5 chapter {* Isabelle/HOL \label{ch:hol} *}
7 section {* Primitive types \label{sec:hol-typedef} *}
10 \begin{matharray}{rcl}
11 @{command_def (HOL) "typedecl"} & : & @{text "theory \<rightarrow> theory"} \\
12 @{command_def (HOL) "typedef"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
16 'typedecl' typespec infix?
18 'typedef' altname? abstype '=' repset
21 altname: '(' (name | 'open' | 'open' name) ')'
23 abstype: typespec infix?
25 repset: term ('morphisms' name name)?
31 \item @{command (HOL) "typedecl"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t"} is similar
32 to the original @{command "typedecl"} of Isabelle/Pure (see
33 \secref{sec:types-pure}), but also declares type arity @{text "t ::
34 (type, \<dots>, type) type"}, making @{text t} an actual HOL type
35 constructor. %FIXME check, update
37 \item @{command (HOL) "typedef"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>n) t = A"} sets up
38 a goal stating non-emptiness of the set @{text A}. After finishing
39 the proof, the theory will be augmented by a Gordon/HOL-style type
40 definition, which establishes a bijection between the representing
41 set @{text A} and the new type @{text t}.
43 Technically, @{command (HOL) "typedef"} defines both a type @{text
44 t} and a set (term constant) of the same name (an alternative base
45 name may be given in parentheses). The injection from type to set
46 is called @{text Rep_t}, its inverse @{text Abs_t} (this may be
47 changed via an explicit @{keyword (HOL) "morphisms"} declaration).
49 Theorems @{text Rep_t}, @{text Rep_t_inverse}, and @{text
50 Abs_t_inverse} provide the most basic characterization as a
51 corresponding injection/surjection pair (in both directions). Rules
52 @{text Rep_t_inject} and @{text Abs_t_inject} provide a slightly
53 more convenient view on the injectivity part, suitable for automated
54 proof tools (e.g.\ in @{attribute simp} or @{attribute iff}
55 declarations). Rules @{text Rep_t_cases}/@{text Rep_t_induct}, and
56 @{text Abs_t_cases}/@{text Abs_t_induct} provide alternative views
57 on surjectivity; these are already declared as set or type rules for
58 the generic @{method cases} and @{method induct} methods.
60 An alternative name may be specified in parentheses; the default is
61 to use @{text t} as indicated before. The ``@{text "(open)"}''
62 declaration suppresses a separate constant definition for the
67 Note that raw type declarations are rarely used in practice; the
68 main application is with experimental (or even axiomatic!) theory
69 fragments. Instead of primitive HOL type definitions, user-level
70 theories usually refer to higher-level packages such as @{command
71 (HOL) "record"} (see \secref{sec:hol-record}) or @{command (HOL)
72 "datatype"} (see \secref{sec:hol-datatype}).
76 section {* Adhoc tuples *}
79 \begin{matharray}{rcl}
80 @{attribute (HOL) split_format}@{text "\<^sup>*"} & : & @{text attribute} \\
84 'split\_format' (((name *) + 'and') | ('(' 'complete' ')'))
90 \item @{attribute (HOL) split_format}~@{text "p\<^sub>1 \<dots> p\<^sub>m \<AND> \<dots>
91 \<AND> q\<^sub>1 \<dots> q\<^sub>n"} puts expressions of low-level tuple types into
92 canonical form as specified by the arguments given; the @{text i}-th
93 collection of arguments refers to occurrences in premise @{text i}
94 of the rule. The ``@{text "(complete)"}'' option causes \emph{all}
95 arguments in function applications to be represented canonically
96 according to their tuple type structure.
98 Note that these operations tend to invent funny names for new local
99 parameters to be introduced.
105 section {* Records \label{sec:hol-record} *}
108 In principle, records merely generalize the concept of tuples, where
109 components may be addressed by labels instead of just position. The
110 logical infrastructure of records in Isabelle/HOL is slightly more
111 advanced, though, supporting truly extensible record schemes. This
112 admits operations that are polymorphic with respect to record
113 extension, yielding ``object-oriented'' effects like (single)
114 inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more
115 details on object-oriented verification and record subtyping in HOL.
119 subsection {* Basic concepts *}
122 Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
123 at the level of terms and types. The notation is as follows:
126 \begin{tabular}{l|l|l}
127 & record terms & record types \\ \hline
128 fixed & @{text "\<lparr>x = a, y = b\<rparr>"} & @{text "\<lparr>x :: A, y :: B\<rparr>"} \\
129 schematic & @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} &
130 @{text "\<lparr>x :: A, y :: B, \<dots> :: M\<rparr>"} \\
134 \noindent The ASCII representation of @{text "\<lparr>x = a\<rparr>"} is @{text
137 A fixed record @{text "\<lparr>x = a, y = b\<rparr>"} has field @{text x} of value
138 @{text a} and field @{text y} of value @{text b}. The corresponding
139 type is @{text "\<lparr>x :: A, y :: B\<rparr>"}, assuming that @{text "a :: A"}
140 and @{text "b :: B"}.
142 A record scheme like @{text "\<lparr>x = a, y = b, \<dots> = m\<rparr>"} contains fields
143 @{text x} and @{text y} as before, but also possibly further fields
144 as indicated by the ``@{text "\<dots>"}'' notation (which is actually part
145 of the syntax). The improper field ``@{text "\<dots>"}'' of a record
146 scheme is called the \emph{more part}. Logically it is just a free
147 variable, which is occasionally referred to as ``row variable'' in
148 the literature. The more part of a record scheme may be
149 instantiated by zero or more further components. For example, the
150 previous scheme may get instantiated to @{text "\<lparr>x = a, y = b, z =
151 c, \<dots> = m'\<rparr>"}, where @{text m'} refers to a different more part.
152 Fixed records are special instances of record schemes, where
153 ``@{text "\<dots>"}'' is properly terminated by the @{text "() :: unit"}
154 element. In fact, @{text "\<lparr>x = a, y = b\<rparr>"} is just an abbreviation
155 for @{text "\<lparr>x = a, y = b, \<dots> = ()\<rparr>"}.
157 \medskip Two key observations make extensible records in a simply
158 typed language like HOL work out:
162 \item the more part is internalized, as a free term or type
165 \item field names are externalized, they cannot be accessed within
166 the logic as first-class values.
170 \medskip In Isabelle/HOL record types have to be defined explicitly,
171 fixing their field names and types, and their (optional) parent
172 record. Afterwards, records may be formed using above syntax, while
173 obeying the canonical order of fields as given by their declaration.
174 The record package provides several standard operations like
175 selectors and updates. The common setup for various generic proof
176 tools enable succinct reasoning patterns. See also the Isabelle/HOL
177 tutorial \cite{isabelle-hol-book} for further instructions on using
182 subsection {* Record specifications *}
185 \begin{matharray}{rcl}
186 @{command_def (HOL) "record"} & : & @{text "theory \<rightarrow> theory"} \\
190 'record' typespec '=' (type '+')? (constdecl +)
196 \item @{command (HOL) "record"}~@{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t = \<tau> + c\<^sub>1 :: \<sigma>\<^sub>1
197 \<dots> c\<^sub>n :: \<sigma>\<^sub>n"} defines extensible record type @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"},
198 derived from the optional parent record @{text "\<tau>"} by adding new
199 field components @{text "c\<^sub>i :: \<sigma>\<^sub>i"} etc.
201 The type variables of @{text "\<tau>"} and @{text "\<sigma>\<^sub>i"} need to be
202 covered by the (distinct) parameters @{text "\<alpha>\<^sub>1, \<dots>,
203 \<alpha>\<^sub>m"}. Type constructor @{text t} has to be new, while @{text
204 \<tau>} needs to specify an instance of an existing record type. At
205 least one new field @{text "c\<^sub>i"} has to be specified.
206 Basically, field names need to belong to a unique record. This is
207 not a real restriction in practice, since fields are qualified by
208 the record name internally.
210 The parent record specification @{text \<tau>} is optional; if omitted
211 @{text t} becomes a root record. The hierarchy of all records
212 declared within a theory context forms a forest structure, i.e.\ a
213 set of trees starting with a root record each. There is no way to
214 merge multiple parent records!
216 For convenience, @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} is made a
217 type abbreviation for the fixed record type @{text "\<lparr>c\<^sub>1 ::
218 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n\<rparr>"}, likewise is @{text
219 "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m, \<zeta>) t_scheme"} made an abbreviation for
220 @{text "\<lparr>c\<^sub>1 :: \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n, \<dots> ::
227 subsection {* Record operations *}
230 Any record definition of the form presented above produces certain
231 standard operations. Selectors and updates are provided for any
232 field, including the improper one ``@{text more}''. There are also
233 cumulative record constructor functions. To simplify the
234 presentation below, we assume for now that @{text "(\<alpha>\<^sub>1, \<dots>,
235 \<alpha>\<^sub>m) t"} is a root record with fields @{text "c\<^sub>1 ::
236 \<sigma>\<^sub>1, \<dots>, c\<^sub>n :: \<sigma>\<^sub>n"}.
238 \medskip \textbf{Selectors} and \textbf{updates} are available for
239 any field (including ``@{text more}''):
241 \begin{matharray}{lll}
242 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
243 @{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>"} \\
246 There is special syntax for application of updates: @{text "r\<lparr>x :=
247 a\<rparr>"} abbreviates term @{text "x_update a r"}. Further notation for
248 repeated updates is also available: @{text "r\<lparr>x := a\<rparr>\<lparr>y := b\<rparr>\<lparr>z :=
249 c\<rparr>"} may be written @{text "r\<lparr>x := a, y := b, z := c\<rparr>"}. Note that
250 because of postfix notation the order of fields shown here is
251 reverse than in the actual term. Since repeated updates are just
252 function applications, fields may be freely permuted in @{text "\<lparr>x
253 := a, y := b, z := c\<rparr>"}, as far as logical equality is concerned.
254 Thus commutativity of independent updates can be proven within the
255 logic for any two fields, but not as a general theorem.
257 \medskip The \textbf{make} operation provides a cumulative record
258 constructor function:
260 \begin{matharray}{lll}
261 @{text "t.make"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
264 \medskip We now reconsider the case of non-root records, which are
265 derived of some parent. In general, the latter may depend on
266 another parent as well, resulting in a list of \emph{ancestor
267 records}. Appending the lists of fields of all ancestors results in
268 a certain field prefix. The record package automatically takes care
269 of this by lifting operations over this context of ancestor fields.
270 Assuming that @{text "(\<alpha>\<^sub>1, \<dots>, \<alpha>\<^sub>m) t"} has ancestor
271 fields @{text "b\<^sub>1 :: \<rho>\<^sub>1, \<dots>, b\<^sub>k :: \<rho>\<^sub>k"},
272 the above record operations will get the following types:
276 @{text "c\<^sub>i"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow> \<sigma>\<^sub>i"} \\
277 @{text "c\<^sub>i_update"} & @{text "::"} & @{text "\<sigma>\<^sub>i \<Rightarrow>
278 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr> \<Rightarrow>
279 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
280 @{text "t.make"} & @{text "::"} & @{text "\<rho>\<^sub>1 \<Rightarrow> \<dots> \<rho>\<^sub>k \<Rightarrow> \<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow>
281 \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
285 \noindent Some further operations address the extension aspect of a
286 derived record scheme specifically: @{text "t.fields"} produces a
287 record fragment consisting of exactly the new fields introduced here
288 (the result may serve as a more part elsewhere); @{text "t.extend"}
289 takes a fixed record and adds a given more part; @{text
290 "t.truncate"} restricts a record scheme to a fixed record.
294 @{text "t.fields"} & @{text "::"} & @{text "\<sigma>\<^sub>1 \<Rightarrow> \<dots> \<sigma>\<^sub>n \<Rightarrow> \<lparr>\<^vec>c :: \<^vec>\<sigma>\<rparr>"} \\
295 @{text "t.extend"} & @{text "::"} & @{text "\<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>\<rparr> \<Rightarrow>
296 \<zeta> \<Rightarrow> \<lparr>\<^vec>b :: \<^vec>\<rho>, \<^vec>c :: \<^vec>\<sigma>, \<dots> :: \<zeta>\<rparr>"} \\
297 @{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>"} \\
301 \noindent Note that @{text "t.make"} and @{text "t.fields"} coincide
306 subsection {* Derived rules and proof tools *}
309 The record package proves several results internally, declaring
310 these facts to appropriate proof tools. This enables users to
311 reason about record structures quite conveniently. Assume that
312 @{text t} is a record type as specified above.
316 \item Standard conversions for selectors or updates applied to
317 record constructor terms are made part of the default Simplifier
318 context; thus proofs by reduction of basic operations merely require
319 the @{method simp} method without further arguments. These rules
320 are available as @{text "t.simps"}, too.
322 \item Selectors applied to updated records are automatically reduced
323 by an internal simplification procedure, which is also part of the
324 standard Simplifier setup.
326 \item Inject equations of a form analogous to @{prop "(x, y) = (x',
327 y') \<equiv> x = x' \<and> y = y'"} are declared to the Simplifier and Classical
328 Reasoner as @{attribute iff} rules. These rules are available as
331 \item The introduction rule for record equality analogous to @{text
332 "x r = x r' \<Longrightarrow> y r = y r' \<dots> \<Longrightarrow> r = r'"} is declared to the Simplifier,
333 and as the basic rule context as ``@{attribute intro}@{text "?"}''.
334 The rule is called @{text "t.equality"}.
336 \item Representations of arbitrary record expressions as canonical
337 constructor terms are provided both in @{method cases} and @{method
338 induct} format (cf.\ the generic proof methods of the same name,
339 \secref{sec:cases-induct}). Several variations are available, for
340 fixed records, record schemes, more parts etc.
342 The generic proof methods are sufficiently smart to pick the most
343 sensible rule according to the type of the indicated record
344 expression: users just need to apply something like ``@{text "(cases
345 r)"}'' to a certain proof problem.
347 \item The derived record operations @{text "t.make"}, @{text
348 "t.fields"}, @{text "t.extend"}, @{text "t.truncate"} are \emph{not}
349 treated automatically, but usually need to be expanded by hand,
350 using the collective fact @{text "t.defs"}.
356 section {* Datatypes \label{sec:hol-datatype} *}
359 \begin{matharray}{rcl}
360 @{command_def (HOL) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\
361 @{command_def (HOL) "rep_datatype"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
365 'datatype' (dtspec + 'and')
367 'rep\_datatype' ('(' (name +) ')')? (term +)
370 dtspec: parname? typespec infix? '=' (cons + '|')
372 cons: name (type *) mixfix?
377 \item @{command (HOL) "datatype"} defines inductive datatypes in
380 \item @{command (HOL) "rep_datatype"} represents existing types as
381 inductive ones, generating the standard infrastructure of derived
382 concepts (primitive recursion etc.).
386 The induction and exhaustion theorems generated provide case names
387 according to the constructors involved, while parameters are named
388 after the types (see also \secref{sec:cases-induct}).
390 See \cite{isabelle-HOL} for more details on datatypes, but beware of
391 the old-style theory syntax being used there! Apart from proper
392 proof methods for case-analysis and induction, there are also
393 emulations of ML tactics @{method (HOL) case_tac} and @{method (HOL)
394 induct_tac} available, see \secref{sec:hol-induct-tac}; these admit
395 to refer directly to the internal structure of subgoals (including
396 internally bound parameters).
400 section {* Recursive functions \label{sec:recursion} *}
403 \begin{matharray}{rcl}
404 @{command_def (HOL) "primrec"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
405 @{command_def (HOL) "fun"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
406 @{command_def (HOL) "function"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
407 @{command_def (HOL) "termination"} & : & @{text "local_theory \<rightarrow> proof(prove)"} \\
411 'primrec' target? fixes 'where' equations
413 equations: (thmdecl? prop + '|')
415 ('fun' | 'function') target? functionopts? fixes 'where' clauses
417 clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
419 functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
421 'termination' ( term )?
426 \item @{command (HOL) "primrec"} defines primitive recursive
427 functions over datatypes, see also \cite{isabelle-HOL}.
429 \item @{command (HOL) "function"} defines functions by general
430 wellfounded recursion. A detailed description with examples can be
431 found in \cite{isabelle-function}. The function is specified by a
432 set of (possibly conditional) recursive equations with arbitrary
433 pattern matching. The command generates proof obligations for the
434 completeness and the compatibility of patterns.
436 The defined function is considered partial, and the resulting
437 simplification rules (named @{text "f.psimps"}) and induction rule
438 (named @{text "f.pinduct"}) are guarded by a generated domain
439 predicate @{text "f_dom"}. The @{command (HOL) "termination"}
440 command can then be used to establish that the function is total.
442 \item @{command (HOL) "fun"} is a shorthand notation for ``@{command
443 (HOL) "function"}~@{text "(sequential)"}, followed by automated
444 proof attempts regarding pattern matching and termination. See
445 \cite{isabelle-function} for further details.
447 \item @{command (HOL) "termination"}~@{text f} commences a
448 termination proof for the previously defined function @{text f}. If
449 this is omitted, the command refers to the most recent function
450 definition. After the proof is closed, the recursive equations and
451 the induction principle is established.
457 Recursive definitions introduced by the @{command (HOL) "function"}
459 reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
460 "c.induct"} (where @{text c} is the name of the function definition)
461 refers to a specific induction rule, with parameters named according
462 to the user-specified equations.
463 For the @{command (HOL) "primrec"} the induction principle coincides
464 with structural recursion on the datatype the recursion is carried
466 Case names of @{command (HOL)
467 "primrec"} are that of the datatypes involved, while those of
468 @{command (HOL) "function"} are numbered (starting from 1).
470 The equations provided by these packages may be referred later as
471 theorem list @{text "f.simps"}, where @{text f} is the (collective)
472 name of the functions defined. Individual equations may be named
475 The @{command (HOL) "function"} command accepts the following
480 \item @{text sequential} enables a preprocessor which disambiguates
481 overlapping patterns by making them mutually disjoint. Earlier
482 equations take precedence over later ones. This allows to give the
483 specification in a format very similar to functional programming.
484 Note that the resulting simplification and induction rules
485 correspond to the transformed specification, not the one given
486 originally. This usually means that each equation given by the user
487 may result in several theroems. Also note that this automatic
488 transformation only works for ML-style datatype patterns.
490 \item @{text domintros} enables the automated generation of
491 introduction rules for the domain predicate. While mostly not
492 needed, they can be helpful in some proofs about partial functions.
494 \item @{text tailrec} generates the unconstrained recursive
495 equations even without a termination proof, provided that the
496 function is tail-recursive. This currently only works
498 \item @{text "default d"} allows to specify a default value for a
499 (partial) function, which will ensure that @{text "f x = d x"}
500 whenever @{text "x \<notin> f_dom"}.
506 subsection {* Proof methods related to recursive definitions *}
509 \begin{matharray}{rcl}
510 @{method_def (HOL) pat_completeness} & : & @{text method} \\
511 @{method_def (HOL) relation} & : & @{text method} \\
512 @{method_def (HOL) lexicographic_order} & : & @{text method} \\
518 'lexicographic\_order' (clasimpmod *)
524 \item @{method (HOL) pat_completeness} is a specialized method to
525 solve goals regarding the completeness of pattern matching, as
526 required by the @{command (HOL) "function"} package (cf.\
527 \cite{isabelle-function}).
529 \item @{method (HOL) relation}~@{text R} introduces a termination
530 proof using the relation @{text R}. The resulting proof state will
531 contain goals expressing that @{text R} is wellfounded, and that the
532 arguments of recursive calls decrease with respect to @{text R}.
533 Usually, this method is used as the initial proof step of manual
536 \item @{method (HOL) "lexicographic_order"} attempts a fully
537 automated termination proof by searching for a lexicographic
538 combination of size measures on the arguments of the function. The
539 method accepts the same arguments as the @{method auto} method,
540 which it uses internally to prove local descents. The same context
541 modifiers as for @{method auto} are accepted, see
542 \secref{sec:clasimp}.
544 In case of failure, extensive information is printed, which can help
545 to analyse the situation (cf.\ \cite{isabelle-function}).
551 subsection {* Old-style recursive function definitions (TFL) *}
554 The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
555 "recdef_tc"} for defining recursive are mostly obsolete; @{command
556 (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
558 \begin{matharray}{rcl}
559 @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
560 @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
564 'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
568 hints: '(' 'hints' (recdefmod *) ')'
570 recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
572 tc: nameref ('(' nat ')')?
578 \item @{command (HOL) "recdef"} defines general well-founded
579 recursive functions (using the TFL package), see also
580 \cite{isabelle-HOL}. The ``@{text "(permissive)"}'' option tells
581 TFL to recover from failed proof attempts, returning unfinished
582 results. The @{text recdef_simp}, @{text recdef_cong}, and @{text
583 recdef_wf} hints refer to auxiliary rules to be used in the internal
584 automated proof process of TFL. Additional @{syntax clasimpmod}
585 declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
586 context of the Simplifier (cf.\ \secref{sec:simplifier}) and
587 Classical reasoner (cf.\ \secref{sec:classical}).
589 \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
590 proof for leftover termination condition number @{text i} (default
591 1) as generated by a @{command (HOL) "recdef"} definition of
594 Note that in most cases, @{command (HOL) "recdef"} is able to finish
595 its internal proofs without manual intervention.
599 \medskip Hints for @{command (HOL) "recdef"} may be also declared
600 globally, using the following attributes.
602 \begin{matharray}{rcl}
603 @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
604 @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
605 @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
609 ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
615 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
618 An \textbf{inductive definition} specifies the least predicate (or
619 set) @{text R} closed under given rules: applying a rule to elements
620 of @{text R} yields a result within @{text R}. For example, a
621 structural operational semantics is an inductive definition of an
624 Dually, a \textbf{coinductive definition} specifies the greatest
625 predicate~/ set @{text R} that is consistent with given rules: every
626 element of @{text R} can be seen as arising by applying a rule to
627 elements of @{text R}. An important example is using bisimulation
628 relations to formalise equivalence of processes and infinite data
631 \medskip The HOL package is related to the ZF one, which is
632 described in a separate paper,\footnote{It appeared in CADE
633 \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
634 which you should refer to in case of difficulties. The package is
635 simpler than that of ZF thanks to implicit type-checking in HOL.
636 The types of the (co)inductive predicates (or sets) determine the
637 domain of the fixedpoint definition, and the package does not have
638 to use inference rules for type-checking.
640 \begin{matharray}{rcl}
641 @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
642 @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
643 @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
644 @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
645 @{attribute_def (HOL) mono} & : & @{text attribute} \\
649 ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
650 ('where' clauses)? ('monos' thmrefs)?
652 clauses: (thmdecl? prop + '|')
654 'mono' (() | 'add' | 'del')
660 \item @{command (HOL) "inductive"} and @{command (HOL)
661 "coinductive"} define (co)inductive predicates from the
662 introduction rules given in the @{keyword "where"} part. The
663 optional @{keyword "for"} part contains a list of parameters of the
664 (co)inductive predicates that remain fixed throughout the
665 definition. The optional @{keyword "monos"} section contains
666 \emph{monotonicity theorems}, which are required for each operator
667 applied to a recursive set in the introduction rules. There
668 \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
669 for each premise @{text "M R\<^sub>i t"} in an introduction rule!
671 \item @{command (HOL) "inductive_set"} and @{command (HOL)
672 "coinductive_set"} are wrappers for to the previous commands,
673 allowing the definition of (co)inductive sets.
675 \item @{attribute (HOL) mono} declares monotonicity rules. These
676 rule are involved in the automated monotonicity proof of @{command
683 subsection {* Derived rules *}
686 Each (co)inductive definition @{text R} adds definitions to the
687 theory and also proves some theorems:
691 \item @{text R.intros} is the list of introduction rules as proven
692 theorems, for the recursive predicates (or sets). The rules are
693 also available individually, using the names given them in the
696 \item @{text R.cases} is the case analysis (or elimination) rule;
698 \item @{text R.induct} or @{text R.coinduct} is the (co)induction
703 When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
704 defined simultaneously, the list of introduction rules is called
705 @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
706 called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
707 of mutual induction rules is called @{text
708 "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
712 subsection {* Monotonicity theorems *}
715 Each theory contains a default set of theorems that are used in
716 monotonicity proofs. New rules can be added to this set via the
717 @{attribute (HOL) mono} attribute. The HOL theory @{text Inductive}
718 shows how this is done. In general, the following monotonicity
719 theorems may be added:
723 \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
724 monotonicity of inductive definitions whose introduction rules have
725 premises involving terms such as @{text "M R\<^sub>i t"}.
727 \item Monotonicity theorems for logical operators, which are of the
728 general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}. For example, in
729 the case of the operator @{text "\<or>"}, the corresponding theorem is
731 \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"}}
734 \item De Morgan style equations for reasoning about the ``polarity''
737 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
738 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
741 \item Equations for reducing complex operators to more primitive
742 ones whose monotonicity can easily be proved, e.g.
744 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
745 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
750 %FIXME: Example of an inductive definition
754 section {* Arithmetic proof support *}
757 \begin{matharray}{rcl}
758 @{method_def (HOL) arith} & : & @{text method} \\
759 @{attribute_def (HOL) arith} & : & @{text attribute} \\
760 @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
763 The @{method (HOL) arith} method decides linear arithmetic problems
764 (on types @{text nat}, @{text int}, @{text real}). Any current
765 facts are inserted into the goal before running the procedure.
767 The @{attribute (HOL) arith} attribute declares facts that are
768 always supplied to the arithmetic provers implicitly.
770 The @{attribute (HOL) arith_split} attribute declares case split
771 rules to be expanded before @{method (HOL) arith} is invoked.
773 Note that a simpler (but faster) arithmetic prover is
774 already invoked by the Simplifier.
778 section {* Intuitionistic proof search *}
781 \begin{matharray}{rcl}
782 @{method_def (HOL) iprover} & : & @{text method} \\
786 'iprover' ('!' ?) (rulemod *)
790 The @{method (HOL) iprover} method performs intuitionistic proof
791 search, depending on specifically declared rules from the context,
792 or given as explicit arguments. Chained facts are inserted into the
793 goal before commencing proof search; ``@{method (HOL) iprover}@{text
794 "!"}'' means to include the current @{fact prems} as well.
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 {* Invoking automated reasoning tools -- The Sledgehammer *}
830 Isabelle/HOL includes a generic \emph{ATP manager} that allows
831 external automated reasoning tools to crunch a pending goal.
832 Supported provers include E\footnote{\url{http://www.eprover.org}},
833 SPASS\footnote{\url{http://www.spass-prover.org/}}, and Vampire.
834 There is also a wrapper to invoke provers remotely via the
835 SystemOnTPTP\footnote{\url{http://www.cs.miami.edu/~tptp/cgi-bin/SystemOnTPTP}}
838 The problem passed to external provers consists of the goal together
839 with a smart selection of lemmas from the current theory context.
840 The result of a successful proof search is some source text that
841 usually reconstructs the proof within Isabelle, without requiring
842 external provers again. The Metis
843 prover\footnote{\url{http://www.gilith.com/software/metis/}} that is
844 integrated into Isabelle/HOL is being used here.
846 In this mode of operation, heavy means of automated reasoning are
847 used as a strong relevance filter, while the main proof checking
848 works via explicit inferences going through the Isabelle kernel.
849 Moreover, rechecking Isabelle proof texts with already specified
850 auxiliary facts is much faster than performing fully automated
851 search over and over again.
853 \begin{matharray}{rcl}
854 @{command_def (HOL) "sledgehammer"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
855 @{command_def (HOL) "print_atps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
856 @{command_def (HOL) "atp_info"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
857 @{command_def (HOL) "atp_kill"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
858 @{command_def (HOL) "atp_messages"}@{text "\<^sup>*"} & : & @{text "any \<rightarrow>"} \\
859 @{method_def (HOL) metis} & : & @{text method} \\
863 'sledgehammer' (nameref *)
865 'atp\_messages' ('(' nat ')')?
874 \item @{command (HOL) sledgehammer}~@{text "prover\<^sub>1 \<dots> prover\<^sub>n"}
875 invokes the specified automated theorem provers on the first
876 subgoal. Provers are run in parallel, the first successful result
877 is displayed, and the other attempts are terminated.
879 Provers are defined in the theory context, see also @{command (HOL)
880 print_atps}. If no provers are given as arguments to @{command
881 (HOL) sledgehammer}, the system refers to the default defined as
882 ``ATP provers'' preference by the user interface.
884 There are additional preferences for timeout (default: 60 seconds),
885 and the maximum number of independent prover processes (default: 5);
886 excessive provers are automatically terminated.
888 \item @{command (HOL) print_atps} prints the list of automated
889 theorem provers available to the @{command (HOL) sledgehammer}
892 \item @{command (HOL) atp_info} prints information about presently
893 running provers, including elapsed runtime, and the remaining time
896 \item @{command (HOL) atp_kill} terminates all presently running
899 \item @{command (HOL) atp_messages} displays recent messages issued
900 by automated theorem provers. This allows to examine results that
901 might have got lost due to the asynchronous nature of default
902 @{command (HOL) sledgehammer} output. An optional message limit may
903 be specified (default 5).
905 \item @{method (HOL) metis}~@{text "facts"} invokes the Metis prover
906 with the given facts. Metis is an automated proof tool of medium
907 strength, but is fully integrated into Isabelle/HOL, with explicit
908 inferences going through the kernel. Thus its results are
909 guaranteed to be ``correct by construction''.
911 Note that all facts used with Metis need to be specified as explicit
912 arguments. There are no rule declarations as for other Isabelle
913 provers, like @{method blast} or @{method fast}.
919 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
922 The following tools of Isabelle/HOL support cases analysis and
923 induction in unstructured tactic scripts; see also
924 \secref{sec:cases-induct} for proper Isar versions of similar ideas.
926 \begin{matharray}{rcl}
927 @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
928 @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
929 @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
930 @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
934 'case\_tac' goalspec? term rule?
936 'induct\_tac' goalspec? (insts * 'and') rule?
938 'ind\_cases' (prop +) ('for' (name +)) ?
940 'inductive\_cases' (thmdecl? (prop +) + 'and')
943 rule: ('rule' ':' thmref)
949 \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
950 to reason about inductive types. Rules are selected according to
951 the declarations by the @{attribute cases} and @{attribute induct}
952 attributes, cf.\ \secref{sec:cases-induct}. The @{command (HOL)
953 datatype} package already takes care of this.
955 These unstructured tactics feature both goal addressing and dynamic
956 instantiation. Note that named rule cases are \emph{not} provided
957 as would be by the proper @{method cases} and @{method induct} proof
958 methods (see \secref{sec:cases-induct}). Unlike the @{method
959 induct} method, @{method induct_tac} does not handle structured rule
960 statements, only the compact object-logic conclusion of the subgoal
963 \item @{method (HOL) ind_cases} and @{command (HOL)
964 "inductive_cases"} provide an interface to the internal @{ML_text
965 mk_cases} operation. Rules are simplified in an unrestricted
968 While @{method (HOL) ind_cases} is a proof method to apply the
969 result immediately as elimination rules, @{command (HOL)
970 "inductive_cases"} provides case split theorems at the theory level
971 for later use. The @{keyword "for"} argument of the @{method (HOL)
972 ind_cases} method allows to specify a list of variables that should
973 be generalized before applying the resulting rule.
979 section {* Executable code *}
982 Isabelle/Pure provides two generic frameworks to support code
983 generation from executable specifications. Isabelle/HOL
984 instantiates these mechanisms in a way that is amenable to end-user
987 One framework generates code from both functional and relational
988 programs to SML. See \cite{isabelle-HOL} for further information
989 (this actually covers the new-style theory format as well).
991 \begin{matharray}{rcl}
992 @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
993 @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
994 @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
995 @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
996 @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\
997 @{attribute_def (HOL) code} & : & @{text attribute} \\
1004 ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
1005 ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
1006 'contains' ( ( name '=' term ) + | term + )
1009 modespec: '(' ( name * ) ')'
1012 'consts\_code' (codespec +)
1015 codespec: const template attachment ?
1018 'types\_code' (tycodespec +)
1021 tycodespec: name template attachment ?
1027 template: '(' string ')'
1030 attachment: 'attach' modespec ? verblbrace text verbrbrace
1039 \item @{command (HOL) "value"}~@{text t} evaluates and prints a term
1040 using the code generator.
1044 \medskip The other framework generates code from functional programs
1045 (including overloading using type classes) to SML \cite{SML}, OCaml
1046 \cite{OCaml} and Haskell \cite{haskell-revised-report}.
1047 Conceptually, code generation is split up in three steps:
1048 \emph{selection} of code theorems, \emph{translation} into an
1049 abstract executable view and \emph{serialization} to a specific
1050 \emph{target language}. See \cite{isabelle-codegen} for an
1051 introduction on how to use it.
1053 \begin{matharray}{rcl}
1054 @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1055 @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1056 @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1057 @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
1058 @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
1059 @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
1060 @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
1061 @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
1062 @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
1063 @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
1064 @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
1065 @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
1066 @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
1067 @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1068 @{attribute_def (HOL) code} & : & @{text attribute} \\
1072 'export\_code' ( constexpr + ) ? \\
1073 ( ( 'in' target ( 'module\_name' string ) ? \\
1074 ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
1077 'code\_thms' ( constexpr + ) ?
1080 'code\_deps' ( constexpr + ) ?
1086 constexpr: ( const | 'name.*' | '*' )
1089 typeconstructor: nameref
1095 target: 'OCaml' | 'SML' | 'Haskell'
1098 'code\_datatype' const +
1101 'code\_const' (const + 'and') \\
1102 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1105 'code\_type' (typeconstructor + 'and') \\
1106 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1109 'code\_class' (class + 'and') \\
1110 ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
1113 'code\_instance' (( typeconstructor '::' class ) + 'and') \\
1114 ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
1117 'code\_monad' const const target
1120 'code\_reserved' target ( string + )
1123 'code\_include' target ( string ( string | '-') )
1126 'code\_modulename' target ( ( string string ) + )
1129 'code\_abort' ( const + )
1132 syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
1135 'code' ( 'inline' ) ? ( 'del' ) ?
1141 \item @{command (HOL) "export_code"} is the canonical interface for
1142 generating and serializing code: for a given list of constants, code
1143 is generated for the specified target languages. Abstract code is
1144 cached incrementally. If no constant is given, the currently cached
1145 code is serialized. If no serialization instruction is given, only
1146 abstract code is cached.
1148 Constants may be specified by giving them literally, referring to
1149 all executable contants within a certain theory by giving @{text
1150 "name.*"}, or referring to \emph{all} executable constants currently
1151 available by giving @{text "*"}.
1153 By default, for each involved theory one corresponding name space
1154 module is generated. Alternativly, a module name may be specified
1155 after the @{keyword "module_name"} keyword; then \emph{all} code is
1156 placed in this module.
1158 For \emph{SML} and \emph{OCaml}, the file specification refers to a
1159 single file; for \emph{Haskell}, it refers to a whole directory,
1160 where code is generated in multiple files reflecting the module
1161 hierarchy. The file specification ``@{text "-"}'' denotes standard
1162 output. For \emph{SML}, omitting the file specification compiles
1163 code internally in the context of the current ML session.
1165 Serializers take an optional list of arguments in parentheses. For
1166 \emph{Haskell} a module name prefix may be given using the ``@{text
1167 "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim
1168 "deriving (Read, Show)"}'' clause to each appropriate datatype
1171 \item @{command (HOL) "code_thms"} prints a list of theorems
1172 representing the corresponding program containing all given
1173 constants; if no constants are given, the currently cached code
1174 theorems are printed.
1176 \item @{command (HOL) "code_deps"} visualizes dependencies of
1177 theorems representing the corresponding program containing all given
1178 constants; if no constants are given, the currently cached code
1179 theorems are visualized.
1181 \item @{command (HOL) "code_datatype"} specifies a constructor set
1184 \item @{command (HOL) "code_const"} associates a list of constants
1185 with target-specific serializations; omitting a serialization
1186 deletes an existing serialization.
1188 \item @{command (HOL) "code_type"} associates a list of type
1189 constructors with target-specific serializations; omitting a
1190 serialization deletes an existing serialization.
1192 \item @{command (HOL) "code_class"} associates a list of classes
1193 with target-specific class names; omitting a serialization deletes
1194 an existing serialization. This applies only to \emph{Haskell}.
1196 \item @{command (HOL) "code_instance"} declares a list of type
1197 constructor / class instance relations as ``already present'' for a
1198 given target. Omitting a ``@{text "-"}'' deletes an existing
1199 ``already present'' declaration. This applies only to
1202 \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
1203 to generate monadic code for Haskell.
1205 \item @{command (HOL) "code_reserved"} declares a list of names as
1206 reserved for a given target, preventing it to be shadowed by any
1209 \item @{command (HOL) "code_include"} adds arbitrary named content
1210 (``include'') to generated code. A ``@{text "-"}'' as last argument
1211 will remove an already added ``include''.
1213 \item @{command (HOL) "code_modulename"} declares aliasings from one
1214 module name onto another.
1216 \item @{command (HOL) "code_abort"} declares constants which are not
1217 required to have a definition by means of code equations; if
1218 needed these are implemented by program abort instead.
1220 \item @{attribute (HOL) code} explicitly selects (or with option
1221 ``@{text "del"}'' deselects) a code equation for code
1222 generation. Usually packages introducing code equations provide
1223 a reasonable default setup for selection.
1225 \item @{attribute (HOL) code}~@{text inline} declares (or with
1226 option ``@{text "del"}'' removes) inlining theorems which are
1227 applied as rewrite rules to any code equation during
1230 \item @{command (HOL) "print_codesetup"} gives an overview on
1231 selected code equations, code generator datatypes and
1238 section {* Definition by specification \label{sec:hol-specification} *}
1241 \begin{matharray}{rcl}
1242 @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1243 @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1247 ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
1249 decl: ((name ':')? term '(' 'overloaded' ')'?)
1254 \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
1255 goal stating the existence of terms with the properties specified to
1256 hold for the constants given in @{text decls}. After finishing the
1257 proof, the theory will be augmented with definitions for the given
1258 constants, as well as with theorems stating the properties for these
1261 \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
1262 a goal stating the existence of terms with the properties specified
1263 to hold for the constants given in @{text decls}. After finishing
1264 the proof, the theory will be augmented with axioms expressing the
1265 properties given in the first place.
1267 \item @{text decl} declares a constant to be defined by the
1268 specification given. The definition for the constant @{text c} is
1269 bound to the name @{text c_def} unless a theorem name is given in
1270 the declaration. Overloaded constants should be declared as such.
1274 Whether to use @{command (HOL) "specification"} or @{command (HOL)
1275 "ax_specification"} is to some extent a matter of style. @{command
1276 (HOL) "specification"} introduces no new axioms, and so by
1277 construction cannot introduce inconsistencies, whereas @{command
1278 (HOL) "ax_specification"} does introduce axioms, but only after the
1279 user has explicitly proven it to be safe. A practical issue must be
1280 considered, though: After introducing two constants with the same
1281 properties using @{command (HOL) "specification"}, one can prove
1282 that the two constants are, in fact, equal. If this might be a
1283 problem, one should use @{command (HOL) "ax_specification"}.