3 \def\isabellecontext{HOL{\isaliteral{5F}{\isacharunderscore}}Specific}%
10 \isacommand{theory}\isamarkupfalse%
11 \ HOL{\isaliteral{5F}{\isacharunderscore}}Specific\isanewline
12 \isakeyword{imports}\ Base\ Main\isanewline
21 \isamarkupchapter{Isabelle/HOL \label{ch:hol}%
25 \isamarkupsection{Higher-Order Logic%
29 \begin{isamarkuptext}%
30 Isabelle/HOL is based on Higher-Order Logic, a polymorphic
31 version of Church's Simple Theory of Types. HOL can be best
32 understood as a simply-typed version of classical set theory. The
33 logic was first implemented in Gordon's HOL system
34 \cite{mgordon-hol}. It extends Church's original logic
35 \cite{church40} by explicit type variables (naive polymorphism) and
36 a sound axiomatization scheme for new types based on subsets of
39 Andrews's book \cite{andrews86} is a full description of the
40 original Church-style higher-order logic, with proofs of correctness
41 and completeness wrt.\ certain set-theoretic interpretations. The
42 particular extensions of Gordon-style HOL are explained semantically
43 in two chapters of the 1993 HOL book \cite{pitts93}.
45 Experience with HOL over decades has demonstrated that higher-order
46 logic is widely applicable in many areas of mathematics and computer
47 science. In a sense, Higher-Order Logic is simpler than First-Order
48 Logic, because there are fewer restrictions and special cases. Note
49 that HOL is \emph{weaker} than FOL with axioms for ZF set theory,
50 which is traditionally considered the standard foundation of regular
51 mathematics, but for most applications this does not matter. If you
52 prefer ML to Lisp, you will probably prefer HOL to ZF.
54 \medskip The syntax of HOL follows \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{22}{\isachardoublequote}}}-calculus and
55 functional programming. Function application is curried. To apply
56 the function \isa{f} of type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{3}}{\isaliteral{22}{\isachardoublequote}}} to the
57 arguments \isa{a} and \isa{b} in HOL, you simply write \isa{{\isaliteral{22}{\isachardoublequote}}f\ a\ b{\isaliteral{22}{\isachardoublequote}}} (as in ML or Haskell). There is no ``apply'' operator; the
58 existing application of the Pure \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}{\isaliteral{22}{\isachardoublequote}}}-calculus is re-used.
59 Note that in HOL \isa{{\isaliteral{22}{\isachardoublequote}}f\ {\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} means ``\isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{22}{\isachardoublequote}}} applied to
60 the pair \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}a{\isaliteral{2C}{\isacharcomma}}\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} (which is notation for \isa{{\isaliteral{22}{\isachardoublequote}}Pair\ a\ b{\isaliteral{22}{\isachardoublequote}}}). The latter typically introduces extra formal efforts that can
61 be avoided by currying functions by default. Explicit tuples are as
62 infrequent in HOL formalizations as in good ML or Haskell programs.
64 \medskip Isabelle/HOL has a distinct feel, compared to other
65 object-logics like Isabelle/ZF. It identifies object-level types
66 with meta-level types, taking advantage of the default
67 type-inference mechanism of Isabelle/Pure. HOL fully identifies
68 object-level functions with meta-level functions, with native
69 abstraction and application.
71 These identifications allow Isabelle to support HOL particularly
72 nicely, but they also mean that HOL requires some sophistication
73 from the user. In particular, an understanding of Hindley-Milner
74 type-inference with type-classes, which are both used extensively in
75 the standard libraries and applications. Beginners can set
76 \hyperlink{attribute.show-types}{\mbox{\isa{show{\isaliteral{5F}{\isacharunderscore}}types}}} or even \hyperlink{attribute.show-sorts}{\mbox{\isa{show{\isaliteral{5F}{\isacharunderscore}}sorts}}} to get more
77 explicit information about the result of type-inference.%
81 \isamarkupsection{Inductive and coinductive definitions \label{sec:hol-inductive}%
85 \begin{isamarkuptext}%
86 An \emph{inductive definition} specifies the least predicate
87 or set \isa{R} closed under given rules: applying a rule to
88 elements of \isa{R} yields a result within \isa{R}. For
89 example, a structural operational semantics is an inductive
90 definition of an evaluation relation.
92 Dually, a \emph{coinductive definition} specifies the greatest
93 predicate or set \isa{R} that is consistent with given rules:
94 every element of \isa{R} can be seen as arising by applying a rule
95 to elements of \isa{R}. An important example is using
96 bisimulation relations to formalise equivalence of processes and
97 infinite data structures.
99 Both inductive and coinductive definitions are based on the
100 Knaster-Tarski fixed-point theorem for complete lattices. The
101 collection of introduction rules given by the user determines a
102 functor on subsets of set-theoretic relations. The required
103 monotonicity of the recursion scheme is proven as a prerequisite to
104 the fixed-point definition and the resulting consequences. This
105 works by pushing inclusion through logical connectives and any other
106 operator that might be wrapped around recursive occurrences of the
107 defined relation: there must be a monotonicity theorem of the form
108 \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C6C653E}{\isasymle}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C4D3E}{\isasymM}}\ A\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{5C3C4D3E}{\isasymM}}\ B{\isaliteral{22}{\isachardoublequote}}}, for each premise \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4D3E}{\isasymM}}\ R\ t{\isaliteral{22}{\isachardoublequote}}} in an
109 introduction rule. The default rule declarations of Isabelle/HOL
110 already take care of most common situations.
112 \begin{matharray}{rcl}
113 \indexdef{HOL}{command}{inductive}\hypertarget{command.HOL.inductive}{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
114 \indexdef{HOL}{command}{inductive\_set}\hypertarget{command.HOL.inductive-set}{\hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
115 \indexdef{HOL}{command}{coinductive}\hypertarget{command.HOL.coinductive}{\hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
116 \indexdef{HOL}{command}{coinductive\_set}\hypertarget{command.HOL.coinductive-set}{\hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isaliteral{5F}{\isacharunderscore}}set}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
117 \indexdef{HOL}{attribute}{mono}\hypertarget{attribute.HOL.mono}{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}} & : & \isa{attribute} \\
123 \rail@term{\hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}}}[]
125 \rail@term{\hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}}}}}[]
127 \rail@term{\hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}}[]
129 \rail@term{\hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isaliteral{5F}{\isacharunderscore}}set}}}}}[]
133 \rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]
136 \rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]
139 \rail@term{\isa{\isakeyword{for}}}[]
140 \rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]
144 \rail@term{\isa{\isakeyword{where}}}[]
145 \rail@nont{\isa{clauses}}[]
150 \rail@term{\isa{\isakeyword{monos}}}[]
151 \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
154 \rail@begin{3}{\isa{clauses}}
158 \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
160 \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
162 \rail@cterm{\isa{{\isaliteral{7C}{\isacharbar}}}}[]
166 \rail@term{\hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}}}[]
169 \rail@term{\isa{add}}[]
171 \rail@term{\isa{del}}[]
179 \item \hyperlink{command.HOL.inductive}{\mbox{\isa{\isacommand{inductive}}}} and \hyperlink{command.HOL.coinductive}{\mbox{\isa{\isacommand{coinductive}}}} define (co)inductive predicates from the introduction
182 The propositions given as \isa{{\isaliteral{22}{\isachardoublequote}}clauses{\isaliteral{22}{\isachardoublequote}}} in the \hyperlink{keyword.where}{\mbox{\isa{\isakeyword{where}}}} part are either rules of the usual \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C416E643E}{\isasymAnd}}{\isaliteral{2F}{\isacharslash}}{\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}{\isaliteral{22}{\isachardoublequote}}} format
183 (with arbitrary nesting), or equalities using \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C65717569763E}{\isasymequiv}}{\isaliteral{22}{\isachardoublequote}}}. The
184 latter specifies extra-logical abbreviations in the sense of
185 \indexref{}{command}{abbreviation}\hyperlink{command.abbreviation}{\mbox{\isa{\isacommand{abbreviation}}}}. Introducing abstract syntax
186 simultaneously with the actual introduction rules is occasionally
187 useful for complex specifications.
189 The optional \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} part contains a list of parameters of
190 the (co)inductive predicates that remain fixed throughout the
191 definition, in contrast to arguments of the relation that may vary
192 in each occurrence within the given \isa{{\isaliteral{22}{\isachardoublequote}}clauses{\isaliteral{22}{\isachardoublequote}}}.
194 The optional \hyperlink{keyword.monos}{\mbox{\isa{\isakeyword{monos}}}} declaration contains additional
195 \emph{monotonicity theorems}, which are required for each operator
196 applied to a recursive set in the introduction rules.
198 \item \hyperlink{command.HOL.inductive-set}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}}}} and \hyperlink{command.HOL.coinductive-set}{\mbox{\isa{\isacommand{coinductive{\isaliteral{5F}{\isacharunderscore}}set}}}} are wrappers for to the previous commands for
199 native HOL predicates. This allows to define (co)inductive sets,
200 where multiple arguments are simulated via tuples.
202 \item \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} declares monotonicity rules in the
203 context. These rule are involved in the automated monotonicity
204 proof of the above inductive and coinductive definitions.
210 \isamarkupsubsection{Derived rules%
214 \begin{isamarkuptext}%
215 A (co)inductive definition of \isa{R} provides the following
220 \item \isa{R{\isaliteral{2E}{\isachardot}}intros} is the list of introduction rules as proven
221 theorems, for the recursive predicates (or sets). The rules are
222 also available individually, using the names given them in the
225 \item \isa{R{\isaliteral{2E}{\isachardot}}cases} is the case analysis (or elimination) rule;
227 \item \isa{R{\isaliteral{2E}{\isachardot}}induct} or \isa{R{\isaliteral{2E}{\isachardot}}coinduct} is the (co)induction
232 When several predicates \isa{{\isaliteral{22}{\isachardoublequote}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ R\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} are
233 defined simultaneously, the list of introduction rules is called
234 \isa{{\isaliteral{22}{\isachardoublequote}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{5F}{\isacharunderscore}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{2E}{\isachardot}}intros{\isaliteral{22}{\isachardoublequote}}}, the case analysis rules are
235 called \isa{{\isaliteral{22}{\isachardoublequote}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2E}{\isachardot}}cases{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ R\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{2E}{\isachardot}}cases{\isaliteral{22}{\isachardoublequote}}}, and the list
236 of mutual induction rules is called \isa{{\isaliteral{22}{\isachardoublequote}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{5F}{\isacharunderscore}}R\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{2E}{\isachardot}}inducts{\isaliteral{22}{\isachardoublequote}}}.%
240 \isamarkupsubsection{Monotonicity theorems%
244 \begin{isamarkuptext}%
245 The context maintains a default set of theorems that are used
246 in monotonicity proofs. New rules can be declared via the
247 \hyperlink{attribute.HOL.mono}{\mbox{\isa{mono}}} attribute. See the main Isabelle/HOL
248 sources for some examples. The general format of such monotonicity
249 theorems is as follows:
253 \item Theorems of the form \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{5C3C6C653E}{\isasymle}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C4D3E}{\isasymM}}\ A\ {\isaliteral{5C3C6C653E}{\isasymle}}\ {\isaliteral{5C3C4D3E}{\isasymM}}\ B{\isaliteral{22}{\isachardoublequote}}}, for proving
254 monotonicity of inductive definitions whose introduction rules have
255 premises involving terms such as \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C4D3E}{\isasymM}}\ R\ t{\isaliteral{22}{\isachardoublequote}}}.
257 \item Monotonicity theorems for logical operators, which are of the
258 general form \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}. For example, in
259 the case of the operator \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6F723E}{\isasymor}}{\isaliteral{22}{\isachardoublequote}}}, the corresponding theorem is
261 \infer{\isa{{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}}}{\isa{{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}P\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{2}}{\isaliteral{22}{\isachardoublequote}}}}
264 \item De Morgan style equations for reasoning about the ``polarity''
267 \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ P{\isaliteral{22}{\isachardoublequote}}} \qquad\qquad
268 \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ {\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C616E643E}{\isasymand}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P\ {\isaliteral{5C3C6F723E}{\isasymor}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ Q{\isaliteral{22}{\isachardoublequote}}}
271 \item Equations for reducing complex operators to more primitive
272 ones whose monotonicity can easily be proved, e.g.
274 \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}P\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ Q{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6C6F6E676C65667472696768746172726F773E}{\isasymlongleftrightarrow}}\ {\isaliteral{5C3C6E6F743E}{\isasymnot}}\ P\ {\isaliteral{5C3C6F723E}{\isasymor}}\ Q{\isaliteral{22}{\isachardoublequote}}} \qquad\qquad
275 \isa{{\isaliteral{22}{\isachardoublequote}}Ball\ A\ P\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ {\isaliteral{5C3C666F72616C6C3E}{\isasymforall}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C6C6F6E6772696768746172726F773E}{\isasymlongrightarrow}}\ P\ x{\isaliteral{22}{\isachardoublequote}}}
282 \isamarkupsubsubsection{Examples%
286 \begin{isamarkuptext}%
287 The finite powerset operator can be defined inductively like this:%
290 \isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}set}\isamarkupfalse%
291 \ Fin\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ set\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ set\ set{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{for}\ A\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ set{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
292 \isakeyword{where}\isanewline
293 \ \ empty{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{7D}{\isacharbraceright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ Fin\ A{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
294 {\isaliteral{7C}{\isacharbar}}\ insert{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}a\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C696E3E}{\isasymin}}\ Fin\ A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ insert\ a\ B\ {\isaliteral{5C3C696E3E}{\isasymin}}\ Fin\ A{\isaliteral{22}{\isachardoublequoteclose}}%
295 \begin{isamarkuptext}%
296 The accessible part of a relation is defined as follows:%
299 \isacommand{inductive}\isamarkupfalse%
300 \ acc\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
301 \ \ \isakeyword{for}\ r\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infix}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C707265633E}{\isasymprec}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
302 \isakeyword{where}\ acc{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C416E643E}{\isasymAnd}}y{\isaliteral{2E}{\isachardot}}\ y\ {\isaliteral{5C3C707265633E}{\isasymprec}}\ x\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ acc\ r\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ acc\ r\ x{\isaliteral{22}{\isachardoublequoteclose}}%
303 \begin{isamarkuptext}%
304 Common logical connectives can be easily characterized as
305 non-recursive inductive definitions with parameters, but without
309 \isacommand{inductive}\isamarkupfalse%
310 \ AND\ \isakeyword{for}\ A\ B\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ bool\isanewline
311 \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ AND\ A\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
313 \isacommand{inductive}\isamarkupfalse%
314 \ OR\ \isakeyword{for}\ A\ B\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ bool\isanewline
315 \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}A\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ OR\ A\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
316 \ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ OR\ A\ B{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
318 \isacommand{inductive}\isamarkupfalse%
319 \ EXISTS\ \isakeyword{for}\ B\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
320 \isakeyword{where}\ {\isaliteral{22}{\isachardoublequoteopen}}B\ a\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ EXISTS\ B{\isaliteral{22}{\isachardoublequoteclose}}%
321 \begin{isamarkuptext}%
322 Here the \isa{{\isaliteral{22}{\isachardoublequote}}cases{\isaliteral{22}{\isachardoublequote}}} or \isa{{\isaliteral{22}{\isachardoublequote}}induct{\isaliteral{22}{\isachardoublequote}}} rules produced by
323 the \hyperlink{command.inductive}{\mbox{\isa{\isacommand{inductive}}}} package coincide with the expected
324 elimination rules for Natural Deduction. Already in the original
325 article by Gerhard Gentzen \cite{Gentzen:1935} there is a hint that
326 each connective can be characterized by its introductions, and the
327 elimination can be constructed systematically.%
331 \isamarkupsection{Recursive functions \label{sec:recursion}%
335 \begin{isamarkuptext}%
336 \begin{matharray}{rcl}
337 \indexdef{HOL}{command}{primrec}\hypertarget{command.HOL.primrec}{\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
338 \indexdef{HOL}{command}{fun}\hypertarget{command.HOL.fun}{\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
339 \indexdef{HOL}{command}{function}\hypertarget{command.HOL.function}{\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
340 \indexdef{HOL}{command}{termination}\hypertarget{command.HOL.termination}{\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
345 \rail@term{\hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}}[]
348 \rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]
350 \rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]
351 \rail@term{\isa{\isakeyword{where}}}[]
352 \rail@nont{\isa{equations}}[]
356 \rail@term{\hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}}}[]
358 \rail@term{\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}}[]
362 \rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]
366 \rail@nont{\isa{functionopts}}[]
368 \rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]
370 \rail@term{\isa{\isakeyword{where}}}[]
371 \rail@nont{\isa{equations}}[]
373 \rail@begin{3}{\isa{equations}}
377 \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
379 \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
381 \rail@cterm{\isa{{\isaliteral{7C}{\isacharbar}}}}[]
384 \rail@begin{3}{\isa{functionopts}}
385 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
388 \rail@term{\isa{sequential}}[]
390 \rail@term{\isa{domintros}}[]
393 \rail@cterm{\isa{{\isaliteral{2C}{\isacharcomma}}}}[]
395 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
398 \rail@term{\hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}}[]
401 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
409 \item \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}} defines primitive recursive
410 functions over datatypes (see also \indexref{HOL}{command}{datatype}\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} and
411 \indexref{HOL}{command}{rep\_datatype}\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isaliteral{5F}{\isacharunderscore}}datatype}}}}). The given \isa{equations}
412 specify reduction rules that are produced by instantiating the
413 generic combinator for primitive recursion that is available for
416 Each equation needs to be of the form:
419 {\isaliteral{22}{\isachardoublequote}}f\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ x\isaliteral{5C3C5E7375623E}{}\isactrlsub m\ {\isaliteral{28}{\isacharparenleft}}C\ y\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ y\isaliteral{5C3C5E7375623E}{}\isactrlsub k{\isaliteral{29}{\isacharparenright}}\ z\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ z\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3D}{\isacharequal}}\ rhs{\isaliteral{22}{\isachardoublequote}}%
422 such that \isa{C} is a datatype constructor, \isa{rhs} contains
423 only the free variables on the left-hand side (or from the context),
424 and all recursive occurrences of \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{22}{\isachardoublequote}}} in \isa{{\isaliteral{22}{\isachardoublequote}}rhs{\isaliteral{22}{\isachardoublequote}}} are of
425 the form \isa{{\isaliteral{22}{\isachardoublequote}}f\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ y\isaliteral{5C3C5E7375623E}{}\isactrlsub i\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}} for some \isa{i}. At most one
426 reduction rule for each constructor can be given. The order does
427 not matter. For missing constructors, the function is defined to
428 return a default value, but this equation is made difficult to
431 The reduction rules are declared as \hyperlink{attribute.simp}{\mbox{\isa{simp}}} by default,
432 which enables standard proof methods like \hyperlink{method.simp}{\mbox{\isa{simp}}} and
433 \hyperlink{method.auto}{\mbox{\isa{auto}}} to normalize expressions of \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{22}{\isachardoublequote}}} applied to
434 datatype constructions, by simulating symbolic computation via
437 \item \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} defines functions by general
438 wellfounded recursion. A detailed description with examples can be
439 found in \cite{isabelle-function}. The function is specified by a
440 set of (possibly conditional) recursive equations with arbitrary
441 pattern matching. The command generates proof obligations for the
442 completeness and the compatibility of patterns.
444 The defined function is considered partial, and the resulting
445 simplification rules (named \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{2E}{\isachardot}}psimps{\isaliteral{22}{\isachardoublequote}}}) and induction rule
446 (named \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{2E}{\isachardot}}pinduct{\isaliteral{22}{\isachardoublequote}}}) are guarded by a generated domain
447 predicate \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{5F}{\isacharunderscore}}dom{\isaliteral{22}{\isachardoublequote}}}. The \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}
448 command can then be used to establish that the function is total.
450 \item \hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}} is a shorthand notation for ``\hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}sequential{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}, followed by automated
451 proof attempts regarding pattern matching and termination. See
452 \cite{isabelle-function} for further details.
454 \item \hyperlink{command.HOL.termination}{\mbox{\isa{\isacommand{termination}}}}~\isa{f} commences a
455 termination proof for the previously defined function \isa{f}. If
456 this is omitted, the command refers to the most recent function
457 definition. After the proof is closed, the recursive equations and
458 the induction principle is established.
462 Recursive definitions introduced by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}}
463 command accommodate reasoning by induction (cf.\ \hyperlink{method.induct}{\mbox{\isa{induct}}}):
464 rule \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{2E}{\isachardot}}induct{\isaliteral{22}{\isachardoublequote}}} refers to a specific induction rule, with
465 parameters named according to the user-specified equations. Cases
466 are numbered starting from 1. For \hyperlink{command.HOL.primrec}{\mbox{\isa{\isacommand{primrec}}}}, the
467 induction principle coincides with structural recursion on the
468 datatype where the recursion is carried out.
470 The equations provided by these packages may be referred later as
471 theorem list \isa{{\isaliteral{22}{\isachardoublequote}}f{\isaliteral{2E}{\isachardot}}simps{\isaliteral{22}{\isachardoublequote}}}, where \isa{f} is the (collective)
472 name of the functions defined. Individual equations may be named
475 The \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} command accepts the following
480 \item \isa{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 theorems. Also note that this automatic
488 transformation only works for ML-style datatype patterns.
490 \item \isa{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.
498 \isamarkupsubsubsection{Example: evaluation of expressions%
502 \begin{isamarkuptext}%
503 Subsequently, we define mutual datatypes for arithmetic and
504 boolean expressions, and use \hyperlink{command.primrec}{\mbox{\isa{\isacommand{primrec}}}} for evaluation
505 functions that follow the same recursive structure.%
508 \isacommand{datatype}\isamarkupfalse%
509 \ {\isaliteral{27}{\isacharprime}}a\ aexp\ {\isaliteral{3D}{\isacharequal}}\isanewline
510 \ \ \ \ IF\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
511 \ \ {\isaliteral{7C}{\isacharbar}}\ Sum\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
512 \ \ {\isaliteral{7C}{\isacharbar}}\ Diff\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
513 \ \ {\isaliteral{7C}{\isacharbar}}\ Var\ {\isaliteral{27}{\isacharprime}}a\isanewline
514 \ \ {\isaliteral{7C}{\isacharbar}}\ Num\ nat\isanewline
515 \isakeyword{and}\ {\isaliteral{27}{\isacharprime}}a\ bexp\ {\isaliteral{3D}{\isacharequal}}\isanewline
516 \ \ \ \ Less\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
517 \ \ {\isaliteral{7C}{\isacharbar}}\ And\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
518 \ \ {\isaliteral{7C}{\isacharbar}}\ Neg\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequoteclose}}%
519 \begin{isamarkuptext}%
520 \medskip Evaluation of arithmetic and boolean expressions%
523 \isacommand{primrec}\isamarkupfalse%
524 \ evala\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ aexp\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
525 \ \ \isakeyword{and}\ evalb\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ bexp\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
526 \isakeyword{where}\isanewline
527 \ \ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}IF\ b\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ evalb\ env\ b\ then\ evala\ env\ a{\isadigit{1}}\ else\ evala\ env\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
528 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}Sum\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evala\ env\ a{\isadigit{1}}\ {\isaliteral{2B}{\isacharplus}}\ evala\ env\ a{\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
529 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}Diff\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evala\ env\ a{\isadigit{1}}\ {\isaliteral{2D}{\isacharminus}}\ evala\ env\ a{\isadigit{2}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
530 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}Var\ v{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ env\ v{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
531 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}Num\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
532 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evalb\ env\ {\isaliteral{28}{\isacharparenleft}}Less\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}evala\ env\ a{\isadigit{1}}\ {\isaliteral{3C}{\isacharless}}\ evala\ env\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
533 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evalb\ env\ {\isaliteral{28}{\isacharparenleft}}And\ b{\isadigit{1}}\ b{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}evalb\ env\ b{\isadigit{1}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ evalb\ env\ b{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
534 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}evalb\ env\ {\isaliteral{28}{\isacharparenleft}}Neg\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6E6F743E}{\isasymnot}}\ evalb\ env\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
535 \begin{isamarkuptext}%
536 Since the value of an expression depends on the value of its
537 variables, the functions \isa{evala} and \isa{evalb} take an
538 additional parameter, an \emph{environment} that maps variables to
541 \medskip Substitution on expressions can be defined similarly. The
542 mapping \isa{f} of type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequote}}} given as a
543 parameter is lifted canonically on the types \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequote}}} and
544 \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequote}}}, respectively.%
547 \isacommand{primrec}\isamarkupfalse%
548 \ substa\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ aexp{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ aexp\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ aexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
549 \ \ \isakeyword{and}\ substb\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ aexp{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ bexp\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ bexp{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
550 \isakeyword{where}\isanewline
551 \ \ {\isaliteral{22}{\isachardoublequoteopen}}substa\ f\ {\isaliteral{28}{\isacharparenleft}}IF\ b\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ IF\ {\isaliteral{28}{\isacharparenleft}}substb\ f\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
552 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substa\ f\ {\isaliteral{28}{\isacharparenleft}}Sum\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Sum\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
553 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substa\ f\ {\isaliteral{28}{\isacharparenleft}}Diff\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Diff\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
554 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substa\ f\ {\isaliteral{28}{\isacharparenleft}}Var\ v{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ v{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
555 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substa\ f\ {\isaliteral{28}{\isacharparenleft}}Num\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Num\ n{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
556 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substb\ f\ {\isaliteral{28}{\isacharparenleft}}Less\ a{\isadigit{1}}\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Less\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substa\ f\ a{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
557 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substb\ f\ {\isaliteral{28}{\isacharparenleft}}And\ b{\isadigit{1}}\ b{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ And\ {\isaliteral{28}{\isacharparenleft}}substb\ f\ b{\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}substb\ f\ b{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
558 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}substb\ f\ {\isaliteral{28}{\isacharparenleft}}Neg\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Neg\ {\isaliteral{28}{\isacharparenleft}}substb\ f\ b{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
559 \begin{isamarkuptext}%
560 In textbooks about semantics one often finds substitution
561 theorems, which express the relationship between substitution and
562 evaluation. For \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ aexp{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ bexp{\isaliteral{22}{\isachardoublequote}}}, we can prove
563 such a theorem by mutual induction, followed by simplification.%
566 \isacommand{lemma}\isamarkupfalse%
567 \ subst{\isaliteral{5F}{\isacharunderscore}}one{\isaliteral{3A}{\isacharcolon}}\isanewline
568 \ \ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}substa\ {\isaliteral{28}{\isacharparenleft}}Var\ {\isaliteral{28}{\isacharparenleft}}v\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evala\ {\isaliteral{28}{\isacharparenleft}}env\ {\isaliteral{28}{\isacharparenleft}}v\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ evala\ env\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ a{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
569 \ \ {\isaliteral{22}{\isachardoublequoteopen}}evalb\ env\ {\isaliteral{28}{\isacharparenleft}}substb\ {\isaliteral{28}{\isacharparenleft}}Var\ {\isaliteral{28}{\isacharparenleft}}v\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evalb\ {\isaliteral{28}{\isacharparenleft}}env\ {\isaliteral{28}{\isacharparenleft}}v\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ evala\ env\ a{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ b{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
576 \isacommand{by}\isamarkupfalse%
577 \ {\isaliteral{28}{\isacharparenleft}}induct\ a\ \isakeyword{and}\ b{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all%
586 \isacommand{lemma}\isamarkupfalse%
587 \ subst{\isaliteral{5F}{\isacharunderscore}}all{\isaliteral{3A}{\isacharcolon}}\isanewline
588 \ \ {\isaliteral{22}{\isachardoublequoteopen}}evala\ env\ {\isaliteral{28}{\isacharparenleft}}substa\ s\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evala\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ evala\ env\ {\isaliteral{28}{\isacharparenleft}}s\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ a{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
589 \ \ {\isaliteral{22}{\isachardoublequoteopen}}evalb\ env\ {\isaliteral{28}{\isacharparenleft}}substb\ s\ b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ evalb\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ evala\ env\ {\isaliteral{28}{\isacharparenleft}}s\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ b{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
596 \isacommand{by}\isamarkupfalse%
597 \ {\isaliteral{28}{\isacharparenleft}}induct\ a\ \isakeyword{and}\ b{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all%
605 \isamarkupsubsubsection{Example: a substitution function for terms%
609 \begin{isamarkuptext}%
610 Functions on datatypes with nested recursion are also defined
611 by mutual primitive recursion.%
614 \isacommand{datatype}\isamarkupfalse%
615 \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{22}{\isachardoublequoteopen}}term{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{3D}{\isacharequal}}\ Var\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{7C}{\isacharbar}}\ App\ {\isaliteral{27}{\isacharprime}}b\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term\ list{\isaliteral{22}{\isachardoublequoteclose}}%
616 \begin{isamarkuptext}%
617 A substitution function on type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term{\isaliteral{22}{\isachardoublequote}}} can be
618 defined as follows, by working simultaneously on \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term\ list{\isaliteral{22}{\isachardoublequote}}}:%
621 \isacommand{primrec}\isamarkupfalse%
622 \ subst{\isaliteral{5F}{\isacharunderscore}}term\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline
623 \ \ subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term\ list\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term\ list{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
624 \isakeyword{where}\isanewline
625 \ \ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term\ f\ {\isaliteral{28}{\isacharparenleft}}Var\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ f\ a{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
626 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term\ f\ {\isaliteral{28}{\isacharparenleft}}App\ b\ ts{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ App\ b\ {\isaliteral{28}{\isacharparenleft}}subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f\ ts{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
627 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
628 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f\ {\isaliteral{28}{\isacharparenleft}}t\ {\isaliteral{23}{\isacharhash}}\ ts{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ subst{\isaliteral{5F}{\isacharunderscore}}term\ f\ t\ {\isaliteral{23}{\isacharhash}}\ subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f\ ts{\isaliteral{22}{\isachardoublequoteclose}}%
629 \begin{isamarkuptext}%
630 The recursion scheme follows the structure of the unfolded
631 definition of type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ term{\isaliteral{22}{\isachardoublequote}}}. To prove properties of this
632 substitution function, mutual induction is needed:%
635 \isacommand{lemma}\isamarkupfalse%
636 \ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term\ {\isaliteral{28}{\isacharparenleft}}subst{\isaliteral{5F}{\isacharunderscore}}term\ f{\isadigit{1}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ f{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ t\ {\isaliteral{3D}{\isacharequal}}\ subst{\isaliteral{5F}{\isacharunderscore}}term\ f{\isadigit{1}}\ {\isaliteral{28}{\isacharparenleft}}subst{\isaliteral{5F}{\isacharunderscore}}term\ f{\isadigit{2}}\ t{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\ \isakeyword{and}\isanewline
637 \ \ {\isaliteral{22}{\isachardoublequoteopen}}subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ {\isaliteral{28}{\isacharparenleft}}subst{\isaliteral{5F}{\isacharunderscore}}term\ f{\isadigit{1}}\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ f{\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ ts\ {\isaliteral{3D}{\isacharequal}}\ subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f{\isadigit{1}}\ {\isaliteral{28}{\isacharparenleft}}subst{\isaliteral{5F}{\isacharunderscore}}term{\isaliteral{5F}{\isacharunderscore}}list\ f{\isadigit{2}}\ ts{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
644 \isacommand{by}\isamarkupfalse%
645 \ {\isaliteral{28}{\isacharparenleft}}induct\ t\ \isakeyword{and}\ ts{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all%
653 \isamarkupsubsubsection{Example: a map function for infinitely branching trees%
657 \begin{isamarkuptext}%
658 Defining functions on infinitely branching datatypes by
659 primitive recursion is just as easy.%
662 \isacommand{datatype}\isamarkupfalse%
663 \ {\isaliteral{27}{\isacharprime}}a\ tree\ {\isaliteral{3D}{\isacharequal}}\ Atom\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{7C}{\isacharbar}}\ Branch\ {\isaliteral{22}{\isachardoublequoteopen}}nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ tree{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
665 \isacommand{primrec}\isamarkupfalse%
666 \ map{\isaliteral{5F}{\isacharunderscore}}tree\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}a\ tree\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ tree{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
667 \isakeyword{where}\isanewline
668 \ \ {\isaliteral{22}{\isachardoublequoteopen}}map{\isaliteral{5F}{\isacharunderscore}}tree\ f\ {\isaliteral{28}{\isacharparenleft}}Atom\ a{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Atom\ {\isaliteral{28}{\isacharparenleft}}f\ a{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
669 {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}map{\isaliteral{5F}{\isacharunderscore}}tree\ f\ {\isaliteral{28}{\isacharparenleft}}Branch\ ts{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ Branch\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C6C616D6264613E}{\isasymlambda}}x{\isaliteral{2E}{\isachardot}}\ map{\isaliteral{5F}{\isacharunderscore}}tree\ f\ {\isaliteral{28}{\isacharparenleft}}ts\ x{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}%
670 \begin{isamarkuptext}%
671 Note that all occurrences of functions such as \isa{ts}
672 above must be applied to an argument. In particular, \isa{{\isaliteral{22}{\isachardoublequote}}map{\isaliteral{5F}{\isacharunderscore}}tree\ f\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ ts{\isaliteral{22}{\isachardoublequote}}} is not allowed here.%
676 \begin{isamarkuptext}%
677 Here is a simple composition lemma for \isa{map{\isaliteral{5F}{\isacharunderscore}}tree}:%
680 \isacommand{lemma}\isamarkupfalse%
681 \ {\isaliteral{22}{\isachardoublequoteopen}}map{\isaliteral{5F}{\isacharunderscore}}tree\ g\ {\isaliteral{28}{\isacharparenleft}}map{\isaliteral{5F}{\isacharunderscore}}tree\ f\ t{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ map{\isaliteral{5F}{\isacharunderscore}}tree\ {\isaliteral{28}{\isacharparenleft}}g\ {\isaliteral{5C3C636972633E}{\isasymcirc}}\ f{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
688 \isacommand{by}\isamarkupfalse%
689 \ {\isaliteral{28}{\isacharparenleft}}induct\ t{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all%
697 \isamarkupsubsection{Proof methods related to recursive definitions%
701 \begin{isamarkuptext}%
702 \begin{matharray}{rcl}
703 \indexdef{HOL}{method}{pat\_completeness}\hypertarget{method.HOL.pat-completeness}{\hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isaliteral{5F}{\isacharunderscore}}completeness}}}} & : & \isa{method} \\
704 \indexdef{HOL}{method}{relation}\hypertarget{method.HOL.relation}{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}} & : & \isa{method} \\
705 \indexdef{HOL}{method}{lexicographic\_order}\hypertarget{method.HOL.lexicographic-order}{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order}}}} & : & \isa{method} \\
706 \indexdef{HOL}{method}{size\_change}\hypertarget{method.HOL.size-change}{\hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isaliteral{5F}{\isacharunderscore}}change}}}} & : & \isa{method} \\
711 \rail@term{\hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}}[]
712 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
715 \rail@term{\hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order}}}}[]
718 \rail@cnont{\hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}}[]
722 \rail@term{\hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isaliteral{5F}{\isacharunderscore}}change}}}}[]
723 \rail@nont{\isa{orders}}[]
726 \rail@cnont{\hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}}[]
729 \rail@begin{4}{\isa{orders}}
733 \rail@term{\isa{max}}[]
735 \rail@term{\isa{min}}[]
737 \rail@term{\isa{ms}}[]
746 \item \hyperlink{method.HOL.pat-completeness}{\mbox{\isa{pat{\isaliteral{5F}{\isacharunderscore}}completeness}}} is a specialized method to
747 solve goals regarding the completeness of pattern matching, as
748 required by the \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} package (cf.\
749 \cite{isabelle-function}).
751 \item \hyperlink{method.HOL.relation}{\mbox{\isa{relation}}}~\isa{R} introduces a termination
752 proof using the relation \isa{R}. The resulting proof state will
753 contain goals expressing that \isa{R} is wellfounded, and that the
754 arguments of recursive calls decrease with respect to \isa{R}.
755 Usually, this method is used as the initial proof step of manual
758 \item \hyperlink{method.HOL.lexicographic-order}{\mbox{\isa{lexicographic{\isaliteral{5F}{\isacharunderscore}}order}}} attempts a fully
759 automated termination proof by searching for a lexicographic
760 combination of size measures on the arguments of the function. The
761 method accepts the same arguments as the \hyperlink{method.auto}{\mbox{\isa{auto}}} method,
762 which it uses internally to prove local descents. The same context
763 modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see
764 \secref{sec:clasimp}.
766 In case of failure, extensive information is printed, which can help
767 to analyse the situation (cf.\ \cite{isabelle-function}).
769 \item \hyperlink{method.HOL.size-change}{\mbox{\isa{size{\isaliteral{5F}{\isacharunderscore}}change}}} also works on termination goals,
770 using a variation of the size-change principle, together with a
771 graph decomposition technique (see \cite{krauss_phd} for details).
772 Three kinds of orders are used internally: \isa{max}, \isa{min},
773 and \isa{ms} (multiset), which is only available when the theory
774 \isa{Multiset} is loaded. When no order kinds are given, they are
775 tried in order. The search for a termination proof uses SAT solving
778 For local descent proofs, the same context modifiers as for \hyperlink{method.auto}{\mbox{\isa{auto}}} are accepted, see \secref{sec:clasimp}.
784 \isamarkupsubsection{Functions with explicit partiality%
788 \begin{isamarkuptext}%
789 \begin{matharray}{rcl}
790 \indexdef{HOL}{command}{partial\_function}\hypertarget{command.HOL.partial-function}{\hyperlink{command.HOL.partial-function}{\mbox{\isa{\isacommand{partial{\isaliteral{5F}{\isacharunderscore}}function}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
791 \indexdef{HOL}{attribute}{partial\_function\_mono}\hypertarget{attribute.HOL.partial-function-mono}{\hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isaliteral{5F}{\isacharunderscore}}function{\isaliteral{5F}{\isacharunderscore}}mono}}}} & : & \isa{attribute} \\
796 \rail@term{\hyperlink{command.HOL.partial-function}{\mbox{\isa{\isacommand{partial{\isaliteral{5F}{\isacharunderscore}}function}}}}}[]
799 \rail@nont{\hyperlink{syntax.target}{\mbox{\isa{target}}}}[]
801 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
802 \rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]
803 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
804 \rail@nont{\hyperlink{syntax.fixes}{\mbox{\isa{fixes}}}}[]
806 \rail@term{\isa{\isakeyword{where}}}[]
809 \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
811 \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
818 \item \hyperlink{command.HOL.partial-function}{\mbox{\isa{\isacommand{partial{\isaliteral{5F}{\isacharunderscore}}function}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}mode{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} defines
819 recursive functions based on fixpoints in complete partial
820 orders. No termination proof is required from the user or
821 constructed internally. Instead, the possibility of non-termination
822 is modelled explicitly in the result type, which contains an
823 explicit bottom element.
825 Pattern matching and mutual recursion are currently not supported.
826 Thus, the specification consists of a single function described by a
827 single recursive equation.
829 There are no fixed syntactic restrictions on the body of the
830 function, but the induced functional must be provably monotonic
831 wrt.\ the underlying order. The monotonicitity proof is performed
832 internally, and the definition is rejected when it fails. The proof
833 can be influenced by declaring hints using the
834 \hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isaliteral{5F}{\isacharunderscore}}function{\isaliteral{5F}{\isacharunderscore}}mono}}} attribute.
836 The mandatory \isa{mode} argument specifies the mode of operation
837 of the command, which directly corresponds to a complete partial
838 order on the result type. By default, the following modes are
842 \item \isa{option} defines functions that map into the \isa{option} type. Here, the value \isa{None} is used to model a
843 non-terminating computation. Monotonicity requires that if \isa{None} is returned by a recursive call, then the overall result
844 must also be \isa{None}. This is best achieved through the use of
845 the monadic operator \isa{{\isaliteral{22}{\isachardoublequote}}Option{\isaliteral{2E}{\isachardot}}bind{\isaliteral{22}{\isachardoublequote}}}.
847 \item \isa{tailrec} defines functions with an arbitrary result
848 type and uses the slightly degenerated partial order where \isa{{\isaliteral{22}{\isachardoublequote}}undefined{\isaliteral{22}{\isachardoublequote}}} is the bottom element. Now, monotonicity requires that
849 if \isa{undefined} is returned by a recursive call, then the
850 overall result must also be \isa{undefined}. In practice, this is
851 only satisfied when each recursive call is a tail call, whose result
852 is directly returned. Thus, this mode of operation allows the
853 definition of arbitrary tail-recursive functions.
856 Experienced users may define new modes by instantiating the locale
857 \isa{{\isaliteral{22}{\isachardoublequote}}partial{\isaliteral{5F}{\isacharunderscore}}function{\isaliteral{5F}{\isacharunderscore}}definitions{\isaliteral{22}{\isachardoublequote}}} appropriately.
859 \item \hyperlink{attribute.HOL.partial-function-mono}{\mbox{\isa{partial{\isaliteral{5F}{\isacharunderscore}}function{\isaliteral{5F}{\isacharunderscore}}mono}}} declares rules for
860 use in the internal monononicity proofs of partial function
867 \isamarkupsubsection{Old-style recursive function definitions (TFL)%
871 \begin{isamarkuptext}%
872 The old TFL commands \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} and \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isaliteral{5F}{\isacharunderscore}}tc}}}} for defining recursive are mostly obsolete; \hyperlink{command.HOL.function}{\mbox{\isa{\isacommand{function}}}} or \hyperlink{command.HOL.fun}{\mbox{\isa{\isacommand{fun}}}} should be used instead.
874 \begin{matharray}{rcl}
875 \indexdef{HOL}{command}{recdef}\hypertarget{command.HOL.recdef}{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
876 \indexdef{HOL}{command}{recdef\_tc}\hypertarget{command.HOL.recdef-tc}{\hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isaliteral{5F}{\isacharunderscore}}tc}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
881 \rail@term{\hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}}}[]
884 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
885 \rail@term{\isa{\isakeyword{permissive}}}[]
886 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
889 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
890 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
892 \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
897 \rail@nont{\isa{hints}}[]
901 \rail@nont{\isa{recdeftc}}[]
904 \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
906 \rail@nont{\isa{tc}}[]
908 \rail@begin{2}{\isa{hints}}
909 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
910 \rail@term{\isa{\isakeyword{hints}}}[]
913 \rail@cnont{\isa{recdefmod}}[]
915 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
917 \rail@begin{4}{\isa{recdefmod}}
920 \rail@term{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}simp}}[]
922 \rail@term{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}cong}}[]
924 \rail@term{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}wf}}[]
928 \rail@term{\isa{add}}[]
930 \rail@term{\isa{del}}[]
932 \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
933 \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
935 \rail@nont{\hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}}[]
938 \rail@begin{2}{\isa{tc}}
939 \rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]
942 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
943 \rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]
944 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
952 \item \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} defines general well-founded
953 recursive functions (using the TFL package), see also
954 \cite{isabelle-HOL}. The ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}permissive{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' option tells
955 TFL to recover from failed proof attempts, returning unfinished
956 results. The \isa{recdef{\isaliteral{5F}{\isacharunderscore}}simp}, \isa{recdef{\isaliteral{5F}{\isacharunderscore}}cong}, and \isa{recdef{\isaliteral{5F}{\isacharunderscore}}wf} hints refer to auxiliary rules to be used in the internal
957 automated proof process of TFL. Additional \hyperlink{syntax.clasimpmod}{\mbox{\isa{clasimpmod}}}
958 declarations (cf.\ \secref{sec:clasimp}) may be given to tune the
959 context of the Simplifier (cf.\ \secref{sec:simplifier}) and
960 Classical reasoner (cf.\ \secref{sec:classical}).
962 \item \hyperlink{command.HOL.recdef-tc}{\mbox{\isa{\isacommand{recdef{\isaliteral{5F}{\isacharunderscore}}tc}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}c\ {\isaliteral{28}{\isacharparenleft}}i{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} recommences the
963 proof for leftover termination condition number \isa{i} (default
964 1) as generated by a \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} definition of
967 Note that in most cases, \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} is able to finish
968 its internal proofs without manual intervention.
972 \medskip Hints for \hyperlink{command.HOL.recdef}{\mbox{\isa{\isacommand{recdef}}}} may be also declared
973 globally, using the following attributes.
975 \begin{matharray}{rcl}
976 \indexdef{HOL}{attribute}{recdef\_simp}\hypertarget{attribute.HOL.recdef-simp}{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}simp}}}} & : & \isa{attribute} \\
977 \indexdef{HOL}{attribute}{recdef\_cong}\hypertarget{attribute.HOL.recdef-cong}{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}cong}}}} & : & \isa{attribute} \\
978 \indexdef{HOL}{attribute}{recdef\_wf}\hypertarget{attribute.HOL.recdef-wf}{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}wf}}}} & : & \isa{attribute} \\
984 \rail@term{\hyperlink{attribute.HOL.recdef-simp}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}simp}}}}[]
986 \rail@term{\hyperlink{attribute.HOL.recdef-cong}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}cong}}}}[]
988 \rail@term{\hyperlink{attribute.HOL.recdef-wf}{\mbox{\isa{recdef{\isaliteral{5F}{\isacharunderscore}}wf}}}}[]
992 \rail@term{\isa{add}}[]
994 \rail@term{\isa{del}}[]
1001 \isamarkupsection{Datatypes \label{sec:hol-datatype}%
1005 \begin{isamarkuptext}%
1006 \begin{matharray}{rcl}
1007 \indexdef{HOL}{command}{datatype}\hypertarget{command.HOL.datatype}{\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
1008 \indexdef{HOL}{command}{rep\_datatype}\hypertarget{command.HOL.rep-datatype}{\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isaliteral{5F}{\isacharunderscore}}datatype}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
1013 \rail@term{\hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}}}[]
1015 \rail@nont{\isa{spec}}[]
1017 \rail@cterm{\isa{\isakeyword{and}}}[]
1021 \rail@term{\hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isaliteral{5F}{\isacharunderscore}}datatype}}}}}[]
1024 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
1026 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
1029 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
1032 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
1036 \rail@begin{2}{\isa{spec}}
1039 \rail@nont{\hyperlink{syntax.parname}{\mbox{\isa{parname}}}}[]
1041 \rail@nont{\hyperlink{syntax.typespec}{\mbox{\isa{typespec}}}}[]
1044 \rail@nont{\hyperlink{syntax.mixfix}{\mbox{\isa{mixfix}}}}[]
1046 \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
1048 \rail@nont{\isa{cons}}[]
1050 \rail@cterm{\isa{{\isaliteral{7C}{\isacharbar}}}}[]
1053 \rail@begin{2}{\isa{cons}}
1054 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
1057 \rail@cnont{\hyperlink{syntax.type}{\mbox{\isa{type}}}}[]
1061 \rail@nont{\hyperlink{syntax.mixfix}{\mbox{\isa{mixfix}}}}[]
1069 \item \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} defines inductive datatypes in
1072 \item \hyperlink{command.HOL.rep-datatype}{\mbox{\isa{\isacommand{rep{\isaliteral{5F}{\isacharunderscore}}datatype}}}} represents existing types as
1075 For foundational reasons, some basic types such as \isa{nat}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C74696D65733E}{\isasymtimes}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{22}{\isachardoublequote}}}, \isa{bool} and \isa{unit} are
1076 introduced by more primitive means using \indexref{}{command}{typedef}\hyperlink{command.typedef}{\mbox{\isa{\isacommand{typedef}}}}. To
1077 recover the rich infrastructure of \hyperlink{command.datatype}{\mbox{\isa{\isacommand{datatype}}}} (e.g.\ rules
1078 for \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} and the primitive recursion
1079 combinators), such types may be represented as actual datatypes
1080 later. This is done by specifying the constructors of the desired
1081 type, and giving a proof of the induction rule, distinctness and
1082 injectivity of constructors.
1084 For example, see \verb|~~/src/HOL/Sum_Type.thy| for the
1085 representation of the primitive sum type as fully-featured datatype.
1089 The generated rules for \hyperlink{method.induct}{\mbox{\isa{induct}}} and \hyperlink{method.cases}{\mbox{\isa{cases}}} provide
1090 case names according to the given constructors, while parameters are
1091 named after the types (see also \secref{sec:cases-induct}).
1093 See \cite{isabelle-HOL} for more details on datatypes, but beware of
1094 the old-style theory syntax being used there! Apart from proper
1095 proof methods for case-analysis and induction, there are also
1096 emulations of ML tactics \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}tac}}} available, see \secref{sec:hol-induct-tac}; these admit
1097 to refer directly to the internal structure of subgoals (including
1098 internally bound parameters).%
1099 \end{isamarkuptext}%
1102 \isamarkupsubsubsection{Examples%
1106 \begin{isamarkuptext}%
1107 We define a type of finite sequences, with slightly different
1108 names than the existing \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{27}{\isacharprime}}a\ list{\isaliteral{22}{\isachardoublequote}}} that is already in \hyperlink{theory.Main}{\mbox{\isa{Main}}}:%
1109 \end{isamarkuptext}%
1111 \isacommand{datatype}\isamarkupfalse%
1112 \ {\isaliteral{27}{\isacharprime}}a\ seq\ {\isaliteral{3D}{\isacharequal}}\ Empty\ {\isaliteral{7C}{\isacharbar}}\ Seq\ {\isaliteral{27}{\isacharprime}}a\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ seq{\isaliteral{22}{\isachardoublequoteclose}}%
1113 \begin{isamarkuptext}%
1114 We can now prove some simple lemma by structural induction:%
1115 \end{isamarkuptext}%
1117 \isacommand{lemma}\isamarkupfalse%
1118 \ {\isaliteral{22}{\isachardoublequoteopen}}Seq\ x\ xs\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ xs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
1125 \isacommand{proof}\isamarkupfalse%
1126 \ {\isaliteral{28}{\isacharparenleft}}induct\ xs\ arbitrary{\isaliteral{3A}{\isacharcolon}}\ x{\isaliteral{29}{\isacharparenright}}\isanewline
1127 \ \ \isacommand{case}\isamarkupfalse%
1129 \begin{isamarkuptxt}%
1130 This case can be proved using the simplifier: the freeness
1131 properties of the datatype are already declared as \hyperlink{attribute.simp}{\mbox{\isa{simp}}} rules.%
1134 \ \ \isacommand{show}\isamarkupfalse%
1135 \ {\isaliteral{22}{\isachardoublequoteopen}}Seq\ x\ Empty\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ Empty{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
1136 \ \ \ \ \isacommand{by}\isamarkupfalse%
1138 \isacommand{next}\isamarkupfalse%
1140 \ \ \isacommand{case}\isamarkupfalse%
1141 \ {\isaliteral{28}{\isacharparenleft}}Seq\ y\ ys{\isaliteral{29}{\isacharparenright}}%
1142 \begin{isamarkuptxt}%
1143 The step case is proved similarly.%
1146 \ \ \isacommand{show}\isamarkupfalse%
1147 \ {\isaliteral{22}{\isachardoublequoteopen}}Seq\ x\ {\isaliteral{28}{\isacharparenleft}}Seq\ y\ ys{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ Seq\ y\ ys{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
1148 \ \ \ \ \isacommand{using}\isamarkupfalse%
1149 \ {\isaliteral{60}{\isacharbackquoteopen}}Seq\ y\ ys\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ ys{\isaliteral{60}{\isacharbackquoteclose}}\ \isacommand{by}\isamarkupfalse%
1151 \isacommand{qed}\isamarkupfalse%
1160 \begin{isamarkuptext}%
1161 Here is a more succinct version of the same proof:%
1162 \end{isamarkuptext}%
1164 \isacommand{lemma}\isamarkupfalse%
1165 \ {\isaliteral{22}{\isachardoublequoteopen}}Seq\ x\ xs\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ xs{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
1172 \isacommand{by}\isamarkupfalse%
1173 \ {\isaliteral{28}{\isacharparenleft}}induct\ xs\ arbitrary{\isaliteral{3A}{\isacharcolon}}\ x{\isaliteral{29}{\isacharparenright}}\ simp{\isaliteral{5F}{\isacharunderscore}}all%
1181 \isamarkupsection{Records \label{sec:hol-record}%
1185 \begin{isamarkuptext}%
1186 In principle, records merely generalize the concept of tuples, where
1187 components may be addressed by labels instead of just position. The
1188 logical infrastructure of records in Isabelle/HOL is slightly more
1189 advanced, though, supporting truly extensible record schemes. This
1190 admits operations that are polymorphic with respect to record
1191 extension, yielding ``object-oriented'' effects like (single)
1192 inheritance. See also \cite{NaraschewskiW-TPHOLs98} for more
1193 details on object-oriented verification and record subtyping in HOL.%
1194 \end{isamarkuptext}%
1197 \isamarkupsubsection{Basic concepts%
1201 \begin{isamarkuptext}%
1202 Isabelle/HOL supports both \emph{fixed} and \emph{schematic} records
1203 at the level of terms and types. The notation is as follows:
1206 \begin{tabular}{l|l|l}
1207 & record terms & record types \\ \hline
1208 fixed & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ B{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
1209 schematic & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ m{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} &
1210 \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ B{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ M{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
1214 \noindent The ASCII representation of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} is \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{7C}{\isacharbar}}\ x\ {\isaliteral{3D}{\isacharequal}}\ a\ {\isaliteral{7C}{\isacharbar}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}.
1216 A fixed record \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} has field \isa{x} of value
1217 \isa{a} and field \isa{y} of value \isa{b}. The corresponding
1218 type is \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ B{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}, assuming that \isa{{\isaliteral{22}{\isachardoublequote}}a\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ A{\isaliteral{22}{\isachardoublequote}}}
1219 and \isa{{\isaliteral{22}{\isachardoublequote}}b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ B{\isaliteral{22}{\isachardoublequote}}}.
1221 A record scheme like \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ m{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} contains fields
1222 \isa{x} and \isa{y} as before, but also possibly further fields
1223 as indicated by the ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}'' notation (which is actually part
1224 of the syntax). The improper field ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}'' of a record
1225 scheme is called the \emph{more part}. Logically it is just a free
1226 variable, which is occasionally referred to as ``row variable'' in
1227 the literature. The more part of a record scheme may be
1228 instantiated by zero or more further components. For example, the
1229 previous scheme may get instantiated to \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ z\ {\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ m{\isaliteral{27}{\isacharprime}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}, where \isa{m{\isaliteral{27}{\isacharprime}}} refers to a different more part.
1230 Fixed records are special instances of record schemes, where
1231 ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{22}{\isachardoublequote}}}'' is properly terminated by the \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ unit{\isaliteral{22}{\isachardoublequote}}}
1232 element. In fact, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} is just an abbreviation
1233 for \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}.
1235 \medskip Two key observations make extensible records in a simply
1236 typed language like HOL work out:
1240 \item the more part is internalized, as a free term or type
1243 \item field names are externalized, they cannot be accessed within
1244 the logic as first-class values.
1248 \medskip In Isabelle/HOL record types have to be defined explicitly,
1249 fixing their field names and types, and their (optional) parent
1250 record. Afterwards, records may be formed using above syntax, while
1251 obeying the canonical order of fields as given by their declaration.
1252 The record package provides several standard operations like
1253 selectors and updates. The common setup for various generic proof
1254 tools enable succinct reasoning patterns. See also the Isabelle/HOL
1255 tutorial \cite{isabelle-hol-book} for further instructions on using
1256 records in practice.%
1257 \end{isamarkuptext}%
1260 \isamarkupsubsection{Record specifications%
1264 \begin{isamarkuptext}%
1265 \begin{matharray}{rcl}
1266 \indexdef{HOL}{command}{record}\hypertarget{command.HOL.record}{\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
1271 \rail@term{\hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}}[]
1272 \rail@nont{\hyperlink{syntax.typespec-sorts}{\mbox{\isa{typespec{\isaliteral{5F}{\isacharunderscore}}sorts}}}}[]
1273 \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
1277 \rail@nont{\hyperlink{syntax.type}{\mbox{\isa{type}}}}[]
1278 \rail@term{\isa{{\isaliteral{2B}{\isacharplus}}}}[]
1281 \rail@nont{\hyperlink{syntax.constdecl}{\mbox{\isa{constdecl}}}}[]
1290 \item \hyperlink{command.HOL.record}{\mbox{\isa{\isacommand{record}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}\ t\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\ {\isaliteral{2B}{\isacharplus}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} defines extensible record type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}},
1291 derived from the optional parent record \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7461753E}{\isasymtau}}{\isaliteral{22}{\isachardoublequote}}} by adding new
1292 field components \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} etc.
1294 The type variables of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7461753E}{\isasymtau}}{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} need to be
1295 covered by the (distinct) parameters \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{22}{\isachardoublequote}}}. Type constructor \isa{t} has to be new, while \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} needs to specify an instance of an existing record type. At
1296 least one new field \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} has to be specified.
1297 Basically, field names need to belong to a unique record. This is
1298 not a real restriction in practice, since fields are qualified by
1299 the record name internally.
1301 The parent record specification \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} is optional; if omitted
1302 \isa{t} becomes a root record. The hierarchy of all records
1303 declared within a theory context forms a forest structure, i.e.\ a
1304 set of trees starting with a root record each. There is no way to
1305 merge multiple parent records!
1307 For convenience, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}} is made a
1308 type abbreviation for the fixed record type \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}, likewise is \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{5F}{\isacharunderscore}}scheme{\isaliteral{22}{\isachardoublequote}}} made an abbreviation for
1309 \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}.
1312 \end{isamarkuptext}%
1315 \isamarkupsubsection{Record operations%
1319 \begin{isamarkuptext}%
1320 Any record definition of the form presented above produces certain
1321 standard operations. Selectors and updates are provided for any
1322 field, including the improper one ``\isa{more}''. There are also
1323 cumulative record constructor functions. To simplify the
1324 presentation below, we assume for now that \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}} is a root record with fields \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ c\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}}.
1326 \medskip \textbf{Selectors} and \textbf{updates} are available for
1327 any field (including ``\isa{more}''):
1329 \begin{matharray}{lll}
1330 \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} \\
1331 \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{5F}{\isacharunderscore}}update{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
1334 There is special syntax for application of updates: \isa{{\isaliteral{22}{\isachardoublequote}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} abbreviates term \isa{{\isaliteral{22}{\isachardoublequote}}x{\isaliteral{5F}{\isacharunderscore}}update\ a\ r{\isaliteral{22}{\isachardoublequote}}}. Further notation for
1335 repeated updates is also available: \isa{{\isaliteral{22}{\isachardoublequote}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}z\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} may be written \isa{{\isaliteral{22}{\isachardoublequote}}r{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ z\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}. Note that
1336 because of postfix notation the order of fields shown here is
1337 reverse than in the actual term. Since repeated updates are just
1338 function applications, fields may be freely permuted in \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ a{\isaliteral{2C}{\isacharcomma}}\ y\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ b{\isaliteral{2C}{\isacharcomma}}\ z\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}\ c{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}}, as far as logical equality is concerned.
1339 Thus commutativity of independent updates can be proven within the
1340 logic for any two fields, but not as a general theorem.
1342 \medskip The \textbf{make} operation provides a cumulative record
1343 constructor function:
1345 \begin{matharray}{lll}
1346 \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}make{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
1349 \medskip We now reconsider the case of non-root records, which are
1350 derived of some parent. In general, the latter may depend on
1351 another parent as well, resulting in a list of \emph{ancestor
1352 records}. Appending the lists of fields of all ancestors results in
1353 a certain field prefix. The record package automatically takes care
1354 of this by lifting operations over this context of ancestor fields.
1355 Assuming that \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub m{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}} has ancestor
1356 fields \isa{{\isaliteral{22}{\isachardoublequote}}b\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C72686F3E}{\isasymrho}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ b\isaliteral{5C3C5E7375623E}{}\isactrlsub k\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C72686F3E}{\isasymrho}}\isaliteral{5C3C5E7375623E}{}\isactrlsub k{\isaliteral{22}{\isachardoublequote}}},
1357 the above record operations will get the following types:
1360 \begin{tabular}{lll}
1361 \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} \\
1362 \isa{{\isaliteral{22}{\isachardoublequote}}c\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{5F}{\isacharunderscore}}update{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub i\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
1363 \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}make{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C72686F3E}{\isasymrho}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C72686F3E}{\isasymrho}}\isaliteral{5C3C5E7375623E}{}\isactrlsub k\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
1367 \noindent Some further operations address the extension aspect of a
1368 derived record scheme specifically: \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}fields{\isaliteral{22}{\isachardoublequote}}} produces a
1369 record fragment consisting of exactly the new fields introduced here
1370 (the result may serve as a more part elsewhere); \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}extend{\isaliteral{22}{\isachardoublequote}}}
1371 takes a fixed record and adds a given more part; \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}truncate{\isaliteral{22}{\isachardoublequote}}} restricts a record scheme to a fixed record.
1374 \begin{tabular}{lll}
1375 \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}fields{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
1376 \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}extend{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
1377 \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}truncate{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7A6574613E}{\isasymzeta}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C6C706172723E}{\isasymlparr}}\isaliteral{5C3C5E7665633E}{}\isactrlvec b\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C72686F3E}{\isasymrho}}{\isaliteral{2C}{\isacharcomma}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec c\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ \isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C7369676D613E}{\isasymsigma}}{\isaliteral{5C3C72706172723E}{\isasymrparr}}{\isaliteral{22}{\isachardoublequote}}} \\
1381 \noindent Note that \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}make{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}fields{\isaliteral{22}{\isachardoublequote}}} coincide
1383 \end{isamarkuptext}%
1386 \isamarkupsubsection{Derived rules and proof tools%
1390 \begin{isamarkuptext}%
1391 The record package proves several results internally, declaring
1392 these facts to appropriate proof tools. This enables users to
1393 reason about record structures quite conveniently. Assume that
1394 \isa{t} is a record type as specified above.
1398 \item Standard conversions for selectors or updates applied to
1399 record constructor terms are made part of the default Simplifier
1400 context; thus proofs by reduction of basic operations merely require
1401 the \hyperlink{method.simp}{\mbox{\isa{simp}}} method without further arguments. These rules
1402 are available as \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}simps{\isaliteral{22}{\isachardoublequote}}}, too.
1404 \item Selectors applied to updated records are automatically reduced
1405 by an internal simplification procedure, which is also part of the
1406 standard Simplifier setup.
1408 \item Inject equations of a form analogous to \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}x{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}x{\isaliteral{27}{\isacharprime}}{\isaliteral{2C}{\isacharcomma}}\ y{\isaliteral{27}{\isacharprime}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C65717569763E}{\isasymequiv}}\ x\ {\isaliteral{3D}{\isacharequal}}\ x{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C616E643E}{\isasymand}}\ y\ {\isaliteral{3D}{\isacharequal}}\ y{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequote}}} are declared to the Simplifier and Classical
1409 Reasoner as \hyperlink{attribute.iff}{\mbox{\isa{iff}}} rules. These rules are available as
1410 \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}iffs{\isaliteral{22}{\isachardoublequote}}}.
1412 \item The introduction rule for record equality analogous to \isa{{\isaliteral{22}{\isachardoublequote}}x\ r\ {\isaliteral{3D}{\isacharequal}}\ x\ r{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ y\ r\ {\isaliteral{3D}{\isacharequal}}\ y\ r{\isaliteral{27}{\isacharprime}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ r\ {\isaliteral{3D}{\isacharequal}}\ r{\isaliteral{27}{\isacharprime}}{\isaliteral{22}{\isachardoublequote}}} is declared to the Simplifier,
1413 and as the basic rule context as ``\hyperlink{attribute.intro}{\mbox{\isa{intro}}}\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequote}}}''.
1414 The rule is called \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}equality{\isaliteral{22}{\isachardoublequote}}}.
1416 \item Representations of arbitrary record expressions as canonical
1417 constructor terms are provided both in \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} format (cf.\ the generic proof methods of the same name,
1418 \secref{sec:cases-induct}). Several variations are available, for
1419 fixed records, record schemes, more parts etc.
1421 The generic proof methods are sufficiently smart to pick the most
1422 sensible rule according to the type of the indicated record
1423 expression: users just need to apply something like ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}cases\ r{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' to a certain proof problem.
1425 \item The derived record operations \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}make{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}fields{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}extend{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}truncate{\isaliteral{22}{\isachardoublequote}}} are \emph{not}
1426 treated automatically, but usually need to be expanded by hand,
1427 using the collective fact \isa{{\isaliteral{22}{\isachardoublequote}}t{\isaliteral{2E}{\isachardot}}defs{\isaliteral{22}{\isachardoublequote}}}.
1430 \end{isamarkuptext}%
1433 \isamarkupsubsubsection{Examples%
1437 \begin{isamarkuptext}%
1438 See \verb|~~/src/HOL/ex/Records.thy|, for example.%
1439 \end{isamarkuptext}%
1442 \isamarkupsection{Adhoc tuples%
1446 \begin{isamarkuptext}%
1447 \begin{matharray}{rcl}
1448 \indexdef{HOL}{attribute}{split\_format}\hypertarget{attribute.HOL.split-format}{\hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isaliteral{5F}{\isacharunderscore}}format}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{attribute} \\
1453 \rail@term{\hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isaliteral{5F}{\isacharunderscore}}format}}}}[]
1456 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
1457 \rail@term{\isa{complete}}[]
1458 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
1466 \item \hyperlink{attribute.HOL.split-format}{\mbox{\isa{split{\isaliteral{5F}{\isacharunderscore}}format}}}\ \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}complete{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} causes
1467 arguments in function applications to be represented canonically
1468 according to their tuple type structure.
1470 Note that this operation tends to invent funny names for new local
1471 parameters introduced.
1474 \end{isamarkuptext}%
1477 \isamarkupsection{Typedef axiomatization \label{sec:hol-typedef}%
1481 \begin{isamarkuptext}%
1482 A Gordon/HOL-style type definition is a certain axiom scheme
1483 that identifies a new type with a subset of an existing type. More
1484 precisely, the new type is defined by exhibiting an existing type
1485 \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}}, a set \isa{{\isaliteral{22}{\isachardoublequote}}A\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{5C3C7461753E}{\isasymtau}}\ set{\isaliteral{22}{\isachardoublequote}}}, and a theorem that proves
1486 \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6578697374733E}{\isasymexists}}x{\isaliteral{2E}{\isachardot}}\ x\ {\isaliteral{5C3C696E3E}{\isasymin}}\ A{\isaliteral{22}{\isachardoublequote}}}. Thus \isa{A} is a non-empty subset of \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}}, and the new type denotes this subset. New functions are
1487 postulated that establish an isomorphism between the new type and
1488 the subset. In general, the type \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} may involve type
1489 variables \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} which means that the type definition
1490 produces a type constructor \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}} depending on
1491 those type arguments.
1493 The axiomatization can be considered a ``definition'' in the sense
1494 of the particular set-theoretic interpretation of HOL
1495 \cite{pitts93}, where the universe of types is required to be
1496 downwards-closed wrt.\ arbitrary non-empty subsets. Thus genuinely
1497 new types introduced by \hyperlink{command.typedef}{\mbox{\isa{\isacommand{typedef}}}} stay within the range
1498 of HOL models by construction. Note that \indexref{}{command}{type\_synonym}\hyperlink{command.type-synonym}{\mbox{\isa{\isacommand{type{\isaliteral{5F}{\isacharunderscore}}synonym}}}} from Isabelle/Pure merely introduces syntactic
1499 abbreviations, without any logical significance.
1501 \begin{matharray}{rcl}
1502 \indexdef{HOL}{command}{typedef}\hypertarget{command.HOL.typedef}{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
1507 \rail@term{\hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}}[]
1510 \rail@nont{\isa{alt{\isaliteral{5F}{\isacharunderscore}}name}}[]
1512 \rail@nont{\isa{abs{\isaliteral{5F}{\isacharunderscore}}type}}[]
1513 \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
1514 \rail@nont{\isa{rep{\isaliteral{5F}{\isacharunderscore}}set}}[]
1516 \rail@begin{3}{\isa{alt{\isaliteral{5F}{\isacharunderscore}}name}}
1517 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
1519 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
1521 \rail@term{\isa{\isakeyword{open}}}[]
1523 \rail@term{\isa{\isakeyword{open}}}[]
1524 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
1526 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
1528 \rail@begin{2}{\isa{abs{\isaliteral{5F}{\isacharunderscore}}type}}
1529 \rail@nont{\hyperlink{syntax.typespec-sorts}{\mbox{\isa{typespec{\isaliteral{5F}{\isacharunderscore}}sorts}}}}[]
1532 \rail@nont{\hyperlink{syntax.mixfix}{\mbox{\isa{mixfix}}}}[]
1535 \rail@begin{2}{\isa{rep{\isaliteral{5F}{\isacharunderscore}}set}}
1536 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
1539 \rail@term{\isa{\isakeyword{morphisms}}}[]
1540 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
1541 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
1549 \item \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}\ t\ {\isaliteral{3D}{\isacharequal}}\ A{\isaliteral{22}{\isachardoublequote}}}
1550 axiomatizes a type definition in the background theory of the
1551 current context, depending on a non-emptiness result of the set
1552 \isa{A} that needs to be proven here. The set \isa{A} may
1553 contain type variables \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{22}{\isachardoublequote}}} as specified on the LHS,
1554 but no term variables.
1556 Even though a local theory specification, the newly introduced type
1557 constructor cannot depend on parameters or assumptions of the
1558 context: this is structurally impossible in HOL. In contrast, the
1559 non-emptiness proof may use local assumptions in unusual situations,
1560 which could result in different interpretations in target contexts:
1561 the meaning of the bijection between the representing set \isa{A}
1562 and the new type \isa{t} may then change in different application
1565 By default, \hyperlink{command.HOL.typedef}{\mbox{\isa{\isacommand{typedef}}}} defines both a type
1566 constructor \isa{t} for the new type, and a term constant \isa{t} for the representing set within the old type. Use the ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}open{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}'' option to suppress a separate constant definition
1567 altogether. The injection from type to set is called \isa{Rep{\isaliteral{5F}{\isacharunderscore}}t},
1568 its inverse \isa{Abs{\isaliteral{5F}{\isacharunderscore}}t}, unless explicit \hyperlink{keyword.HOL.morphisms}{\mbox{\isa{\isakeyword{morphisms}}}} specification provides alternative names.
1570 The core axiomatization uses the locale predicate \isa{type{\isaliteral{5F}{\isacharunderscore}}definition} as defined in Isabelle/HOL. Various basic
1571 consequences of that are instantiated accordingly, re-using the
1572 locale facts with names derived from the new type constructor. Thus
1573 the generic \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep} is turned into the specific
1574 \isa{{\isaliteral{22}{\isachardoublequote}}Rep{\isaliteral{5F}{\isacharunderscore}}t{\isaliteral{22}{\isachardoublequote}}}, for example.
1576 Theorems \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep}, \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep{\isaliteral{5F}{\isacharunderscore}}inverse}, and \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Abs{\isaliteral{5F}{\isacharunderscore}}inverse}
1577 provide the most basic characterization as a corresponding
1578 injection/surjection pair (in both directions). The derived rules
1579 \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep{\isaliteral{5F}{\isacharunderscore}}inject} and \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Abs{\isaliteral{5F}{\isacharunderscore}}inject} provide a more convenient version of
1580 injectivity, suitable for automated proof tools (e.g.\ in
1581 declarations involving \hyperlink{attribute.simp}{\mbox{\isa{simp}}} or \hyperlink{attribute.iff}{\mbox{\isa{iff}}}).
1582 Furthermore, the rules \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep{\isaliteral{5F}{\isacharunderscore}}cases}~/ \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Rep{\isaliteral{5F}{\isacharunderscore}}induct}, and \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Abs{\isaliteral{5F}{\isacharunderscore}}cases}~/
1583 \isa{type{\isaliteral{5F}{\isacharunderscore}}definition{\isaliteral{2E}{\isachardot}}Abs{\isaliteral{5F}{\isacharunderscore}}induct} provide alternative views on
1584 surjectivity. These rules are already declared as set or type rules
1585 for the generic \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} methods,
1588 An alternative name for the set definition (and other derived
1589 entities) may be specified in parentheses; the default is to use
1595 If you introduce a new type axiomatically, i.e.\ via \indexref{}{command}{typedecl}\hyperlink{command.typedecl}{\mbox{\isa{\isacommand{typedecl}}}} and \indexref{}{command}{axiomatization}\hyperlink{command.axiomatization}{\mbox{\isa{\isacommand{axiomatization}}}}, the minimum requirement
1596 is that it has a non-empty model, to avoid immediate collapse of the
1597 HOL logic. Moreover, one needs to demonstrate that the
1598 interpretation of such free-form axiomatizations can coexist with
1599 that of the regular \indexdef{}{command}{typedef}\hypertarget{command.typedef}{\hyperlink{command.typedef}{\mbox{\isa{\isacommand{typedef}}}}} scheme, and any extension
1600 that other people might have introduced elsewhere (e.g.\ in HOLCF
1601 \cite{MuellerNvOS99}).
1603 \end{isamarkuptext}%
1606 \isamarkupsubsubsection{Examples%
1610 \begin{isamarkuptext}%
1611 Type definitions permit the introduction of abstract data
1612 types in a safe way, namely by providing models based on already
1613 existing types. Given some abstract axiomatic description \isa{P}
1614 of a type, this involves two steps:
1618 \item Find an appropriate type \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} and subset \isa{A} which
1619 has the desired properties \isa{P}, and make a type definition
1620 based on this representation.
1622 \item Prove that \isa{P} holds for \isa{{\isaliteral{5C3C7461753E}{\isasymtau}}} by lifting \isa{P}
1623 from the representation.
1627 You can later forget about the representation and work solely in
1628 terms of the abstract properties \isa{P}.
1630 \medskip The following trivial example pulls a three-element type
1631 into existence within the formal logical environment of HOL.%
1632 \end{isamarkuptext}%
1634 \isacommand{typedef}\isamarkupfalse%
1635 \ three\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{7B}{\isacharbraceleft}}{\isaliteral{28}{\isacharparenleft}}True{\isaliteral{2C}{\isacharcomma}}\ True{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}True{\isaliteral{2C}{\isacharcomma}}\ False{\isaliteral{29}{\isacharparenright}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{28}{\isacharparenleft}}False{\isaliteral{2C}{\isacharcomma}}\ True{\isaliteral{29}{\isacharparenright}}{\isaliteral{7D}{\isacharbraceright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
1642 \isacommand{by}\isamarkupfalse%
1652 \isacommand{definition}\isamarkupfalse%
1653 \ {\isaliteral{22}{\isachardoublequoteopen}}One\ {\isaliteral{3D}{\isacharequal}}\ Abs{\isaliteral{5F}{\isacharunderscore}}three\ {\isaliteral{28}{\isacharparenleft}}True{\isaliteral{2C}{\isacharcomma}}\ True{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
1654 \isacommand{definition}\isamarkupfalse%
1655 \ {\isaliteral{22}{\isachardoublequoteopen}}Two\ {\isaliteral{3D}{\isacharequal}}\ Abs{\isaliteral{5F}{\isacharunderscore}}three\ {\isaliteral{28}{\isacharparenleft}}True{\isaliteral{2C}{\isacharcomma}}\ False{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
1656 \isacommand{definition}\isamarkupfalse%
1657 \ {\isaliteral{22}{\isachardoublequoteopen}}Three\ {\isaliteral{3D}{\isacharequal}}\ Abs{\isaliteral{5F}{\isacharunderscore}}three\ {\isaliteral{28}{\isacharparenleft}}False{\isaliteral{2C}{\isacharcomma}}\ True{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
1659 \isacommand{lemma}\isamarkupfalse%
1660 \ three{\isaliteral{5F}{\isacharunderscore}}distinct{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}One\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ Two{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}One\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ Three{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{22}{\isachardoublequoteopen}}Two\ {\isaliteral{5C3C6E6F7465713E}{\isasymnoteq}}\ Three{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
1667 \isacommand{by}\isamarkupfalse%
1668 \ {\isaliteral{28}{\isacharparenleft}}simp{\isaliteral{5F}{\isacharunderscore}}all\ add{\isaliteral{3A}{\isacharcolon}}\ One{\isaliteral{5F}{\isacharunderscore}}def\ Two{\isaliteral{5F}{\isacharunderscore}}def\ Three{\isaliteral{5F}{\isacharunderscore}}def\ Abs{\isaliteral{5F}{\isacharunderscore}}three{\isaliteral{5F}{\isacharunderscore}}inject\ three{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
1677 \isacommand{lemma}\isamarkupfalse%
1678 \ three{\isaliteral{5F}{\isacharunderscore}}cases{\isaliteral{3A}{\isacharcolon}}\isanewline
1679 \ \ \isakeyword{fixes}\ x\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ three\ \isakeyword{obtains}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ One{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ Two{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}x\ {\isaliteral{3D}{\isacharequal}}\ Three{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
1686 \isacommand{by}\isamarkupfalse%
1687 \ {\isaliteral{28}{\isacharparenleft}}cases\ x{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{28}{\isacharparenleft}}auto\ simp{\isaliteral{3A}{\isacharcolon}}\ One{\isaliteral{5F}{\isacharunderscore}}def\ Two{\isaliteral{5F}{\isacharunderscore}}def\ Three{\isaliteral{5F}{\isacharunderscore}}def\ Abs{\isaliteral{5F}{\isacharunderscore}}three{\isaliteral{5F}{\isacharunderscore}}inject\ three{\isaliteral{5F}{\isacharunderscore}}def{\isaliteral{29}{\isacharparenright}}%
1695 \begin{isamarkuptext}%
1696 Note that such trivial constructions are better done with
1697 derived specification mechanisms such as \hyperlink{command.datatype}{\mbox{\isa{\isacommand{datatype}}}}:%
1698 \end{isamarkuptext}%
1700 \isacommand{datatype}\isamarkupfalse%
1701 \ three{\isaliteral{27}{\isacharprime}}\ {\isaliteral{3D}{\isacharequal}}\ One{\isaliteral{27}{\isacharprime}}\ {\isaliteral{7C}{\isacharbar}}\ Two{\isaliteral{27}{\isacharprime}}\ {\isaliteral{7C}{\isacharbar}}\ Three{\isaliteral{27}{\isacharprime}}%
1702 \begin{isamarkuptext}%
1703 This avoids re-doing basic definitions and proofs from the
1704 primitive \hyperlink{command.typedef}{\mbox{\isa{\isacommand{typedef}}}} above.%
1705 \end{isamarkuptext}%
1708 \isamarkupsection{Functorial structure of types%
1712 \begin{isamarkuptext}%
1713 \begin{matharray}{rcl}
1714 \indexdef{HOL}{command}{enriched\_type}\hypertarget{command.HOL.enriched-type}{\hyperlink{command.HOL.enriched-type}{\mbox{\isa{\isacommand{enriched{\isaliteral{5F}{\isacharunderscore}}type}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}}
1719 \rail@term{\hyperlink{command.HOL.enriched-type}{\mbox{\isa{\isacommand{enriched{\isaliteral{5F}{\isacharunderscore}}type}}}}}[]
1722 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
1723 \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
1725 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
1732 \item \hyperlink{command.HOL.enriched-type}{\mbox{\isa{\isacommand{enriched{\isaliteral{5F}{\isacharunderscore}}type}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}prefix{\isaliteral{3A}{\isacharcolon}}\ m{\isaliteral{22}{\isachardoublequote}}} allows to
1733 prove and register properties about the functorial structure of type
1734 constructors. These properties then can be used by other packages
1735 to deal with those type constructors in certain type constructions.
1736 Characteristic theorems are noted in the current local theory. By
1737 default, they are prefixed with the base name of the type
1738 constructor, an explicit prefix can be given alternatively.
1740 The given term \isa{{\isaliteral{22}{\isachardoublequote}}m{\isaliteral{22}{\isachardoublequote}}} is considered as \emph{mapper} for the
1741 corresponding type constructor and must conform to the following
1744 \begin{matharray}{lll}
1745 \isa{{\isaliteral{22}{\isachardoublequote}}m{\isaliteral{22}{\isachardoublequote}}} & \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} &
1746 \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}\ {\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub k\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}\isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}\ t\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}\isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}\ t{\isaliteral{22}{\isachardoublequote}}} \\
1749 \noindent where \isa{t} is the type constructor, \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}} and \isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7665633E}{}\isactrlvec {\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}} are distinct
1750 type variables free in the local theory and \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}},
1751 \ldots, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C7369676D613E}{\isasymsigma}}\isaliteral{5C3C5E697375623E}{}\isactrlisub k{\isaliteral{22}{\isachardoublequote}}} is a subsequence of \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{22}{\isachardoublequote}}}, \ldots,
1752 \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C626574613E}{\isasymbeta}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{5C3C616C7068613E}{\isasymalpha}}\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{22}{\isachardoublequote}}}.
1755 \end{isamarkuptext}%
1758 \isamarkupsection{Arithmetic proof support%
1762 \begin{isamarkuptext}%
1763 \begin{matharray}{rcl}
1764 \indexdef{HOL}{method}{arith}\hypertarget{method.HOL.arith}{\hyperlink{method.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{method} \\
1765 \indexdef{HOL}{attribute}{arith}\hypertarget{attribute.HOL.arith}{\hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}}} & : & \isa{attribute} \\
1766 \indexdef{HOL}{attribute}{arith\_split}\hypertarget{attribute.HOL.arith-split}{\hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isaliteral{5F}{\isacharunderscore}}split}}}} & : & \isa{attribute} \\
1769 The \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} method decides linear arithmetic problems
1770 (on types \isa{nat}, \isa{int}, \isa{real}). Any current
1771 facts are inserted into the goal before running the procedure.
1773 The \hyperlink{attribute.HOL.arith}{\mbox{\isa{arith}}} attribute declares facts that are
1774 always supplied to the arithmetic provers implicitly.
1776 The \hyperlink{attribute.HOL.arith-split}{\mbox{\isa{arith{\isaliteral{5F}{\isacharunderscore}}split}}} attribute declares case split
1777 rules to be expanded before \hyperlink{method.HOL.arith}{\mbox{\isa{arith}}} is invoked.
1779 Note that a simpler (but faster) arithmetic prover is
1780 already invoked by the Simplifier.%
1781 \end{isamarkuptext}%
1784 \isamarkupsection{Intuitionistic proof search%
1788 \begin{isamarkuptext}%
1789 \begin{matharray}{rcl}
1790 \indexdef{HOL}{method}{iprover}\hypertarget{method.HOL.iprover}{\hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}}} & : & \isa{method} \\
1795 \rail@term{\hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}}}[]
1798 \rail@cnont{\hyperlink{syntax.rulemod}{\mbox{\isa{rulemod}}}}[]
1804 The \hyperlink{method.HOL.iprover}{\mbox{\isa{iprover}}} method performs intuitionistic proof
1805 search, depending on specifically declared rules from the context,
1806 or given as explicit arguments. Chained facts are inserted into the
1807 goal before commencing proof search.
1809 Rules need to be classified as \hyperlink{attribute.Pure.intro}{\mbox{\isa{intro}}},
1810 \hyperlink{attribute.Pure.elim}{\mbox{\isa{elim}}}, or \hyperlink{attribute.Pure.dest}{\mbox{\isa{dest}}}; here the
1811 ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{21}{\isacharbang}}{\isaliteral{22}{\isachardoublequote}}}'' indicator refers to ``safe'' rules, which may be
1812 applied aggressively (without considering back-tracking later).
1813 Rules declared with ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequote}}}'' are ignored in proof search (the
1814 single-step \hyperlink{method.Pure.rule}{\mbox{\isa{rule}}} method still observes these). An
1815 explicit weight annotation may be given as well; otherwise the
1816 number of rule premises will be taken into account here.%
1817 \end{isamarkuptext}%
1820 \isamarkupsection{Coherent Logic%
1824 \begin{isamarkuptext}%
1825 \begin{matharray}{rcl}
1826 \indexdef{HOL}{method}{coherent}\hypertarget{method.HOL.coherent}{\hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}}} & : & \isa{method} \\
1831 \rail@term{\hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}}}[]
1834 \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
1840 The \hyperlink{method.HOL.coherent}{\mbox{\isa{coherent}}} method solves problems of
1841 \emph{Coherent Logic} \cite{Bezem-Coquand:2005}, which covers
1842 applications in confluence theory, lattice theory and projective
1843 geometry. See \verb|~~/src/HOL/ex/Coherent.thy| for some
1845 \end{isamarkuptext}%
1848 \isamarkupsection{Proving propositions%
1852 \begin{isamarkuptext}%
1853 In addition to the standard proof methods, a number of diagnosis
1854 tools search for proofs and provide an Isar proof snippet on success.
1855 These tools are available via the following commands.
1857 \begin{matharray}{rcl}
1858 \indexdef{HOL}{command}{solve\_direct}\hypertarget{command.HOL.solve-direct}{\hyperlink{command.HOL.solve-direct}{\mbox{\isa{\isacommand{solve{\isaliteral{5F}{\isacharunderscore}}direct}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
1859 \indexdef{HOL}{command}{try}\hypertarget{command.HOL.try}{\hyperlink{command.HOL.try}{\mbox{\isa{\isacommand{try}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
1860 \indexdef{HOL}{command}{sledgehammer}\hypertarget{command.HOL.sledgehammer}{\hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
1861 \indexdef{HOL}{command}{sledgehammer\_params}\hypertarget{command.HOL.sledgehammer-params}{\hyperlink{command.HOL.sledgehammer-params}{\mbox{\isa{\isacommand{sledgehammer{\isaliteral{5F}{\isacharunderscore}}params}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}}
1866 \rail@term{\hyperlink{command.HOL.try}{\mbox{\isa{\isacommand{try}}}}}[]
1871 \rail@term{\isa{simp}}[]
1873 \rail@term{\isa{intro}}[]
1875 \rail@term{\isa{elim}}[]
1877 \rail@term{\isa{dest}}[]
1879 \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
1880 \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
1886 \rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]
1890 \rail@term{\hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}}}[]
1893 \rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[]
1894 \rail@nont{\isa{args}}[]
1895 \rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[]
1899 \rail@nont{\isa{facts}}[]
1903 \rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]
1907 \rail@term{\hyperlink{command.HOL.sledgehammer-params}{\mbox{\isa{\isacommand{sledgehammer{\isaliteral{5F}{\isacharunderscore}}params}}}}}[]
1910 \rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[]
1911 \rail@nont{\isa{args}}[]
1912 \rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[]
1915 \rail@begin{2}{\isa{args}}
1917 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
1918 \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
1919 \rail@nont{\isa{value}}[]
1921 \rail@cterm{\isa{{\isaliteral{2C}{\isacharcomma}}}}[]
1924 \rail@begin{5}{\isa{facts}}
1925 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
1932 \rail@term{\isa{add}}[]
1934 \rail@term{\isa{del}}[]
1936 \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
1938 \rail@nont{\hyperlink{syntax.thmrefs}{\mbox{\isa{thmrefs}}}}[]
1942 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
1945 % FIXME try: proper clasimpmod!?
1946 % FIXME check args "value"
1950 \item \hyperlink{command.HOL.solve-direct}{\mbox{\isa{\isacommand{solve{\isaliteral{5F}{\isacharunderscore}}direct}}}} checks whether the current subgoals can
1951 be solved directly by an existing theorem. Duplicate lemmas can be detected
1954 \item \hyperlink{command.HOL.try}{\mbox{\isa{\isacommand{try}}}} attempts to prove a subgoal using a combination
1955 of standard proof methods (\isa{auto}, \isa{simp}, \isa{blast}, etc.).
1956 Additional facts supplied via \isa{{\isaliteral{22}{\isachardoublequote}}simp{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}, \isa{{\isaliteral{22}{\isachardoublequote}}intro{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}},
1957 \isa{{\isaliteral{22}{\isachardoublequote}}elim{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}, and \isa{{\isaliteral{22}{\isachardoublequote}}dest{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}} are passed to the appropriate proof
1960 \item \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}} attempts to prove a subgoal using external
1961 automatic provers (resolution provers and SMT solvers). See the Sledgehammer
1962 manual \cite{isabelle-sledgehammer} for details.
1964 \item \hyperlink{command.HOL.sledgehammer-params}{\mbox{\isa{\isacommand{sledgehammer{\isaliteral{5F}{\isacharunderscore}}params}}}} changes
1965 \hyperlink{command.HOL.sledgehammer}{\mbox{\isa{\isacommand{sledgehammer}}}} configuration options persistently.
1968 \end{isamarkuptext}%
1971 \isamarkupsection{Checking and refuting propositions%
1975 \begin{isamarkuptext}%
1976 Identifying incorrect propositions usually involves evaluation of
1977 particular assignments and systematic counterexample search. This
1978 is supported by the following commands.
1980 \begin{matharray}{rcl}
1981 \indexdef{HOL}{command}{value}\hypertarget{command.HOL.value}{\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
1982 \indexdef{HOL}{command}{quickcheck}\hypertarget{command.HOL.quickcheck}{\hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
1983 \indexdef{HOL}{command}{refute}\hypertarget{command.HOL.refute}{\hyperlink{command.HOL.refute}{\mbox{\isa{\isacommand{refute}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
1984 \indexdef{HOL}{command}{nitpick}\hypertarget{command.HOL.nitpick}{\hyperlink{command.HOL.nitpick}{\mbox{\isa{\isacommand{nitpick}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}proof\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
1985 \indexdef{HOL}{command}{quickcheck\_params}\hypertarget{command.HOL.quickcheck-params}{\hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isaliteral{5F}{\isacharunderscore}}params}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
1986 \indexdef{HOL}{command}{refute\_params}\hypertarget{command.HOL.refute-params}{\hyperlink{command.HOL.refute-params}{\mbox{\isa{\isacommand{refute{\isaliteral{5F}{\isacharunderscore}}params}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
1987 \indexdef{HOL}{command}{nitpick\_params}\hypertarget{command.HOL.nitpick-params}{\hyperlink{command.HOL.nitpick-params}{\mbox{\isa{\isacommand{nitpick{\isaliteral{5F}{\isacharunderscore}}params}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}}
1992 \rail@term{\hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}}[]
1995 \rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[]
1996 \rail@nont{\isa{name}}[]
1997 \rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[]
2001 \rail@nont{\isa{modes}}[]
2003 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
2007 \rail@term{\hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}}[]
2009 \rail@term{\hyperlink{command.HOL.refute}{\mbox{\isa{\isacommand{refute}}}}}[]
2011 \rail@term{\hyperlink{command.HOL.nitpick}{\mbox{\isa{\isacommand{nitpick}}}}}[]
2015 \rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[]
2016 \rail@nont{\isa{args}}[]
2017 \rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[]
2021 \rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]
2026 \rail@term{\hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isaliteral{5F}{\isacharunderscore}}params}}}}}[]
2028 \rail@term{\hyperlink{command.HOL.refute-params}{\mbox{\isa{\isacommand{refute{\isaliteral{5F}{\isacharunderscore}}params}}}}}[]
2030 \rail@term{\hyperlink{command.HOL.nitpick-params}{\mbox{\isa{\isacommand{nitpick{\isaliteral{5F}{\isacharunderscore}}params}}}}}[]
2034 \rail@term{\isa{{\isaliteral{5B}{\isacharbrackleft}}}}[]
2035 \rail@nont{\isa{args}}[]
2036 \rail@term{\isa{{\isaliteral{5D}{\isacharbrackright}}}}[]
2039 \rail@begin{2}{\isa{modes}}
2040 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
2042 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
2045 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
2047 \rail@begin{2}{\isa{args}}
2049 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
2050 \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
2051 \rail@nont{\isa{value}}[]
2053 \rail@cterm{\isa{{\isaliteral{2C}{\isacharcomma}}}}[]
2057 % FIXME check "value"
2061 \item \hyperlink{command.HOL.value}{\mbox{\isa{\isacommand{value}}}}~\isa{t} evaluates and prints a
2062 term; optionally \isa{modes} can be specified, which are
2063 appended to the current print mode (see also \cite{isabelle-ref}).
2064 Internally, the evaluation is performed by registered evaluators,
2065 which are invoked sequentially until a result is returned.
2066 Alternatively a specific evaluator can be selected using square
2067 brackets; typical evaluators use the current set of code equations
2068 to normalize and include \isa{simp} for fully symbolic evaluation
2069 using the simplifier, \isa{nbe} for \emph{normalization by evaluation}
2070 and \emph{code} for code generation in SML.
2072 \item \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}} tests the current goal for
2073 counterexamples using a series of assignments for its
2074 free variables; by default the first subgoal is tested, an other
2075 can be selected explicitly using an optional goal index.
2076 Assignments can be chosen exhausting the search space upto a given
2077 size or using a fixed number of random assignments in the search space.
2078 By default, quickcheck uses exhaustive testing.
2079 A number of configuration options are supported for
2080 \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}}, notably:
2084 \item[\isa{tester}] specifies how to explore the search space
2085 (e.g. exhaustive or random).
2086 An unknown configuration option is treated as an argument to tester,
2087 making \isa{{\isaliteral{22}{\isachardoublequote}}tester\ {\isaliteral{3D}{\isacharequal}}{\isaliteral{22}{\isachardoublequote}}} optional.
2088 \item[\isa{size}] specifies the maximum size of the search space
2089 for assignment values.
2091 \item[\isa{eval}] takes a term or a list of terms and evaluates
2092 these terms under the variable assignment found by quickcheck.
2094 \item[\isa{iterations}] sets how many sets of assignments are
2095 generated for each particular size.
2097 \item[\isa{no{\isaliteral{5F}{\isacharunderscore}}assms}] specifies whether assumptions in
2098 structured proofs should be ignored.
2100 \item[\isa{timeout}] sets the time limit in seconds.
2102 \item[\isa{default{\isaliteral{5F}{\isacharunderscore}}type}] sets the type(s) generally used to
2103 instantiate type variables.
2105 \item[\isa{report}] if set quickcheck reports how many tests
2106 fulfilled the preconditions.
2108 \item[\isa{quiet}] if not set quickcheck informs about the
2109 current size for assignment values.
2111 \item[\isa{expect}] can be used to check if the user's
2112 expectation was met (\isa{no{\isaliteral{5F}{\isacharunderscore}}expectation}, \isa{no{\isaliteral{5F}{\isacharunderscore}}counterexample}, or \isa{counterexample}).
2116 These option can be given within square brackets.
2118 \item \hyperlink{command.HOL.quickcheck-params}{\mbox{\isa{\isacommand{quickcheck{\isaliteral{5F}{\isacharunderscore}}params}}}} changes
2119 \hyperlink{command.HOL.quickcheck}{\mbox{\isa{\isacommand{quickcheck}}}} configuration options persistently.
2121 \item \hyperlink{command.HOL.refute}{\mbox{\isa{\isacommand{refute}}}} tests the current goal for
2122 counterexamples using a reduction to SAT. The following configuration
2123 options are supported:
2127 \item[\isa{minsize}] specifies the minimum size (cardinality) of the
2128 models to search for.
2130 \item[\isa{maxsize}] specifies the maximum size (cardinality) of the
2131 models to search for. Nonpositive values mean $\infty$.
2133 \item[\isa{maxvars}] specifies the maximum number of Boolean variables
2134 to use when transforming the term into a propositional formula.
2135 Nonpositive values mean $\infty$.
2137 \item[\isa{satsolver}] specifies the SAT solver to use.
2139 \item[\isa{no{\isaliteral{5F}{\isacharunderscore}}assms}] specifies whether assumptions in
2140 structured proofs should be ignored.
2142 \item[\isa{maxtime}] sets the time limit in seconds.
2144 \item[\isa{expect}] can be used to check if the user's
2145 expectation was met (\isa{genuine}, \isa{potential},
2146 \isa{none}, or \isa{unknown}).
2150 These option can be given within square brackets.
2152 \item \hyperlink{command.HOL.refute-params}{\mbox{\isa{\isacommand{refute{\isaliteral{5F}{\isacharunderscore}}params}}}} changes
2153 \hyperlink{command.HOL.refute}{\mbox{\isa{\isacommand{refute}}}} configuration options persistently.
2155 \item \hyperlink{command.HOL.nitpick}{\mbox{\isa{\isacommand{nitpick}}}} tests the current goal for counterexamples
2156 using a reduction to first-order relational logic. See the Nitpick manual
2157 \cite{isabelle-nitpick} for details.
2159 \item \hyperlink{command.HOL.nitpick-params}{\mbox{\isa{\isacommand{nitpick{\isaliteral{5F}{\isacharunderscore}}params}}}} changes
2160 \hyperlink{command.HOL.nitpick}{\mbox{\isa{\isacommand{nitpick}}}} configuration options persistently.
2163 \end{isamarkuptext}%
2166 \isamarkupsection{Unstructured case analysis and induction \label{sec:hol-induct-tac}%
2170 \begin{isamarkuptext}%
2171 The following tools of Isabelle/HOL support cases analysis and
2172 induction in unstructured tactic scripts; see also
2173 \secref{sec:cases-induct} for proper Isar versions of similar ideas.
2175 \begin{matharray}{rcl}
2176 \indexdef{HOL}{method}{case\_tac}\hypertarget{method.HOL.case-tac}{\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\
2177 \indexdef{HOL}{method}{induct\_tac}\hypertarget{method.HOL.induct-tac}{\hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\
2178 \indexdef{HOL}{method}{ind\_cases}\hypertarget{method.HOL.ind-cases}{\hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\
2179 \indexdef{HOL}{command}{inductive\_cases}\hypertarget{command.HOL.inductive-cases}{\hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}cases}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}local{\isaliteral{5F}{\isacharunderscore}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ local{\isaliteral{5F}{\isacharunderscore}}theory{\isaliteral{22}{\isachardoublequote}}} \\
2184 \rail@term{\hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]
2187 \rail@nont{\hyperlink{syntax.goal-spec}{\mbox{\isa{goal{\isaliteral{5F}{\isacharunderscore}}spec}}}}[]
2189 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
2192 \rail@nont{\isa{rule}}[]
2196 \rail@term{\hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}tac}}}}[]
2199 \rail@nont{\hyperlink{syntax.goal-spec}{\mbox{\isa{goal{\isaliteral{5F}{\isacharunderscore}}spec}}}}[]
2204 \rail@nont{\hyperlink{syntax.insts}{\mbox{\isa{insts}}}}[]
2206 \rail@cterm{\isa{\isakeyword{and}}}[]
2211 \rail@nont{\isa{rule}}[]
2215 \rail@term{\hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases}}}}[]
2217 \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
2222 \rail@term{\isa{\isakeyword{for}}}[]
2224 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
2230 \rail@term{\hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}cases}}}}}[]
2234 \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
2237 \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
2241 \rail@cterm{\isa{\isakeyword{and}}}[]
2244 \rail@begin{1}{\isa{rule}}
2245 \rail@term{\isa{rule}}[]
2246 \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
2247 \rail@nont{\hyperlink{syntax.thmref}{\mbox{\isa{thmref}}}}[]
2254 \item \hyperlink{method.HOL.case-tac}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}tac}}} and \hyperlink{method.HOL.induct-tac}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}tac}}} admit
2255 to reason about inductive types. Rules are selected according to
2256 the declarations by the \hyperlink{attribute.cases}{\mbox{\isa{cases}}} and \hyperlink{attribute.induct}{\mbox{\isa{induct}}}
2257 attributes, cf.\ \secref{sec:cases-induct}. The \hyperlink{command.HOL.datatype}{\mbox{\isa{\isacommand{datatype}}}} package already takes care of this.
2259 These unstructured tactics feature both goal addressing and dynamic
2260 instantiation. Note that named rule cases are \emph{not} provided
2261 as would be by the proper \hyperlink{method.cases}{\mbox{\isa{cases}}} and \hyperlink{method.induct}{\mbox{\isa{induct}}} proof
2262 methods (see \secref{sec:cases-induct}). Unlike the \hyperlink{method.induct}{\mbox{\isa{induct}}} method, \hyperlink{method.induct-tac}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}tac}}} does not handle structured rule
2263 statements, only the compact object-logic conclusion of the subgoal
2266 \item \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases}}} and \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}cases}}}} provide an interface to the internal \verb|mk_cases| operation. Rules are simplified in an unrestricted
2269 While \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases}}} is a proof method to apply the
2270 result immediately as elimination rules, \hyperlink{command.HOL.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}cases}}}} provides case split theorems at the theory level
2271 for later use. The \hyperlink{keyword.for}{\mbox{\isa{\isakeyword{for}}}} argument of the \hyperlink{method.HOL.ind-cases}{\mbox{\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases}}} method allows to specify a list of variables that should
2272 be generalized before applying the resulting rule.
2275 \end{isamarkuptext}%
2278 \isamarkupsection{Executable code%
2282 \begin{isamarkuptext}%
2283 For validation purposes, it is often useful to \emph{execute}
2284 specifications. In principle, execution could be simulated by
2285 Isabelle's inference kernel, i.e. by a combination of resolution and
2286 simplification. Unfortunately, this approach is rather inefficient.
2287 A more efficient way of executing specifications is to translate
2288 them into a functional programming language such as ML.
2290 Isabelle provides two generic frameworks to support code generation
2291 from executable specifications. Isabelle/HOL instantiates these
2292 mechanisms in a way that is amenable to end-user applications.%
2293 \end{isamarkuptext}%
2296 \isamarkupsubsection{The new code generator (F. Haftmann)%
2300 \begin{isamarkuptext}%
2301 This framework generates code from functional programs
2302 (including overloading using type classes) to SML \cite{SML}, OCaml
2303 \cite{OCaml}, Haskell \cite{haskell-revised-report} and Scala
2304 \cite{scala-overview-tech-report}. Conceptually, code generation is
2305 split up in three steps: \emph{selection} of code theorems,
2306 \emph{translation} into an abstract executable view and
2307 \emph{serialization} to a specific \emph{target language}.
2308 Inductive specifications can be executed using the predicate
2309 compiler which operates within HOL. See \cite{isabelle-codegen} for
2312 \begin{matharray}{rcl}
2313 \indexdef{HOL}{command}{export\_code}\hypertarget{command.HOL.export-code}{\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isaliteral{5F}{\isacharunderscore}}code}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
2314 \indexdef{HOL}{attribute}{code}\hypertarget{attribute.HOL.code}{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
2315 \indexdef{HOL}{command}{code\_abort}\hypertarget{command.HOL.code-abort}{\hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}abort}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2316 \indexdef{HOL}{command}{code\_datatype}\hypertarget{command.HOL.code-datatype}{\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}datatype}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2317 \indexdef{HOL}{command}{print\_codesetup}\hypertarget{command.HOL.print-codesetup}{\hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}codesetup}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
2318 \indexdef{HOL}{attribute}{code\_inline}\hypertarget{attribute.HOL.code-inline}{\hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}inline}}}} & : & \isa{attribute} \\
2319 \indexdef{HOL}{attribute}{code\_post}\hypertarget{attribute.HOL.code-post}{\hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}post}}}} & : & \isa{attribute} \\
2320 \indexdef{HOL}{command}{print\_codeproc}\hypertarget{command.HOL.print-codeproc}{\hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}codeproc}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
2321 \indexdef{HOL}{command}{code\_thms}\hypertarget{command.HOL.code-thms}{\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}thms}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
2322 \indexdef{HOL}{command}{code\_deps}\hypertarget{command.HOL.code-deps}{\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}deps}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
2323 \indexdef{HOL}{command}{code\_const}\hypertarget{command.HOL.code-const}{\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}const}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2324 \indexdef{HOL}{command}{code\_type}\hypertarget{command.HOL.code-type}{\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}type}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2325 \indexdef{HOL}{command}{code\_class}\hypertarget{command.HOL.code-class}{\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}class}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2326 \indexdef{HOL}{command}{code\_instance}\hypertarget{command.HOL.code-instance}{\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}instance}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2327 \indexdef{HOL}{command}{code\_reserved}\hypertarget{command.HOL.code-reserved}{\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reserved}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2328 \indexdef{HOL}{command}{code\_monad}\hypertarget{command.HOL.code-monad}{\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}monad}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2329 \indexdef{HOL}{command}{code\_include}\hypertarget{command.HOL.code-include}{\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}include}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2330 \indexdef{HOL}{command}{code\_modulename}\hypertarget{command.HOL.code-modulename}{\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}modulename}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2331 \indexdef{HOL}{command}{code\_reflect}\hypertarget{command.HOL.code-reflect}{\hyperlink{command.HOL.code-reflect}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reflect}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}}
2336 \rail@term{\hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isaliteral{5F}{\isacharunderscore}}code}}}}}[]
2338 \rail@nont{\isa{constexpr}}[]
2345 \rail@term{\isa{\isakeyword{in}}}[]
2346 \rail@nont{\isa{target}}[]
2349 \rail@term{\isa{\isakeyword{module{\isaliteral{5F}{\isacharunderscore}}name}}}[]
2350 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2355 \rail@term{\isa{\isakeyword{file}}}[]
2357 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2359 \rail@term{\isa{{\isaliteral{2D}{\isacharminus}}}}[]
2364 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
2365 \rail@nont{\isa{args}}[]
2366 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
2372 \rail@begin{1}{\isa{const}}
2373 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
2375 \rail@begin{3}{\isa{constexpr}}
2377 \rail@nont{\isa{const}}[]
2379 \rail@term{\isa{name{\isaliteral{2E}{\isachardot}}{\isaliteral{5F}{\isacharunderscore}}}}[]
2381 \rail@term{\isa{{\isaliteral{5F}{\isacharunderscore}}}}[]
2384 \rail@begin{1}{\isa{typeconstructor}}
2385 \rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]
2387 \rail@begin{1}{\isa{class}}
2388 \rail@nont{\hyperlink{syntax.nameref}{\mbox{\isa{nameref}}}}[]
2390 \rail@begin{4}{\isa{target}}
2392 \rail@term{\isa{SML}}[]
2394 \rail@term{\isa{OCaml}}[]
2396 \rail@term{\isa{Haskell}}[]
2398 \rail@term{\isa{Scala}}[]
2402 \rail@term{\hyperlink{attribute.HOL.code}{\mbox{\isa{code}}}}[]
2406 \rail@term{\isa{del}}[]
2408 \rail@term{\isa{abstype}}[]
2410 \rail@term{\isa{abstract}}[]
2415 \rail@term{\hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}abort}}}}}[]
2417 \rail@nont{\isa{const}}[]
2422 \rail@term{\hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}datatype}}}}}[]
2424 \rail@nont{\isa{const}}[]
2429 \rail@term{\hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}inline}}}}[]
2432 \rail@term{\isa{del}}[]
2436 \rail@term{\hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}post}}}}[]
2439 \rail@term{\isa{del}}[]
2443 \rail@term{\hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}thms}}}}}[]
2447 \rail@nont{\isa{constexpr}}[]
2453 \rail@term{\hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}deps}}}}}[]
2457 \rail@nont{\isa{constexpr}}[]
2463 \rail@term{\hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}const}}}}}[]
2465 \rail@nont{\isa{const}}[]
2467 \rail@cterm{\isa{\isakeyword{and}}}[]
2471 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
2472 \rail@nont{\isa{target}}[]
2476 \rail@nont{\isa{syntax}}[]
2479 \rail@cterm{\isa{\isakeyword{and}}}[]
2481 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
2486 \rail@term{\hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}type}}}}}[]
2488 \rail@nont{\isa{typeconstructor}}[]
2490 \rail@cterm{\isa{\isakeyword{and}}}[]
2494 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
2495 \rail@nont{\isa{target}}[]
2499 \rail@nont{\isa{syntax}}[]
2502 \rail@cterm{\isa{\isakeyword{and}}}[]
2504 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
2509 \rail@term{\hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}class}}}}}[]
2511 \rail@nont{\isa{class}}[]
2513 \rail@cterm{\isa{\isakeyword{and}}}[]
2517 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
2518 \rail@nont{\isa{target}}[]
2523 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2526 \rail@cterm{\isa{\isakeyword{and}}}[]
2528 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
2533 \rail@term{\hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}instance}}}}}[]
2535 \rail@nont{\isa{typeconstructor}}[]
2536 \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}}}[]
2537 \rail@nont{\isa{class}}[]
2539 \rail@cterm{\isa{\isakeyword{and}}}[]
2543 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
2544 \rail@nont{\isa{target}}[]
2548 \rail@term{\isa{{\isaliteral{2D}{\isacharminus}}}}[]
2551 \rail@cterm{\isa{\isakeyword{and}}}[]
2553 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
2558 \rail@term{\hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reserved}}}}}[]
2559 \rail@nont{\isa{target}}[]
2561 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2566 \rail@term{\hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}monad}}}}}[]
2567 \rail@nont{\isa{const}}[]
2568 \rail@nont{\isa{const}}[]
2569 \rail@nont{\isa{target}}[]
2572 \rail@term{\hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}include}}}}}[]
2573 \rail@nont{\isa{target}}[]
2574 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2576 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2578 \rail@term{\isa{{\isaliteral{2D}{\isacharminus}}}}[]
2582 \rail@term{\hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}modulename}}}}}[]
2583 \rail@nont{\isa{target}}[]
2585 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2586 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2591 \rail@term{\hyperlink{command.HOL.code-reflect}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reflect}}}}}[]
2592 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2596 \rail@term{\isa{\isakeyword{datatypes}}}[]
2597 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2598 \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
2600 \rail@term{\isa{{\isaliteral{5F}{\isacharunderscore}}}}[]
2604 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2606 \rail@cterm{\isa{{\isaliteral{7C}{\isacharbar}}}}[]
2609 \rail@cterm{\isa{\isakeyword{and}}}[]
2616 \rail@term{\isa{\isakeyword{functions}}}[]
2618 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2624 \rail@term{\isa{\isakeyword{file}}}[]
2625 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2628 \rail@begin{4}{\isa{syntax}}
2630 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2633 \rail@term{\isa{\isakeyword{infix}}}[]
2635 \rail@term{\isa{\isakeyword{infixl}}}[]
2637 \rail@term{\isa{\isakeyword{infixr}}}[]
2639 \rail@nont{\hyperlink{syntax.nat}{\mbox{\isa{nat}}}}[]
2640 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2648 \item \hyperlink{command.HOL.export-code}{\mbox{\isa{\isacommand{export{\isaliteral{5F}{\isacharunderscore}}code}}}} generates code for a given list
2649 of constants in the specified target language(s). If no
2650 serialization instruction is given, only abstract code is generated
2653 Constants may be specified by giving them literally, referring to
2654 all executable contants within a certain theory by giving \isa{{\isaliteral{22}{\isachardoublequote}}name{\isaliteral{2E}{\isachardot}}{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}}, or referring to \emph{all} executable constants currently
2655 available by giving \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}}.
2657 By default, for each involved theory one corresponding name space
2658 module is generated. Alternativly, a module name may be specified
2659 after the \hyperlink{keyword.module-name}{\mbox{\isa{\isakeyword{module{\isaliteral{5F}{\isacharunderscore}}name}}}} keyword; then \emph{all} code is
2660 placed in this module.
2662 For \emph{SML}, \emph{OCaml} and \emph{Scala} the file specification
2663 refers to a single file; for \emph{Haskell}, it refers to a whole
2664 directory, where code is generated in multiple files reflecting the
2665 module hierarchy. Omitting the file specification denotes standard
2668 Serializers take an optional list of arguments in parentheses. For
2669 \emph{SML} and \emph{OCaml}, ``\isa{no{\isaliteral{5F}{\isacharunderscore}}signatures}`` omits
2670 explicit module signatures.
2672 For \emph{Haskell} a module name prefix may be given using the
2673 ``\isa{{\isaliteral{22}{\isachardoublequote}}root{\isaliteral{3A}{\isacharcolon}}{\isaliteral{22}{\isachardoublequote}}}'' argument; ``\isa{string{\isaliteral{5F}{\isacharunderscore}}classes}'' adds a
2674 ``\verb|deriving (Read, Show)|'' clause to each appropriate
2675 datatype declaration.
2677 \item \hyperlink{attribute.HOL.code}{\mbox{\isa{code}}} explicitly selects (or with option
2678 ``\isa{{\isaliteral{22}{\isachardoublequote}}del{\isaliteral{22}{\isachardoublequote}}}'' deselects) a code equation for code generation.
2679 Usually packages introducing code equations provide a reasonable
2680 default setup for selection. Variants \isa{{\isaliteral{22}{\isachardoublequote}}code\ abstype{\isaliteral{22}{\isachardoublequote}}} and
2681 \isa{{\isaliteral{22}{\isachardoublequote}}code\ abstract{\isaliteral{22}{\isachardoublequote}}} declare abstract datatype certificates or
2682 code equations on abstract datatype representations respectively.
2684 \item \hyperlink{command.HOL.code-abort}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}abort}}}} declares constants which are not
2685 required to have a definition by means of code equations; if needed
2686 these are implemented by program abort instead.
2688 \item \hyperlink{command.HOL.code-datatype}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}datatype}}}} specifies a constructor set
2691 \item \hyperlink{command.HOL.print-codesetup}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}codesetup}}}} gives an overview on
2692 selected code equations and code generator datatypes.
2694 \item \hyperlink{attribute.HOL.code-inline}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}inline}}} declares (or with option
2695 ``\isa{{\isaliteral{22}{\isachardoublequote}}del{\isaliteral{22}{\isachardoublequote}}}'' removes) inlining theorems which are applied as
2696 rewrite rules to any code equation during preprocessing.
2698 \item \hyperlink{attribute.HOL.code-post}{\mbox{\isa{code{\isaliteral{5F}{\isacharunderscore}}post}}} declares (or with option ``\isa{{\isaliteral{22}{\isachardoublequote}}del{\isaliteral{22}{\isachardoublequote}}}'' removes) theorems which are applied as rewrite rules to any
2699 result of an evaluation.
2701 \item \hyperlink{command.HOL.print-codeproc}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}codeproc}}}} prints the setup of the code
2702 generator preprocessor.
2704 \item \hyperlink{command.HOL.code-thms}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}thms}}}} prints a list of theorems
2705 representing the corresponding program containing all given
2706 constants after preprocessing.
2708 \item \hyperlink{command.HOL.code-deps}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}deps}}}} visualizes dependencies of
2709 theorems representing the corresponding program containing all given
2710 constants after preprocessing.
2712 \item \hyperlink{command.HOL.code-const}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}const}}}} associates a list of constants
2713 with target-specific serializations; omitting a serialization
2714 deletes an existing serialization.
2716 \item \hyperlink{command.HOL.code-type}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}type}}}} associates a list of type
2717 constructors with target-specific serializations; omitting a
2718 serialization deletes an existing serialization.
2720 \item \hyperlink{command.HOL.code-class}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}class}}}} associates a list of classes
2721 with target-specific class names; omitting a serialization deletes
2722 an existing serialization. This applies only to \emph{Haskell}.
2724 \item \hyperlink{command.HOL.code-instance}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}instance}}}} declares a list of type
2725 constructor / class instance relations as ``already present'' for a
2726 given target. Omitting a ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2D}{\isacharminus}}{\isaliteral{22}{\isachardoublequote}}}'' deletes an existing
2727 ``already present'' declaration. This applies only to
2730 \item \hyperlink{command.HOL.code-reserved}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reserved}}}} declares a list of names as
2731 reserved for a given target, preventing it to be shadowed by any
2734 \item \hyperlink{command.HOL.code-monad}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}monad}}}} provides an auxiliary mechanism
2735 to generate monadic code for Haskell.
2737 \item \hyperlink{command.HOL.code-include}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}include}}}} adds arbitrary named content
2738 (``include'') to generated code. A ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2D}{\isacharminus}}{\isaliteral{22}{\isachardoublequote}}}'' as last argument
2739 will remove an already added ``include''.
2741 \item \hyperlink{command.HOL.code-modulename}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}modulename}}}} declares aliasings from one
2742 module name onto another.
2744 \item \hyperlink{command.HOL.code-reflect}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}reflect}}}} without a ``\isa{{\isaliteral{22}{\isachardoublequote}}file{\isaliteral{22}{\isachardoublequote}}}''
2745 argument compiles code into the system runtime environment and
2746 modifies the code generator setup that future invocations of system
2747 runtime code generation referring to one of the ``\isa{{\isaliteral{22}{\isachardoublequote}}datatypes{\isaliteral{22}{\isachardoublequote}}}'' or ``\isa{{\isaliteral{22}{\isachardoublequote}}functions{\isaliteral{22}{\isachardoublequote}}}'' entities use these precompiled
2748 entities. With a ``\isa{{\isaliteral{22}{\isachardoublequote}}file{\isaliteral{22}{\isachardoublequote}}}'' argument, the corresponding code
2749 is generated into that specified file without modifying the code
2753 \end{isamarkuptext}%
2756 \isamarkupsubsection{The old code generator (S. Berghofer)%
2760 \begin{isamarkuptext}%
2761 This framework generates code from both functional and
2762 relational programs to SML, as explained below.
2764 \begin{matharray}{rcl}
2765 \indexdef{}{command}{code\_module}\hypertarget{command.code-module}{\hyperlink{command.code-module}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2766 \indexdef{}{command}{code\_library}\hypertarget{command.code-library}{\hyperlink{command.code-library}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}library}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2767 \indexdef{}{command}{consts\_code}\hypertarget{command.consts-code}{\hyperlink{command.consts-code}{\mbox{\isa{\isacommand{consts{\isaliteral{5F}{\isacharunderscore}}code}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2768 \indexdef{}{command}{types\_code}\hypertarget{command.types-code}{\hyperlink{command.types-code}{\mbox{\isa{\isacommand{types{\isaliteral{5F}{\isacharunderscore}}code}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
2769 \indexdef{}{attribute}{code}\hypertarget{attribute.code}{\hyperlink{attribute.code}{\mbox{\isa{code}}}} & : & \isa{attribute} \\
2775 \rail@term{\hyperlink{command.code-module}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}}}}}[]
2777 \rail@term{\hyperlink{command.code-library}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}library}}}}}[]
2781 \rail@nont{\isa{modespec}}[]
2785 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
2790 \rail@term{\isa{\isakeyword{file}}}[]
2791 \rail@nont{\isa{name}}[]
2795 \rail@term{\isa{\isakeyword{imports}}}[]
2797 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
2802 \rail@term{\isa{\isakeyword{contains}}}[]
2805 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
2806 \rail@term{\isa{{\isaliteral{3D}{\isacharequal}}}}[]
2807 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
2812 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
2817 \rail@begin{2}{\isa{modespec}}
2818 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
2821 \rail@cnont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
2823 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
2826 \rail@term{\hyperlink{command.HOL.consts-code}{\mbox{\isa{\isacommand{consts{\isaliteral{5F}{\isacharunderscore}}code}}}}}[]
2828 \rail@nont{\isa{codespec}}[]
2832 \rail@begin{2}{\isa{codespec}}
2833 \rail@nont{\isa{const}}[]
2834 \rail@nont{\isa{template}}[]
2837 \rail@nont{\isa{attachment}}[]
2841 \rail@term{\hyperlink{command.HOL.types-code}{\mbox{\isa{\isacommand{types{\isaliteral{5F}{\isacharunderscore}}code}}}}}[]
2843 \rail@nont{\isa{tycodespec}}[]
2847 \rail@begin{2}{\isa{tycodespec}}
2848 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
2849 \rail@nont{\isa{template}}[]
2852 \rail@nont{\isa{attachment}}[]
2855 \rail@begin{1}{\isa{const}}
2856 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
2858 \rail@begin{1}{\isa{template}}
2859 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
2860 \rail@nont{\hyperlink{syntax.string}{\mbox{\isa{string}}}}[]
2861 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
2863 \rail@begin{2}{\isa{attachment}}
2864 \rail@term{\isa{attach}}[]
2867 \rail@nont{\isa{modespec}}[]
2869 \rail@term{\isa{{\isaliteral{7B}{\isacharbraceleft}}}}[]
2870 \rail@nont{\hyperlink{syntax.text}{\mbox{\isa{text}}}}[]
2871 \rail@term{\isa{{\isaliteral{7D}{\isacharbraceright}}}}[]
2874 \rail@term{\hyperlink{attribute.code}{\mbox{\isa{code}}}}[]
2877 \rail@nont{\isa{name}}[]
2881 \end{isamarkuptext}%
2884 \isamarkupsubsubsection{Invoking the code generator%
2888 \begin{isamarkuptext}%
2889 The code generator is invoked via the \hyperlink{command.code-module}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}}}}
2890 and \hyperlink{command.code-library}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}library}}}} commands, which correspond to
2891 \emph{incremental} and \emph{modular} code generation, respectively.
2895 \item [Modular] For each theory, an ML structure is generated,
2896 containing the code generated from the constants defined in this
2899 \item [Incremental] All the generated code is emitted into the same
2900 structure. This structure may import code from previously generated
2901 structures, which can be specified via \hyperlink{keyword.imports}{\mbox{\isa{\isakeyword{imports}}}}.
2902 Moreover, the generated structure may also be referred to in later
2903 invocations of the code generator.
2907 After the \hyperlink{command.code-module}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}}}} and \hyperlink{command.code-library}{\mbox{\isa{\isacommand{code{\isaliteral{5F}{\isacharunderscore}}library}}}}
2908 keywords, the user may specify an optional list of ``modes'' in
2909 parentheses. These can be used to instruct the code generator to
2910 emit additional code for special purposes, e.g.\ functions for
2911 converting elements of generated datatypes to Isabelle terms, or
2912 test data generators. The list of modes is followed by a module
2913 name. The module name is optional for modular code generation, but
2914 must be specified for incremental code generation.
2916 The code can either be written to a file, in which case a file name
2917 has to be specified after the \hyperlink{keyword.file}{\mbox{\isa{\isakeyword{file}}}} keyword, or be loaded
2918 directly into Isabelle's ML environment. In the latter case, the
2919 \hyperlink{command.ML}{\mbox{\isa{\isacommand{ML}}}} theory command can be used to inspect the results
2920 interactively, for example.
2922 The terms from which to generate code can be specified after the
2923 \hyperlink{keyword.contains}{\mbox{\isa{\isakeyword{contains}}}} keyword, either as a list of bindings, or just
2924 as a list of terms. In the latter case, the code generator just
2925 produces code for all constants and types occuring in the term, but
2926 does not bind the compiled terms to ML identifiers.
2928 Here is an example:%
2929 \end{isamarkuptext}%
2931 \isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}\isamarkupfalse%
2933 \isakeyword{contains}\ test\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}foldl\ op\ {\isaliteral{2B}{\isacharplus}}\ {\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ int{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{2}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{3}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{4}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{5}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}%
2934 \begin{isamarkuptext}%
2935 \noindent This binds the result of compiling the given term to
2936 the ML identifier \verb|Test.test|.%
2937 \end{isamarkuptext}%
2945 \isacommand{ML}\isamarkupfalse%
2946 \ {\isaliteral{7B2A}{\isacharverbatimopen}}\ %
2950 \ {\isaliteral{28}{\isacharparenleft}}Test{\isaliteral{2E}{\isachardot}}test\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{1}}{\isadigit{5}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}%
2958 \isamarkupsubsubsection{Configuring the code generator%
2962 \begin{isamarkuptext}%
2963 When generating code for a complex term, the code generator
2964 recursively calls itself for all subterms. When it arrives at a
2965 constant, the default strategy of the code generator is to look up
2966 its definition and try to generate code for it. Constants which
2967 have no definitions that are immediately executable, may be
2968 associated with a piece of ML code manually using the \indexref{}{command}{consts\_code}\hyperlink{command.consts-code}{\mbox{\isa{\isacommand{consts{\isaliteral{5F}{\isacharunderscore}}code}}}} command. It takes a list whose elements consist of a
2969 constant (given in usual term syntax -- an explicit type constraint
2970 accounts for overloading), and a mixfix template describing the ML
2971 code. The latter is very much the same as the mixfix templates used
2972 when declaring new constants. The most notable difference is that
2973 terms may be included in the ML template using antiquotation
2974 brackets \verb|{|\verb|*|~\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{2E}{\isachardot}}{\isaliteral{22}{\isachardoublequote}}}~\verb|*|\verb|}|.
2976 A similar mechanism is available for types: \indexref{}{command}{types\_code}\hyperlink{command.types-code}{\mbox{\isa{\isacommand{types{\isaliteral{5F}{\isacharunderscore}}code}}}} associates type constructors with specific ML code.
2978 For example, the following declarations copied from \verb|~~/src/HOL/Product_Type.thy| describe how the product type of
2979 Isabelle/HOL should be compiled to ML.%
2980 \end{isamarkuptext}%
2982 \isacommand{typedecl}\isamarkupfalse%
2983 \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ prod\isanewline
2984 \isacommand{consts}\isamarkupfalse%
2985 \ Pair\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{27}{\isacharprime}}a\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{27}{\isacharprime}}b\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{27}{\isacharprime}}a{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{27}{\isacharprime}}b{\isaliteral{29}{\isacharparenright}}\ prod{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
2987 \isacommand{types{\isaliteral{5F}{\isacharunderscore}}code}\isamarkupfalse%
2988 \ prod\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5F}{\isacharunderscore}}\ {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{2F}{\isacharslash}}\ {\isaliteral{5F}{\isacharunderscore}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\isanewline
2989 \isacommand{consts{\isaliteral{5F}{\isacharunderscore}}code}\isamarkupfalse%
2990 \ Pair\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{2F}{\isacharslash}}\ {\isaliteral{5F}{\isacharunderscore}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}%
2991 \begin{isamarkuptext}%
2992 Sometimes, the code associated with a constant or type may
2993 need to refer to auxiliary functions, which have to be emitted when
2994 the constant is used. Code for such auxiliary functions can be
2995 declared using \hyperlink{keyword.attach}{\mbox{\isa{\isakeyword{attach}}}}. For example, the \isa{wfrec}
2996 function can be implemented as follows:%
2997 \end{isamarkuptext}%
2999 \isacommand{consts{\isaliteral{5F}{\isacharunderscore}}code}\isamarkupfalse%
3000 \ wfrec\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6D6F64756C653E}{\isasymmodule}}wfrec{\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequoteclose}}{\isaliteral{29}{\isacharparenright}}\ \ \isanewline
3001 \isakeyword{attach}\ {\isaliteral{7B2A}{\isacharverbatimopen}}\ fun\ wfrec\ f\ x\ {\isaliteral{3D}{\isacharequal}}\ f\ {\isaliteral{28}{\isacharparenleft}}wfrec\ f{\isaliteral{29}{\isacharparenright}}\ x\ {\isaliteral{2A7D}{\isacharverbatimclose}}%
3002 \begin{isamarkuptext}%
3003 If the code containing a call to \isa{wfrec} resides in an
3004 ML structure different from the one containing the function
3005 definition attached to \isa{wfrec}, the name of the ML structure
3006 (followed by a ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{2E}{\isachardot}}{\isaliteral{22}{\isachardoublequote}}}'') is inserted in place of ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5C3C6D6F64756C653E}{\isasymmodule}}{\isaliteral{22}{\isachardoublequote}}}'' in the above template. The ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{3F}{\isacharquery}}{\isaliteral{22}{\isachardoublequote}}}'' means that
3007 the code generator should ignore the first argument of \isa{wfrec}, i.e.\ the termination relation, which is usually not
3010 \medskip Another possibility of configuring the code generator is to
3011 register theorems to be used for code generation. Theorems can be
3012 registered via the \hyperlink{attribute.code}{\mbox{\isa{code}}} attribute. It takes an optional
3013 name as an argument, which indicates the format of the
3014 theorem. Currently supported formats are equations (this is the
3015 default when no name is specified) and horn clauses (this is
3016 indicated by the name \texttt{ind}). The left-hand sides of
3017 equations may only contain constructors and distinct variables,
3018 whereas horn clauses must have the same format as introduction rules
3019 of inductive definitions.
3021 The following example specifies three equations from which to
3022 generate code for \isa{{\isaliteral{22}{\isachardoublequote}}op\ {\isaliteral{3C}{\isacharless}}{\isaliteral{22}{\isachardoublequote}}} on natural numbers (see also
3023 \verb|~~/src/HOL/Nat.thy|).%
3024 \end{isamarkuptext}%
3026 \isacommand{lemma}\isamarkupfalse%
3027 \ {\isaliteral{5B}{\isacharbrackleft}}code{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}Suc\ m\ {\isaliteral{3C}{\isacharless}}\ Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}m\ {\isaliteral{3C}{\isacharless}}\ n{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3028 \ \ \isakeyword{and}\ {\isaliteral{5B}{\isacharbrackleft}}code{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}n{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}nat{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3C}{\isacharless}}\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ False{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3029 \ \ \isakeyword{and}\ {\isaliteral{5B}{\isacharbrackleft}}code{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}{\isadigit{0}}\ {\isaliteral{3C}{\isacharless}}\ Suc\ n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{3D}{\isacharequal}}\ True{\isaliteral{22}{\isachardoublequoteclose}}%
3035 \isacommand{by}\isamarkupfalse%
3036 \ simp{\isaliteral{5F}{\isacharunderscore}}all%
3044 \isamarkupsubsubsection{Specific HOL code generators%
3048 \begin{isamarkuptext}%
3049 The basic code generator framework offered by Isabelle/Pure
3050 has already been extended with additional code generators for
3051 specific HOL constructs. These include datatypes, recursive
3052 functions and inductive relations. The code generator for inductive
3053 relations can handle expressions of the form \isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{28}{\isacharparenleft}}t\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}\ {\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}\ t\isaliteral{5C3C5E7375623E}{}\isactrlsub n{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C696E3E}{\isasymin}}\ r{\isaliteral{22}{\isachardoublequote}}}, where \isa{{\isaliteral{22}{\isachardoublequote}}r{\isaliteral{22}{\isachardoublequote}}} is an inductively defined relation. If at
3054 least one of the \isa{{\isaliteral{22}{\isachardoublequote}}t\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} is a dummy pattern ``\isa{{\isaliteral{22}{\isachardoublequote}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{22}{\isachardoublequote}}}'',
3055 the above expression evaluates to a sequence of possible answers. If
3056 all of the \isa{{\isaliteral{22}{\isachardoublequote}}t\isaliteral{5C3C5E7375623E}{}\isactrlsub i{\isaliteral{22}{\isachardoublequote}}} are proper terms, the expression evaluates
3059 The following example demonstrates this for beta-reduction on lambda
3060 terms (see also \verb|~~/src/HOL/Proofs/Lambda/Lambda.thy|).%
3061 \end{isamarkuptext}%
3063 \isacommand{datatype}\isamarkupfalse%
3064 \ dB\ {\isaliteral{3D}{\isacharequal}}\isanewline
3065 \ \ \ \ Var\ nat\isanewline
3066 \ \ {\isaliteral{7C}{\isacharbar}}\ App\ dB\ dB\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C6465677265653E}{\isasymdegree}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{2}}{\isadigit{0}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
3067 \ \ {\isaliteral{7C}{\isacharbar}}\ Abs\ dB\isanewline
3069 \isacommand{primrec}\isamarkupfalse%
3070 \ lift\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dB\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ dB{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3071 \isakeyword{where}\isanewline
3072 \ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}lift\ {\isaliteral{28}{\isacharparenleft}}Var\ i{\isaliteral{29}{\isacharparenright}}\ k\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{28}{\isacharparenleft}}if\ i\ {\isaliteral{3C}{\isacharless}}\ k\ then\ Var\ i\ else\ Var\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3073 \ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}lift\ {\isaliteral{28}{\isacharparenleft}}s\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ t{\isaliteral{29}{\isacharparenright}}\ k\ {\isaliteral{3D}{\isacharequal}}\ lift\ s\ k\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ lift\ t\ k{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3074 \ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}lift\ {\isaliteral{28}{\isacharparenleft}}Abs\ s{\isaliteral{29}{\isacharparenright}}\ k\ {\isaliteral{3D}{\isacharequal}}\ Abs\ {\isaliteral{28}{\isacharparenleft}}lift\ s\ {\isaliteral{28}{\isacharparenleft}}k\ {\isaliteral{2B}{\isacharplus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3076 \isacommand{primrec}\isamarkupfalse%
3077 \ subst\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dB\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ dB\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ nat\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ dB{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{5B}{\isacharbrackleft}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{27}{\isacharprime}}{\isaliteral{2F}{\isacharslash}}{\isaliteral{5F}{\isacharunderscore}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isaliteral{5B}{\isacharbrackleft}}{\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{2C}{\isacharcomma}}\ {\isadigit{0}}{\isaliteral{5D}{\isacharbrackright}}\ {\isadigit{3}}{\isadigit{0}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
3078 \isakeyword{where}\isanewline
3079 \ \ \ \ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}Var\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{5B}{\isacharbrackleft}}s{\isaliteral{2F}{\isacharslash}}k{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\isanewline
3080 \ \ \ \ \ \ {\isaliteral{28}{\isacharparenleft}}if\ k\ {\isaliteral{3C}{\isacharless}}\ i\ then\ Var\ {\isaliteral{28}{\isacharparenleft}}i\ {\isaliteral{2D}{\isacharminus}}\ {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ else\ if\ i\ {\isaliteral{3D}{\isacharequal}}\ k\ then\ s\ else\ Var\ i{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3081 \ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}t\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ u{\isaliteral{29}{\isacharparenright}}{\isaliteral{5B}{\isacharbrackleft}}s{\isaliteral{2F}{\isacharslash}}k{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ t{\isaliteral{5B}{\isacharbrackleft}}s{\isaliteral{2F}{\isacharslash}}k{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ u{\isaliteral{5B}{\isacharbrackleft}}s{\isaliteral{2F}{\isacharslash}}k{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3082 \ \ {\isaliteral{7C}{\isacharbar}}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{28}{\isacharparenleft}}Abs\ t{\isaliteral{29}{\isacharparenright}}{\isaliteral{5B}{\isacharbrackleft}}s{\isaliteral{2F}{\isacharslash}}k{\isaliteral{5D}{\isacharbrackright}}\ {\isaliteral{3D}{\isacharequal}}\ Abs\ {\isaliteral{28}{\isacharparenleft}}t{\isaliteral{5B}{\isacharbrackleft}}lift\ s\ {\isadigit{0}}\ {\isaliteral{2F}{\isacharslash}}\ k{\isaliteral{2B}{\isacharplus}}{\isadigit{1}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3084 \isacommand{inductive}\isamarkupfalse%
3085 \ beta\ {\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}dB\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ dB\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ bool{\isaliteral{22}{\isachardoublequoteclose}}\ \ {\isaliteral{28}{\isacharparenleft}}\isakeyword{infixl}\ {\isaliteral{22}{\isachardoublequoteopen}}{\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}{\isaliteral{22}{\isachardoublequoteclose}}\ {\isadigit{5}}{\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\isanewline
3086 \isakeyword{where}\isanewline
3087 \ \ \ \ beta{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}Abs\ s\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ t\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ s{\isaliteral{5B}{\isacharbrackleft}}t{\isaliteral{2F}{\isacharslash}}{\isadigit{0}}{\isaliteral{5D}{\isacharbrackright}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3088 \ \ {\isaliteral{7C}{\isacharbar}}\ appL{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ t\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ s\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ u\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ t\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ u{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3089 \ \ {\isaliteral{7C}{\isacharbar}}\ appR{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ t\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ u\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ s\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ u\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ t{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3090 \ \ {\isaliteral{7C}{\isacharbar}}\ abs{\isaliteral{3A}{\isacharcolon}}\ {\isaliteral{22}{\isachardoublequoteopen}}s\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ t\ {\isaliteral{5C3C4C6F6E6772696768746172726F773E}{\isasymLongrightarrow}}\ Abs\ s\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ Abs\ t{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3092 \isacommand{code{\isaliteral{5F}{\isacharunderscore}}module}\isamarkupfalse%
3094 \isakeyword{contains}\isanewline
3095 \ \ test{\isadigit{1}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}Abs\ {\isaliteral{28}{\isacharparenleft}}Var\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ Var\ {\isadigit{0}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ Var\ {\isadigit{0}}{\isaliteral{22}{\isachardoublequoteclose}}\isanewline
3096 \ \ test{\isadigit{2}}\ {\isaliteral{3D}{\isacharequal}}\ {\isaliteral{22}{\isachardoublequoteopen}}Abs\ {\isaliteral{28}{\isacharparenleft}}Abs\ {\isaliteral{28}{\isacharparenleft}}Var\ {\isadigit{0}}\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ Var\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ {\isaliteral{28}{\isacharparenleft}}Abs\ {\isaliteral{28}{\isacharparenleft}}Var\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C6465677265653E}{\isasymdegree}}\ Var\ {\isadigit{0}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\isaliteral{5C3C5E7375623E}{}\isactrlsub {\isaliteral{5C3C626574613E}{\isasymbeta}}\ {\isaliteral{5F}{\isacharunderscore}}{\isaliteral{22}{\isachardoublequoteclose}}%
3097 \begin{isamarkuptext}%
3098 In the above example, \verb|Test.test1| evaluates to a boolean,
3099 whereas \verb|Test.test2| is a lazy sequence whose elements can be
3100 inspected separately.%
3101 \end{isamarkuptext}%
3109 \isacommand{ML}\isamarkupfalse%
3110 \ {\isaliteral{7B2A}{\isacharverbatimopen}}\ %
3114 \ Test{\isaliteral{2E}{\isachardot}}test{\isadigit{1}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}\isanewline
3115 \isacommand{ML}\isamarkupfalse%
3116 \ {\isaliteral{7B2A}{\isacharverbatimopen}}\ val\ results\ {\isaliteral{3D}{\isacharequal}}\ DSeq{\isaliteral{2E}{\isachardot}}list{\isaliteral{5F}{\isacharunderscore}}of\ Test{\isaliteral{2E}{\isachardot}}test{\isadigit{2}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}\isanewline
3117 \isacommand{ML}\isamarkupfalse%
3118 \ {\isaliteral{7B2A}{\isacharverbatimopen}}\ %
3122 \ {\isaliteral{28}{\isacharparenleft}}length\ results\ {\isaliteral{3D}{\isacharequal}}\ {\isadigit{2}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{2A7D}{\isacharverbatimclose}}%
3130 \begin{isamarkuptext}%
3131 \medskip The theory underlying the HOL code generator is described
3132 more detailed in \cite{Berghofer-Nipkow:2002}. More examples that
3133 illustrate the usage of the code generator can be found e.g.\ in
3134 \verb|~~/src/HOL/MicroJava/J/JListExample.thy| and \verb|~~/src/HOL/MicroJava/JVM/JVMListExample.thy|.%
3135 \end{isamarkuptext}%
3138 \isamarkupsection{Definition by specification \label{sec:hol-specification}%
3142 \begin{isamarkuptext}%
3143 \begin{matharray}{rcl}
3144 \indexdef{HOL}{command}{specification}\hypertarget{command.HOL.specification}{\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
3145 \indexdef{HOL}{command}{ax\_specification}\hypertarget{command.HOL.ax-specification}{\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ proof{\isaliteral{28}{\isacharparenleft}}prove{\isaliteral{29}{\isacharparenright}}{\isaliteral{22}{\isachardoublequote}}} \\
3151 \rail@term{\hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}}[]
3153 \rail@term{\hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}}}[]
3155 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
3157 \rail@nont{\isa{decl}}[]
3160 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
3165 \rail@nont{\hyperlink{syntax.thmdecl}{\mbox{\isa{thmdecl}}}}[]
3167 \rail@nont{\hyperlink{syntax.prop}{\mbox{\isa{prop}}}}[]
3171 \rail@begin{2}{\isa{decl}}
3174 \rail@nont{\hyperlink{syntax.name}{\mbox{\isa{name}}}}[]
3175 \rail@term{\isa{{\isaliteral{3A}{\isacharcolon}}}}[]
3177 \rail@nont{\hyperlink{syntax.term}{\mbox{\isa{term}}}}[]
3178 \rail@term{\isa{{\isaliteral{28}{\isacharparenleft}}}}[]
3179 \rail@term{\isa{\isakeyword{overloaded}}}[]
3182 \rail@term{\isa{{\isaliteral{29}{\isacharparenright}}}}[]
3190 \item \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}decls\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} sets up a
3191 goal stating the existence of terms with the properties specified to
3192 hold for the constants given in \isa{decls}. After finishing the
3193 proof, the theory will be augmented with definitions for the given
3194 constants, as well as with theorems stating the properties for these
3197 \item \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}}~\isa{{\isaliteral{22}{\isachardoublequote}}decls\ {\isaliteral{5C3C7068693E}{\isasymphi}}{\isaliteral{22}{\isachardoublequote}}} sets up
3198 a goal stating the existence of terms with the properties specified
3199 to hold for the constants given in \isa{decls}. After finishing
3200 the proof, the theory will be augmented with axioms expressing the
3201 properties given in the first place.
3203 \item \isa{decl} declares a constant to be defined by the
3204 specification given. The definition for the constant \isa{c} is
3205 bound to the name \isa{c{\isaliteral{5F}{\isacharunderscore}}def} unless a theorem name is given in
3206 the declaration. Overloaded constants should be declared as such.
3210 Whether to use \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} or \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}} is to some extent a matter of style. \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}} introduces no new axioms, and so by
3211 construction cannot introduce inconsistencies, whereas \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}} does introduce axioms, but only after the
3212 user has explicitly proven it to be safe. A practical issue must be
3213 considered, though: After introducing two constants with the same
3214 properties using \hyperlink{command.HOL.specification}{\mbox{\isa{\isacommand{specification}}}}, one can prove
3215 that the two constants are, in fact, equal. If this might be a
3216 problem, one should use \hyperlink{command.HOL.ax-specification}{\mbox{\isa{\isacommand{ax{\isaliteral{5F}{\isacharunderscore}}specification}}}}.%
3217 \end{isamarkuptext}%
3225 \isacommand{end}\isamarkupfalse%
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