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 equations: (thmdecl? prop + '|')
406 ('fun' | 'function') target? functionopts? fixes 'where' clauses
408 clauses: (thmdecl? prop ('(' 'otherwise' ')')? + '|')
410 functionopts: '(' (('sequential' | 'domintros' | 'tailrec' | 'default' term) + ',') ')'
412 'termination' ( term )?
417 \item @{command (HOL) "primrec"} defines primitive recursive
418 functions over datatypes, see also \cite{isabelle-HOL}.
420 \item @{command (HOL) "function"} defines functions by general
421 wellfounded recursion. A detailed description with examples can be
422 found in \cite{isabelle-function}. The function is specified by a
423 set of (possibly conditional) recursive equations with arbitrary
424 pattern matching. The command generates proof obligations for the
425 completeness and the compatibility of patterns.
427 The defined function is considered partial, and the resulting
428 simplification rules (named @{text "f.psimps"}) and induction rule
429 (named @{text "f.pinduct"}) are guarded by a generated domain
430 predicate @{text "f_dom"}. The @{command (HOL) "termination"}
431 command can then be used to establish that the function is total.
433 \item @{command (HOL) "fun"} is a shorthand notation for ``@{command
434 (HOL) "function"}~@{text "(sequential)"}, followed by automated
435 proof attempts regarding pattern matching and termination. See
436 \cite{isabelle-function} for further details.
438 \item @{command (HOL) "termination"}~@{text f} commences a
439 termination proof for the previously defined function @{text f}. If
440 this is omitted, the command refers to the most recent function
441 definition. After the proof is closed, the recursive equations and
442 the induction principle is established.
446 Recursive definitions introduced by the @{command (HOL) "function"}
448 reasoning by induction (cf.\ \secref{sec:cases-induct}): rule @{text
449 "c.induct"} (where @{text c} is the name of the function definition)
450 refers to a specific induction rule, with parameters named according
451 to the user-specified equations. Cases are numbered (starting from 1).
453 For @{command (HOL) "primrec"}, the induction principle coincides
454 with structural recursion on the datatype the recursion is carried
457 The equations provided by these packages may be referred later as
458 theorem list @{text "f.simps"}, where @{text f} is the (collective)
459 name of the functions defined. Individual equations may be named
462 The @{command (HOL) "function"} command accepts the following
467 \item @{text sequential} enables a preprocessor which disambiguates
468 overlapping patterns by making them mutually disjoint. Earlier
469 equations take precedence over later ones. This allows to give the
470 specification in a format very similar to functional programming.
471 Note that the resulting simplification and induction rules
472 correspond to the transformed specification, not the one given
473 originally. This usually means that each equation given by the user
474 may result in several theorems. Also note that this automatic
475 transformation only works for ML-style datatype patterns.
477 \item @{text domintros} enables the automated generation of
478 introduction rules for the domain predicate. While mostly not
479 needed, they can be helpful in some proofs about partial functions.
481 \item @{text tailrec} generates the unconstrained recursive
482 equations even without a termination proof, provided that the
483 function is tail-recursive. This currently only works
485 \item @{text "default d"} allows to specify a default value for a
486 (partial) function, which will ensure that @{text "f x = d x"}
487 whenever @{text "x \<notin> f_dom"}.
493 subsection {* Proof methods related to recursive definitions *}
496 \begin{matharray}{rcl}
497 @{method_def (HOL) pat_completeness} & : & @{text method} \\
498 @{method_def (HOL) relation} & : & @{text method} \\
499 @{method_def (HOL) lexicographic_order} & : & @{text method} \\
500 @{method_def (HOL) size_change} & : & @{text method} \\
506 'lexicographic\_order' ( clasimpmod * )
508 'size\_change' ( orders ( clasimpmod * ) )
510 orders: ( 'max' | 'min' | 'ms' ) *
515 \item @{method (HOL) pat_completeness} is a specialized method to
516 solve goals regarding the completeness of pattern matching, as
517 required by the @{command (HOL) "function"} package (cf.\
518 \cite{isabelle-function}).
520 \item @{method (HOL) relation}~@{text R} introduces a termination
521 proof using the relation @{text R}. The resulting proof state will
522 contain goals expressing that @{text R} is wellfounded, and that the
523 arguments of recursive calls decrease with respect to @{text R}.
524 Usually, this method is used as the initial proof step of manual
527 \item @{method (HOL) "lexicographic_order"} attempts a fully
528 automated termination proof by searching for a lexicographic
529 combination of size measures on the arguments of the function. The
530 method accepts the same arguments as the @{method auto} method,
531 which it uses internally to prove local descents. The same context
532 modifiers as for @{method auto} are accepted, see
533 \secref{sec:clasimp}.
535 In case of failure, extensive information is printed, which can help
536 to analyse the situation (cf.\ \cite{isabelle-function}).
538 \item @{method (HOL) "size_change"} also works on termination goals,
539 using a variation of the size-change principle, together with a
540 graph decomposition technique (see \cite{krauss_phd} for details).
541 Three kinds of orders are used internally: @{text max}, @{text min},
542 and @{text ms} (multiset), which is only available when the theory
543 @{text Multiset} is loaded. When no order kinds are given, they are
544 tried in order. The search for a termination proof uses SAT solving
547 For local descent proofs, the same context modifiers as for @{method
548 auto} are accepted, see \secref{sec:clasimp}.
554 subsection {* Old-style recursive function definitions (TFL) *}
557 The old TFL commands @{command (HOL) "recdef"} and @{command (HOL)
558 "recdef_tc"} for defining recursive are mostly obsolete; @{command
559 (HOL) "function"} or @{command (HOL) "fun"} should be used instead.
561 \begin{matharray}{rcl}
562 @{command_def (HOL) "recdef"} & : & @{text "theory \<rightarrow> theory)"} \\
563 @{command_def (HOL) "recdef_tc"}@{text "\<^sup>*"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
567 'recdef' ('(' 'permissive' ')')? \\ name term (prop +) hints?
571 hints: '(' 'hints' ( recdefmod * ) ')'
573 recdefmod: (('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del') ':' thmrefs) | clasimpmod
575 tc: nameref ('(' nat ')')?
581 \item @{command (HOL) "recdef"} defines general well-founded
582 recursive functions (using the TFL package), see also
583 \cite{isabelle-HOL}. The ``@{text "(permissive)"}'' option tells
584 TFL to recover from failed proof attempts, returning unfinished
585 results. The @{text recdef_simp}, @{text recdef_cong}, and @{text
586 recdef_wf} hints refer to auxiliary rules to be used in the internal
587 automated proof process of TFL. Additional @{syntax clasimpmod}
588 declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
589 context of the Simplifier (cf.\ \secref{sec:simplifier}) and
590 Classical reasoner (cf.\ \secref{sec:classical}).
592 \item @{command (HOL) "recdef_tc"}~@{text "c (i)"} recommences the
593 proof for leftover termination condition number @{text i} (default
594 1) as generated by a @{command (HOL) "recdef"} definition of
597 Note that in most cases, @{command (HOL) "recdef"} is able to finish
598 its internal proofs without manual intervention.
602 \medskip Hints for @{command (HOL) "recdef"} may be also declared
603 globally, using the following attributes.
605 \begin{matharray}{rcl}
606 @{attribute_def (HOL) recdef_simp} & : & @{text attribute} \\
607 @{attribute_def (HOL) recdef_cong} & : & @{text attribute} \\
608 @{attribute_def (HOL) recdef_wf} & : & @{text attribute} \\
612 ('recdef\_simp' | 'recdef\_cong' | 'recdef\_wf') (() | 'add' | 'del')
618 section {* Inductive and coinductive definitions \label{sec:hol-inductive} *}
621 An \textbf{inductive definition} specifies the least predicate (or
622 set) @{text R} closed under given rules: applying a rule to elements
623 of @{text R} yields a result within @{text R}. For example, a
624 structural operational semantics is an inductive definition of an
627 Dually, a \textbf{coinductive definition} specifies the greatest
628 predicate~/ set @{text R} that is consistent with given rules: every
629 element of @{text R} can be seen as arising by applying a rule to
630 elements of @{text R}. An important example is using bisimulation
631 relations to formalise equivalence of processes and infinite data
634 \medskip The HOL package is related to the ZF one, which is
635 described in a separate paper,\footnote{It appeared in CADE
636 \cite{paulson-CADE}; a longer version is distributed with Isabelle.}
637 which you should refer to in case of difficulties. The package is
638 simpler than that of ZF thanks to implicit type-checking in HOL.
639 The types of the (co)inductive predicates (or sets) determine the
640 domain of the fixedpoint definition, and the package does not have
641 to use inference rules for type-checking.
643 \begin{matharray}{rcl}
644 @{command_def (HOL) "inductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
645 @{command_def (HOL) "inductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
646 @{command_def (HOL) "coinductive"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
647 @{command_def (HOL) "coinductive_set"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
648 @{attribute_def (HOL) mono} & : & @{text attribute} \\
652 ('inductive' | 'inductive\_set' | 'coinductive' | 'coinductive\_set') target? fixes ('for' fixes)? \\
653 ('where' clauses)? ('monos' thmrefs)?
655 clauses: (thmdecl? prop + '|')
657 'mono' (() | 'add' | 'del')
663 \item @{command (HOL) "inductive"} and @{command (HOL)
664 "coinductive"} define (co)inductive predicates from the
665 introduction rules given in the @{keyword "where"} part. The
666 optional @{keyword "for"} part contains a list of parameters of the
667 (co)inductive predicates that remain fixed throughout the
668 definition. The optional @{keyword "monos"} section contains
669 \emph{monotonicity theorems}, which are required for each operator
670 applied to a recursive set in the introduction rules. There
671 \emph{must} be a theorem of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"},
672 for each premise @{text "M R\<^sub>i t"} in an introduction rule!
674 \item @{command (HOL) "inductive_set"} and @{command (HOL)
675 "coinductive_set"} are wrappers for to the previous commands,
676 allowing the definition of (co)inductive sets.
678 \item @{attribute (HOL) mono} declares monotonicity rules. These
679 rule are involved in the automated monotonicity proof of @{command
686 subsection {* Derived rules *}
689 Each (co)inductive definition @{text R} adds definitions to the
690 theory and also proves some theorems:
694 \item @{text R.intros} is the list of introduction rules as proven
695 theorems, for the recursive predicates (or sets). The rules are
696 also available individually, using the names given them in the
699 \item @{text R.cases} is the case analysis (or elimination) rule;
701 \item @{text R.induct} or @{text R.coinduct} is the (co)induction
706 When several predicates @{text "R\<^sub>1, \<dots>, R\<^sub>n"} are
707 defined simultaneously, the list of introduction rules is called
708 @{text "R\<^sub>1_\<dots>_R\<^sub>n.intros"}, the case analysis rules are
709 called @{text "R\<^sub>1.cases, \<dots>, R\<^sub>n.cases"}, and the list
710 of mutual induction rules is called @{text
711 "R\<^sub>1_\<dots>_R\<^sub>n.inducts"}.
715 subsection {* Monotonicity theorems *}
718 Each theory contains a default set of theorems that are used in
719 monotonicity proofs. New rules can be added to this set via the
720 @{attribute (HOL) mono} attribute. The HOL theory @{text Inductive}
721 shows how this is done. In general, the following monotonicity
722 theorems may be added:
726 \item Theorems of the form @{text "A \<le> B \<Longrightarrow> M A \<le> M B"}, for proving
727 monotonicity of inductive definitions whose introduction rules have
728 premises involving terms such as @{text "M R\<^sub>i t"}.
730 \item Monotonicity theorems for logical operators, which are of the
731 general form @{text "(\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> (\<dots> \<longrightarrow> \<dots>) \<Longrightarrow> \<dots> \<longrightarrow> \<dots>"}. For example, in
732 the case of the operator @{text "\<or>"}, the corresponding theorem is
734 \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"}}
737 \item De Morgan style equations for reasoning about the ``polarity''
740 @{prop "\<not> \<not> P \<longleftrightarrow> P"} \qquad\qquad
741 @{prop "\<not> (P \<and> Q) \<longleftrightarrow> \<not> P \<or> \<not> Q"}
744 \item Equations for reducing complex operators to more primitive
745 ones whose monotonicity can easily be proved, e.g.
747 @{prop "(P \<longrightarrow> Q) \<longleftrightarrow> \<not> P \<or> Q"} \qquad\qquad
748 @{prop "Ball A P \<equiv> \<forall>x. x \<in> A \<longrightarrow> P x"}
753 %FIXME: Example of an inductive definition
757 section {* Arithmetic proof support *}
760 \begin{matharray}{rcl}
761 @{method_def (HOL) arith} & : & @{text method} \\
762 @{attribute_def (HOL) arith} & : & @{text attribute} \\
763 @{attribute_def (HOL) arith_split} & : & @{text attribute} \\
766 The @{method (HOL) arith} method decides linear arithmetic problems
767 (on types @{text nat}, @{text int}, @{text real}). Any current
768 facts are inserted into the goal before running the procedure.
770 The @{attribute (HOL) arith} attribute declares facts that are
771 always supplied to the arithmetic provers implicitly.
773 The @{attribute (HOL) arith_split} attribute declares case split
774 rules to be expanded before @{method (HOL) arith} is invoked.
776 Note that a simpler (but faster) arithmetic prover is
777 already invoked by the Simplifier.
781 section {* Intuitionistic proof search *}
784 \begin{matharray}{rcl}
785 @{method_def (HOL) iprover} & : & @{text method} \\
789 'iprover' ( rulemod * )
793 The @{method (HOL) iprover} method performs intuitionistic proof
794 search, depending on specifically declared rules from the context,
795 or given as explicit arguments. Chained facts are inserted into the
796 goal before commencing proof search.
798 Rules need to be classified as @{attribute (Pure) intro},
799 @{attribute (Pure) elim}, or @{attribute (Pure) dest}; here the
800 ``@{text "!"}'' indicator refers to ``safe'' rules, which may be
801 applied aggressively (without considering back-tracking later).
802 Rules declared with ``@{text "?"}'' are ignored in proof search (the
803 single-step @{method rule} method still observes these). An
804 explicit weight annotation may be given as well; otherwise the
805 number of rule premises will be taken into account here.
809 section {* Coherent Logic *}
812 \begin{matharray}{rcl}
813 @{method_def (HOL) "coherent"} & : & @{text method} \\
821 The @{method (HOL) coherent} method solves problems of
822 \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
823 applications in confluence theory, lattice theory and projective
824 geometry. See @{"file" "~~/src/HOL/ex/Coherent.thy"} for some
829 section {* Checking and refuting propositions *}
832 Identifying incorrect propositions usually involves evaluation of
833 particular assignments and systematic counter example search. This
834 is supported by the following commands.
836 \begin{matharray}{rcl}
837 @{command_def (HOL) "value"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
838 @{command_def (HOL) "quickcheck"}@{text "\<^sup>*"} & : & @{text "proof \<rightarrow>"} \\
839 @{command_def (HOL) "quickcheck_params"} & : & @{text "theory \<rightarrow> theory"}
843 'value' ( ( '[' name ']' ) ? ) modes? term
846 'quickcheck' ( ( '[' args ']' ) ? ) nat?
849 'quickcheck_params' ( ( '[' args ']' ) ? )
852 modes: '(' (name + ) ')'
855 args: ( name '=' value + ',' )
861 \item @{command (HOL) "value"}~@{text t} evaluates and prints a
862 term; optionally @{text modes} can be specified, which are
863 appended to the current print mode (see also \cite{isabelle-ref}).
864 Internally, the evaluation is performed by registered evaluators,
865 which are invoked sequentially until a result is returned.
866 Alternatively a specific evaluator can be selected using square
867 brackets; available evaluators include @{text nbe} for
868 \emph{normalization by evaluation} and \emph{code} for code
871 \item @{command (HOL) "quickcheck"} tests the current goal for
872 counter examples using a series of arbitrary assignments for its
873 free variables; by default the first subgoal is tested, an other
874 can be selected explicitly using an optional goal index.
875 A number of configuration options are supported for
876 @{command (HOL) "quickcheck"}, notably:
880 \item[size] specifies the maximum size of the search space for
883 \item[iterations] sets how many sets of assignments are
884 generated for each particular size.
886 \item[no\_assms] specifies whether assumptions in
887 structured proofs should be ignored.
891 These option can be given within square brackets.
893 \item @{command (HOL) "quickcheck_params"} changes quickcheck
894 configuration options persitently.
900 section {* Unstructured case analysis and induction \label{sec:hol-induct-tac} *}
903 The following tools of Isabelle/HOL support cases analysis and
904 induction in unstructured tactic scripts; see also
905 \secref{sec:cases-induct} for proper Isar versions of similar ideas.
907 \begin{matharray}{rcl}
908 @{method_def (HOL) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
909 @{method_def (HOL) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
910 @{method_def (HOL) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
911 @{command_def (HOL) "inductive_cases"}@{text "\<^sup>*"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
915 'case\_tac' goalspec? term rule?
917 'induct\_tac' goalspec? (insts * 'and') rule?
919 'ind\_cases' (prop +) ('for' (name +)) ?
921 'inductive\_cases' (thmdecl? (prop +) + 'and')
924 rule: ('rule' ':' thmref)
930 \item @{method (HOL) case_tac} and @{method (HOL) induct_tac} admit
931 to reason about inductive types. Rules are selected according to
932 the declarations by the @{attribute cases} and @{attribute induct}
933 attributes, cf.\ \secref{sec:cases-induct}. The @{command (HOL)
934 datatype} package already takes care of this.
936 These unstructured tactics feature both goal addressing and dynamic
937 instantiation. Note that named rule cases are \emph{not} provided
938 as would be by the proper @{method cases} and @{method induct} proof
939 methods (see \secref{sec:cases-induct}). Unlike the @{method
940 induct} method, @{method induct_tac} does not handle structured rule
941 statements, only the compact object-logic conclusion of the subgoal
944 \item @{method (HOL) ind_cases} and @{command (HOL)
945 "inductive_cases"} provide an interface to the internal @{ML_text
946 mk_cases} operation. Rules are simplified in an unrestricted
949 While @{method (HOL) ind_cases} is a proof method to apply the
950 result immediately as elimination rules, @{command (HOL)
951 "inductive_cases"} provides case split theorems at the theory level
952 for later use. The @{keyword "for"} argument of the @{method (HOL)
953 ind_cases} method allows to specify a list of variables that should
954 be generalized before applying the resulting rule.
960 section {* Executable code *}
963 Isabelle/Pure provides two generic frameworks to support code
964 generation from executable specifications. Isabelle/HOL
965 instantiates these mechanisms in a way that is amenable to end-user
968 One framework generates code from both functional and relational
969 programs to SML. See \cite{isabelle-HOL} for further information
970 (this actually covers the new-style theory format as well).
972 \begin{matharray}{rcl}
973 @{command_def (HOL) "code_module"} & : & @{text "theory \<rightarrow> theory"} \\
974 @{command_def (HOL) "code_library"} & : & @{text "theory \<rightarrow> theory"} \\
975 @{command_def (HOL) "consts_code"} & : & @{text "theory \<rightarrow> theory"} \\
976 @{command_def (HOL) "types_code"} & : & @{text "theory \<rightarrow> theory"} \\
977 @{attribute_def (HOL) code} & : & @{text attribute} \\
981 ( 'code\_module' | 'code\_library' ) modespec ? name ? \\
982 ( 'file' name ) ? ( 'imports' ( name + ) ) ? \\
983 'contains' ( ( name '=' term ) + | term + )
986 modespec: '(' ( name * ) ')'
989 'consts\_code' (codespec +)
992 codespec: const template attachment ?
995 'types\_code' (tycodespec +)
998 tycodespec: name template attachment ?
1004 template: '(' string ')'
1007 attachment: 'attach' modespec ? verblbrace text verbrbrace
1014 \medskip The other framework generates code from functional programs
1015 (including overloading using type classes) to SML \cite{SML}, OCaml
1016 \cite{OCaml} and Haskell \cite{haskell-revised-report}.
1017 Conceptually, code generation is split up in three steps:
1018 \emph{selection} of code theorems, \emph{translation} into an
1019 abstract executable view and \emph{serialization} to a specific
1020 \emph{target language}. See \cite{isabelle-codegen} for an
1021 introduction on how to use it.
1023 \begin{matharray}{rcl}
1024 @{command_def (HOL) "export_code"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1025 @{command_def (HOL) "code_thms"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1026 @{command_def (HOL) "code_deps"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1027 @{command_def (HOL) "code_datatype"} & : & @{text "theory \<rightarrow> theory"} \\
1028 @{command_def (HOL) "code_const"} & : & @{text "theory \<rightarrow> theory"} \\
1029 @{command_def (HOL) "code_type"} & : & @{text "theory \<rightarrow> theory"} \\
1030 @{command_def (HOL) "code_class"} & : & @{text "theory \<rightarrow> theory"} \\
1031 @{command_def (HOL) "code_instance"} & : & @{text "theory \<rightarrow> theory"} \\
1032 @{command_def (HOL) "code_monad"} & : & @{text "theory \<rightarrow> theory"} \\
1033 @{command_def (HOL) "code_reserved"} & : & @{text "theory \<rightarrow> theory"} \\
1034 @{command_def (HOL) "code_include"} & : & @{text "theory \<rightarrow> theory"} \\
1035 @{command_def (HOL) "code_modulename"} & : & @{text "theory \<rightarrow> theory"} \\
1036 @{command_def (HOL) "code_abort"} & : & @{text "theory \<rightarrow> theory"} \\
1037 @{command_def (HOL) "print_codesetup"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1038 @{command_def (HOL) "print_codeproc"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
1039 @{attribute_def (HOL) code} & : & @{text attribute} \\
1043 'export\_code' ( constexpr + ) \\
1044 ( ( 'in' target ( 'module\_name' string ) ? \\
1045 ( 'file' ( string | '-' ) ) ? ( '(' args ')' ) ?) + ) ?
1048 'code\_thms' ( constexpr + ) ?
1051 'code\_deps' ( constexpr + ) ?
1057 constexpr: ( const | 'name.*' | '*' )
1060 typeconstructor: nameref
1066 target: 'OCaml' | 'SML' | 'Haskell'
1069 'code\_datatype' const +
1072 'code\_const' (const + 'and') \\
1073 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1076 'code\_type' (typeconstructor + 'and') \\
1077 ( ( '(' target ( syntax ? + 'and' ) ')' ) + )
1080 'code\_class' (class + 'and') \\
1081 ( ( '(' target \\ ( string ? + 'and' ) ')' ) + )
1084 'code\_instance' (( typeconstructor '::' class ) + 'and') \\
1085 ( ( '(' target ( '-' ? + 'and' ) ')' ) + )
1088 'code\_monad' const const target
1091 'code\_reserved' target ( string + )
1094 'code\_include' target ( string ( string | '-') )
1097 'code\_modulename' target ( ( string string ) + )
1100 'code\_abort' ( const + )
1103 syntax: string | ( 'infix' | 'infixl' | 'infixr' ) nat string
1109 'code_unfold' ( 'del' ) ?
1112 'code_post' ( 'del' ) ?
1118 \item @{command (HOL) "export_code"} is the canonical interface for
1119 generating and serializing code: for a given list of constants, code
1120 is generated for the specified target languages. If no serialization
1121 instruction is given, only abstract code is generated internally.
1123 Constants may be specified by giving them literally, referring to
1124 all executable contants within a certain theory by giving @{text
1125 "name.*"}, or referring to \emph{all} executable constants currently
1126 available by giving @{text "*"}.
1128 By default, for each involved theory one corresponding name space
1129 module is generated. Alternativly, a module name may be specified
1130 after the @{keyword "module_name"} keyword; then \emph{all} code is
1131 placed in this module.
1133 For \emph{SML} and \emph{OCaml}, the file specification refers to a
1134 single file; for \emph{Haskell}, it refers to a whole directory,
1135 where code is generated in multiple files reflecting the module
1136 hierarchy. The file specification ``@{text "-"}'' denotes standard
1137 output. For \emph{SML}, omitting the file specification compiles
1138 code internally in the context of the current ML session.
1140 Serializers take an optional list of arguments in parentheses. For
1141 \emph{SML} and \emph{OCaml}, ``@{text no_signatures}`` omits
1142 explicit module signatures.
1144 For \emph{Haskell} a module name prefix may be given using the ``@{text
1145 "root:"}'' argument; ``@{text string_classes}'' adds a ``@{verbatim
1146 "deriving (Read, Show)"}'' clause to each appropriate datatype
1149 \item @{command (HOL) "code_thms"} prints a list of theorems
1150 representing the corresponding program containing all given
1153 \item @{command (HOL) "code_deps"} visualizes dependencies of
1154 theorems representing the corresponding program containing all given
1157 \item @{command (HOL) "code_datatype"} specifies a constructor set
1160 \item @{command (HOL) "code_const"} associates a list of constants
1161 with target-specific serializations; omitting a serialization
1162 deletes an existing serialization.
1164 \item @{command (HOL) "code_type"} associates a list of type
1165 constructors with target-specific serializations; omitting a
1166 serialization deletes an existing serialization.
1168 \item @{command (HOL) "code_class"} associates a list of classes
1169 with target-specific class names; omitting a serialization deletes
1170 an existing serialization. This applies only to \emph{Haskell}.
1172 \item @{command (HOL) "code_instance"} declares a list of type
1173 constructor / class instance relations as ``already present'' for a
1174 given target. Omitting a ``@{text "-"}'' deletes an existing
1175 ``already present'' declaration. This applies only to
1178 \item @{command (HOL) "code_monad"} provides an auxiliary mechanism
1179 to generate monadic code for Haskell.
1181 \item @{command (HOL) "code_reserved"} declares a list of names as
1182 reserved for a given target, preventing it to be shadowed by any
1185 \item @{command (HOL) "code_include"} adds arbitrary named content
1186 (``include'') to generated code. A ``@{text "-"}'' as last argument
1187 will remove an already added ``include''.
1189 \item @{command (HOL) "code_modulename"} declares aliasings from one
1190 module name onto another.
1192 \item @{command (HOL) "code_abort"} declares constants which are not
1193 required to have a definition by means of code equations; if
1194 needed these are implemented by program abort instead.
1196 \item @{attribute (HOL) code} explicitly selects (or with option
1197 ``@{text "del"}'' deselects) a code equation for code
1198 generation. Usually packages introducing code equations provide
1199 a reasonable default setup for selection.
1201 \item @{attribute (HOL) code_inline} declares (or with
1202 option ``@{text "del"}'' removes) inlining theorems which are
1203 applied as rewrite rules to any code equation during
1206 \item @{attribute (HOL) code_post} declares (or with
1207 option ``@{text "del"}'' removes) theorems which are
1208 applied as rewrite rules to any result of an evaluation.
1210 \item @{command (HOL) "print_codesetup"} gives an overview on
1211 selected code equations and code generator datatypes.
1213 \item @{command (HOL) "print_codeproc"} prints the setup
1214 of the code generator preprocessor.
1220 section {* Definition by specification \label{sec:hol-specification} *}
1223 \begin{matharray}{rcl}
1224 @{command_def (HOL) "specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1225 @{command_def (HOL) "ax_specification"} & : & @{text "theory \<rightarrow> proof(prove)"} \\
1229 ('specification' | 'ax\_specification') '(' (decl +) ')' \\ (thmdecl? prop +)
1231 decl: ((name ':')? term '(' 'overloaded' ')'?)
1236 \item @{command (HOL) "specification"}~@{text "decls \<phi>"} sets up a
1237 goal stating the existence of terms with the properties specified to
1238 hold for the constants given in @{text decls}. After finishing the
1239 proof, the theory will be augmented with definitions for the given
1240 constants, as well as with theorems stating the properties for these
1243 \item @{command (HOL) "ax_specification"}~@{text "decls \<phi>"} sets up
1244 a goal stating the existence of terms with the properties specified
1245 to hold for the constants given in @{text decls}. After finishing
1246 the proof, the theory will be augmented with axioms expressing the
1247 properties given in the first place.
1249 \item @{text decl} declares a constant to be defined by the
1250 specification given. The definition for the constant @{text c} is
1251 bound to the name @{text c_def} unless a theorem name is given in
1252 the declaration. Overloaded constants should be declared as such.
1256 Whether to use @{command (HOL) "specification"} or @{command (HOL)
1257 "ax_specification"} is to some extent a matter of style. @{command
1258 (HOL) "specification"} introduces no new axioms, and so by
1259 construction cannot introduce inconsistencies, whereas @{command
1260 (HOL) "ax_specification"} does introduce axioms, but only after the
1261 user has explicitly proven it to be safe. A practical issue must be
1262 considered, though: After introducing two constants with the same
1263 properties using @{command (HOL) "specification"}, one can prove
1264 that the two constants are, in fact, equal. If this might be a
1265 problem, one should use @{command (HOL) "ax_specification"}.