysis (or elimination) rule;
699 \item @{text R.induct} or @{text R.coinduct} is the (co)induction
704 When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
705 defined simultaneously, the list of introduction rules is called
706 @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
707 called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
708 of mutual induction rules is called @{text
709 "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
713 subsection {* Monotonicity theorems *}
716 Each theory contains a default set of theorems that are used in
717 monotonicity proofs. New rules can be added to this set via the
718 @{attribute (HOL) mono} attribute. The HOL theory @{text Inductive}
719 shows how this is done. In general, the following monotonicity
720 theorems may be added:
724 \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
725 monotonicity of inductive definitions whose introduction rules have
726 premises involving terms such as @{text "M R\<^sub>i t"}.
728 \item Monotonicity theorems for logical operators, which are of the
729 general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}. For example, in
730 the case of the operator @{text "\<or>"}, the corresponding theorem is
732 \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"}}
735 \item De Morgan style equations for reasoning about the ``polarity''
738 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
739 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
742 \item Equations for reducing complex operators to more primitive
743 ones whose monotonicity can easily be proved, e.g.
745 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
746 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
751 %FIXME: Example of an inductive definition
755 section {* Arithmetic proof support *}
758 \begin{matharray}{rcl}
759 @{method_def (HOL) arith} & : & @{text method} \\
760 @{attribute_def (HOL) arith} & : & @{text attribute} \\
761 @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
764 The @{method (HOL) arith} method decides linear arithmetic problems
765 (on types @{text nat}, @{text int}, @{text real}). Any current
766 facts are inserted into the goal before running the procedure.
768 The @{attribute (HOL) arith} attribute declares facts that are
769 always supplied to the arithmetic provers implicitly.
771 The @{attribute (HOL) arith_split} attribute declares case split
772 rules to be expanded before @{method (HOL) arith} is invoked.
774 Note that a simpler (but faster) arithmetic prover is
775 already invoked by the Simplifier.
779 section {* Intuitionistic proof search *}
782 \begin{matharray}{rcl}
783 @{method_def (HOL) iprover} & : & @{text method} \\
787 'iprover' ( rulemod * )
791 The @{method (HOL) iprover} method performs intuitionistic proof
792 search, depending on specifically declared rules from the context,
793 or given as explicit arguments. Chained facts are inserted into the
794 goal before commencing proof search.
796 Rules need to be classified as @{attribute (Pure) intro},
797 @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
798 ``@{text "!"}'' indicator refers to ``safe'' rules, which may be
799 applied aggressively (without considering back-tracking later).
800 Rules declared with ``@{text "?"}'' are ignored in proof search (the
801 single-step @{method rule} method still observes these). An
802 explicit weight annotation may be given as well; otherwise the
803 number of rule premises will be taken into account here.
807 section {* Coherent Logic *}
810 \begin{matharray}{rcl}
811 @{method_def (HOL) "coherent"} & : & @{text method} \\
819 The @{method (HOL) coherent} method solves problems of
820 \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
821 applications in confluence theory, lattice theory and projective
822 geometry. See @{"file" "~~/src/HOL/ex/Coherent.thy"} for some
827 section {* Checking and refuting propositions *}
830 Identifying incorrect propositions usually involves evaluation of
831 particular assignments and systematic counter example search. This
832 is supported by the following commands.
834 \begin{matharray}{rcl}
835 @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
836 @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
837 @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"}
841 'value' ( ( '[' name ']' ) ? ) modes? term
844 'quickcheck' ( ( '[' args ']' ) ? ) nat?
847 'quickcheck_params' ( ( '[' args ']' ) ? )
850 modes: '(' (name + ) ')'
853 args: ( name '=' value + ',' )
859 \item @{command (HOL) "value"}~@{text t} evaluates and prints a
860 term; optionally @{text modes} can be specified, which are
861 appended to the current print mode (see also \cite{isabelle-ref}).
862 Internally, the evaluation is performed by registered evaluators,
863 which are invoked sequentially until a result is returned.
864 Alternatively a specific evaluator can be selected using square
865 brackets; typical evaluators use the current set of code equations
866 to normalize and include @{text simp} for fully symbolic evaluation
867 using the simplifier, @{text nbe} for \emph{normalization by evaluation}
868 and \emph{code} for code generation in SML.
870 \item @{command (HOL) "quickcheck"} tests the current goal for
871 counter examples using a series of arbitrary assignments for its
872 free variables; by default the first subgoal is tested, an other
873 can be selected explicitly using an optional goal index.
874 A number of configuration options are supported for
875 @{command (HOL) "quickcheck"}, notably:
879 \item[size] specifies the maximum size of the search space for
882 \item[iterations] sets how many sets of assignments are
883 generated for each particular size.
885 \item[no\_assms] specifies whether assumptions in
886 structured proofs should be ignored.
890 These option can be given within square brackets.
892 \item @{command (HOL) "quickcheck_params"} changes quickcheck
893 configuration options persitently.
899 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
902 The following tools of Isabelle/HOL support cases analysis and
903 induction in unstructured tactic scripts; see also
904 \secref{sec:cases-induct} for proper Isar versions of similar ideas.
906 \begin{matharray}{rcl}
907 @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
908 @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
909 @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
910 @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
914 'case\_tac' goalspec? term rule?
916 'induct\_tac' goalspec? (insts * 'and') rule?
918 'ind\_cases' (prop +) ('for' (name +)) ?
920 'inductive\_cases' (thmdecl? (prop +) + 'and')
923 rule: ('rule' ':' thmref)
929 \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
930 to reason about inductive types. Rules are selected according to
931 the declarations by the @{attribute cases} and @{attribute induct}
932 attributes, cf.\ \secref{sec:cases-induct}. The @{command (HOL)
933 datatype} package already takes care of this.
935 These unstructured tactics feature both goal addressing and dynamic
936 instantiation. Note that named rule cases are \emph{not} provided
937 as would be by the proper @{method cases} and @{method induct} proof
938 methods (see \secref{sec:cases-induct}). Unlike the @{method
939 induct} method, @{method induct_tac} does not handle structured rule
940 statements, only the compact object-logic conclusion of the subgoal
943 \item @{method (HOL) ind_cases} and @{command (HOL)
944 "inductive_cases"} provide an interface to the internal @{ML_text
945 mk_cases} operation. Rules are simplified in an unrestricted
948 While @{method (HOL) ind_cases} is a proof method to apply the
949 result immediately as elimination rules, @{command (HOL)
950 "inductive_cases"} provides case split theorems at the theory level
951 for later use. The @{keyword "for"} argument of the @{method (HOL)
952 ind_cases} method allows to specify a list of variables that should
953 be generalized before applying the resulting rule.
959 section {* Executable code *}
962 Isabelle/Pure provides two generic frameworks to support code
963 generation from executable specifications. Isabelle/HOL
964 instantiates these mechanisms in a way that is amenable to end-user
967 \medskip One framework generates code from functional programs
968 (including overloading using type classes) to SML \cite{SML}, OCaml
969 \cite{OCaml}, Haskell \cite{haskell-revised-report} and Scala
970 \cite{scala-overview-tech-report}.
971 Conceptually, code generation is split up in three steps:
972 \emph{selection} of code theorems, \emph{translation} into an
973 abstract executable view and \emph{serialization} to a specific
974 \emph{target language}. Inductive specifications can be executed
975 using the predicate compiler which operates within HOL.
976 See \cite{isabelle-codegen} for an introduction.
978 \begin{matharray}{rcl}
979 @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
980 @{attribute_def (HOL) code} & : & @{text attribute} \\
981 @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
982 @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
983 @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
984 @{attribute_def (HOL) code_inline} & : & @{text attribute} \\
985 @{attribute_def (HOL) code_post} & : & @{text attribute} \\
986 @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
987 @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
988 @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
989 @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
990 @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
991 @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
992 @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
993 @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
994 @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
995 @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
996 @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
997 @{command_def (HOL) "code_reflect"} & : & @{text "theory \<rightarrow> theory"}
1001 'export\_code' ( constexpr + ) \\
1002 ( ( 'in' target ( 'module\_name' string ) ? \\
1003 ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
1009 constexpr: ( const | 'name.*' | '*' )
1012 typeconstructor: nameref
1018 target: 'SML' | 'OCaml' | 'Haskell' | 'Scala'
1021 'code' ( 'del' | 'abstype' | 'abstract' ) ?
1024 'code\_abort' ( const + )
1027 'code\_datatype' ( const + )
1030 'code_inline' ( 'del' ) ?
1033 'code_post' ( 'del' ) ?
1036 'code\_thms' ( constexpr + ) ?
1039 'code\_deps' ( constexpr + ) ?
1042 'code\_const' (const + 'and') \\
1043 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1046 'code\_type' (typeconstructor + 'and') \\
1047 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1050 'code\_class' (class + 'and') \\
1051 ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
1054 'code\_instance' (( typeconstructor '::' class ) + 'and') \\
1055 ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
1058 'code\_reserved' target ( string + )
1061 'code\_monad' const const target
1064 'code\_include' target ( string ( string | '-') )
1067 'code\_modulename' target ( ( string string ) + )
1070 'code\_reflect' string ( 'datatypes' ( string '=' ( string + '|' ) + 'and' ) ) ? \\
1071 ( 'functions' ( string + ) ) ? ( 'file' string ) ?
1074 syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
1081 \item @{command (HOL) "export_code"} generates code for a given list
1082 of constants in the specified target language(s). If no
1083 serialization instruction is given, only abstract code is generated
1086 Constants may be specified by giving them literally, referring to
1087 all executable contants within a certain theory by giving @{text
1088 "name.*"}, or referring to \emph{all} executable constants currently
1089 available by giving @{text "*"}.
1091 By default, for each involved theory one corresponding name space
1092 module is generated. Alternativly, a module name may be specified
1093 after the @{keyword "module_name"} keyword; then \emph{all} code is
1094 placed in this module.
1096 For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification
1097 refers to a single file; for \emph{Haskell}, it refers to a whole
1098 directory, where code is generated in multiple files reflecting the
1099 module hierarchy. Omitting the file specification denotes standard
1102 Serializers take an optional list of arguments in parentheses. For
1103 \emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits
1104 explicit module signatures.
1106 For \emph{Haskell} a module name prefix may be given using the
1107 ``@{text "root:"}'' argument; ``@{text string_classes}'' adds a
1108 ``@{verbatim "deriving (Read, Show)"}'' clause to each appropriate
1109 datatype declaration.
1111 \item @{attribute (HOL) code} explicitly selects (or with option
1112 ``@{text "del"}'' deselects) a code equation for code generation.
1113 Usually packages introducing code equations provide a reasonable
1114 default setup for selection. Variants @{text "code abstype"} and
1115 @{text "code abstract"} declare abstract datatype certificates or
1116 code equations on abstract datatype representations respectively.
1118 \item @{command (HOL) "code_abort"} declares constants which are not
1119 required to have a definition by means of code equations; if needed
1120 these are implemented by program abort instead.
1122 \item @{command (HOL) "code_datatype"} specifies a constructor set
1125 \item @{command (HOL) "print_codesetup"} gives an overview on
1126 selected code equations and code generator datatypes.
1128 \item @{attribute (HOL) code_inline} declares (or with option
1129 ``@{text "del"}'' removes) inlining theorems which are applied as
1130 rewrite rules to any code equation during preprocessing.
1132 \item @{attribute (HOL) code_post} declares (or with option ``@{text
1133 "del"}'' removes) theorems which are applied as rewrite rules to any
1134 result of an evaluation.
1136 \item @{command (HOL) "print_codeproc"} prints the setup of the code
1137 generator preprocessor.
1139 \item @{command (HOL) "code_thms"} prints a list of theorems
1140 representing the corresponding program containing all given
1141 constants after preprocessing.
1143 \item @{command (HOL) "code_deps"} visualizes dependencies of
1144 theorems representing the corresponding program containing all given
1145 constants after preprocessing.
1147 \item @{command (HOL) "code_const"} associates a list of constants
1148 with target-specific serializations; omitting a serialization
1149 deletes an existing serialization.
1151 \item @{command (HOL) "code_type"} associates a list of type
1152 constructors with target-specific serializations; omitting a
1153 serialization deletes an existing serialization.
1155 \item @{command (HOL) "code_class"} associates a list of classes
1156 with target-specific class names; omitting a serialization deletes
1157 an existing serialization. This applies only to \emph{Haskell}.
1159 \item @{command (HOL) "code_instance"} declares a list of type
1160 constructor / class instance relations as ``already present'' for a
1161 given target. Omitting a ``@{text "-"}'' deletes an existing
1162 ``already present'' declaration. This applies only to
1165 \item @{command (HOL) "code_reserved"} declares a list of names as
1166 reserved for a given target, preventing it to be shadowed by any
1169 \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
1170 to generate monadic code for Haskell.
1172 \item @{command (HOL) "code_include"} adds arbitrary named content
1173 (``include'') to generated code. A ``@{text "-"}'' as last argument
1174 will remove an already added ``include''.
1176 \item @{command (HOL) "code_modulename"} declares aliasings from one
1177 module name onto another.
1179 \item @{command (HOL) "code_reflect"} without a ``@{text "file"}''
1180 argument compiles code into the system runtime environment and
1181 modifies the code generator setup that future invocations of system
1182 runtime code generation referring to one of the ``@{text
1183 "datatypes"}'' or ``@{text "functions"}'' entities use these precompiled
1184 entities. With a ``@{text "file"}'' argument, the corresponding code
1185 is generated into that specified file without modifying the code
1190 The other framework generates code from both functional and
1191 relational programs to SML. See \cite{isabelle-HOL} for further
1192 information (this actually covers the new-style theory format as
1195 \begin{matharray}{rcl}
1196 @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
1197 @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
1198 @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
1199 @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\
1200 @{attribute_def (HOL) code} & : & @{text attribute} \\
1204 ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
1205 ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
1206 'contains' ( ( name '=' term ) + | term + )
1209 modespec: '(' ( name * ) ')'
1212 'consts\_code' (codespec +)
1215 codespec: const template attachment ?
1218 'types\_code' (tycodespec +)
1221 tycodespec: name template attachment ?
1227 template: '(' string ')'
1230 attachment: 'attach' modespec ? verblbrace text verbrbrace
1240 section {* Definition by specification \label{sec:hol-specification} *}
1243 \begin{matharray}{rcl}
1244 @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1245 @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1249 ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
1251 decl: ((name ':')? term '(' 'overloaded' ')'?)
1256 \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
1257 goal stating the existence of terms with the properties specified to
1258 hold for the constants given in @{text decls}. After finishing the
1259 proof, the theory will be augmented with definitions for the given
1260 constants, as well as with theorems stating the properties for these
1263 \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
1264 a goal stating the existence of terms with the properties specified
1265 to hold for the constants given in @{text decls}. After finishing
1266 the proof, the theory will be augmented with axioms expressing the
1267 properties given in the first place.
1269 \item @{text decl} declares a constant to be defined by the
1270 specification given. The definition for the constant @{text c} is
1271 bound to the name @{text c_def} unless a theorem name is given in
1272 the declaration. Overloaded constants should be declared as such.
1276 Whether to use @{command (HOL) "specification"} or @{command (HOL)
1277 "ax_specification"} is to some extent a matter of style. @{command
1278 (HOL) "specification"} introduces no new axioms, and so by
1279 construction cannot introduce inconsistencies, whereas @{command
1280 (HOL) "ax_specification"} does introduce axioms, but only after the
1281 user has explicitly proven it to be safe. A practical issue must be
1282 considered, though: After introducing two constants with the same
1283 properties using @{command (HOL) "specification"}, one can prove
1284 that the two constants are, in fact, equal. If this might be a
1285 problem, one should use @{command (HOL) "ax_specification"}.