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\begin{isabellebody}%
\def\isabellecontext{ZF{\isaliteral{5F}{\isacharunderscore}}Specific}%
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\isadelimtheory
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\isatagtheory
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\ ZF{\isaliteral{5F}{\isacharunderscore}}Specific\isanewline
\isakeyword{imports}\ Main\isanewline
\isakeyword{begin}%
\endisatagtheory
{\isafoldtheory}%
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\isadelimtheory
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\endisadelimtheory
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\isamarkupchapter{Isabelle/ZF \label{ch:zf}%
}
\isamarkuptrue%
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\isamarkupsection{Type checking%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
The ZF logic is essentially untyped, so the concept of ``type
checking'' is performed as logical reasoning about set-membership
statements. A special method assists users in this task; a version
of this is already declared as a ``solver'' in the standard
Simplifier setup.
\begin{matharray}{rcl}
\indexdef{ZF}{command}{print\_tcset}\hypertarget{command.ZF.print-tcset}{\hyperlink{command.ZF.print-tcset}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}tcset}}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}context\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}{\isaliteral{22}{\isachardoublequote}}} \\
\indexdef{ZF}{method}{typecheck}\hypertarget{method.ZF.typecheck}{\hyperlink{method.ZF.typecheck}{\mbox{\isa{typecheck}}}} & : & \isa{method} \\
\indexdef{ZF}{attribute}{TC}\hypertarget{attribute.ZF.TC}{\hyperlink{attribute.ZF.TC}{\mbox{\isa{TC}}}} & : & \isa{attribute} \\
\end{matharray}
\begin{rail}
'TC' (() | 'add' | 'del')
;
\end{rail}
\begin{description}
\item \hyperlink{command.ZF.print-tcset}{\mbox{\isa{\isacommand{print{\isaliteral{5F}{\isacharunderscore}}tcset}}}} prints the collection of
typechecking rules of the current context.
\item \hyperlink{method.ZF.typecheck}{\mbox{\isa{typecheck}}} attempts to solve any pending
type-checking problems in subgoals.
\item \hyperlink{attribute.ZF.TC}{\mbox{\isa{TC}}} adds or deletes type-checking rules from
the context.
\end{description}%
\end{isamarkuptext}%
\isamarkuptrue%
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\isamarkupsection{(Co)Inductive sets and datatypes%
}
\isamarkuptrue%
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\isamarkupsubsection{Set definitions%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
In ZF everything is a set. The generic inductive package also
provides a specific view for ``datatype'' specifications.
Coinductive definitions are available in both cases, too.
\begin{matharray}{rcl}
\indexdef{ZF}{command}{inductive}\hypertarget{command.ZF.inductive}{\hyperlink{command.ZF.inductive}{\mbox{\isa{\isacommand{inductive}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
\indexdef{ZF}{command}{coinductive}\hypertarget{command.ZF.coinductive}{\hyperlink{command.ZF.coinductive}{\mbox{\isa{\isacommand{coinductive}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
\indexdef{ZF}{command}{datatype}\hypertarget{command.ZF.datatype}{\hyperlink{command.ZF.datatype}{\mbox{\isa{\isacommand{datatype}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
\indexdef{ZF}{command}{codatatype}\hypertarget{command.ZF.codatatype}{\hyperlink{command.ZF.codatatype}{\mbox{\isa{\isacommand{codatatype}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
\end{matharray}
\begin{rail}
('inductive' | 'coinductive') domains intros hints
;
domains: 'domains' (term + '+') ('<=' | subseteq) term
;
intros: 'intros' (thmdecl? prop +)
;
hints: monos? condefs? typeintros? typeelims?
;
monos: ('monos' thmrefs)?
;
condefs: ('con_defs' thmrefs)?
;
typeintros: ('type_intros' thmrefs)?
;
typeelims: ('type_elims' thmrefs)?
;
\end{rail}
In the following syntax specification \isa{{\isaliteral{22}{\isachardoublequote}}monos{\isaliteral{22}{\isachardoublequote}}}, \isa{typeintros}, and \isa{typeelims} are the same as above.
\begin{rail}
('datatype' | 'codatatype') domain? (dtspec + 'and') hints
;
domain: ('<=' | subseteq) term
;
dtspec: term '=' (con + '|')
;
con: name ('(' (term ',' +) ')')?
;
hints: monos? typeintros? typeelims?
;
\end{rail}
See \cite{isabelle-ZF} for further information on inductive
definitions in ZF, but note that this covers the old-style theory
format.%
\end{isamarkuptext}%
\isamarkuptrue%
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\isamarkupsubsection{Primitive recursive functions%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
\begin{matharray}{rcl}
\indexdef{ZF}{command}{primrec}\hypertarget{command.ZF.primrec}{\hyperlink{command.ZF.primrec}{\mbox{\isa{\isacommand{primrec}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
\end{matharray}
\begin{rail}
'primrec' (thmdecl? prop +)
;
\end{rail}%
\end{isamarkuptext}%
\isamarkuptrue%
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\isamarkupsubsection{Cases and induction: emulating tactic scripts%
}
\isamarkuptrue%
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\begin{isamarkuptext}%
The following important tactical tools of Isabelle/ZF have been
ported to Isar. These should not be used in proper proof texts.
\begin{matharray}{rcl}
\indexdef{ZF}{method}{case\_tac}\hypertarget{method.ZF.case-tac}{\hyperlink{method.ZF.case-tac}{\mbox{\isa{case{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\
\indexdef{ZF}{method}{induct\_tac}\hypertarget{method.ZF.induct-tac}{\hyperlink{method.ZF.induct-tac}{\mbox{\isa{induct{\isaliteral{5F}{\isacharunderscore}}tac}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\
\indexdef{ZF}{method}{ind\_cases}\hypertarget{method.ZF.ind-cases}{\hyperlink{method.ZF.ind-cases}{\mbox{\isa{ind{\isaliteral{5F}{\isacharunderscore}}cases}}}}\isa{{\isaliteral{22}{\isachardoublequote}}\isaliteral{5C3C5E7375703E}{}\isactrlsup {\isaliteral{2A}{\isacharasterisk}}{\isaliteral{22}{\isachardoublequote}}} & : & \isa{method} \\
\indexdef{ZF}{command}{inductive\_cases}\hypertarget{command.ZF.inductive-cases}{\hyperlink{command.ZF.inductive-cases}{\mbox{\isa{\isacommand{inductive{\isaliteral{5F}{\isacharunderscore}}cases}}}}} & : & \isa{{\isaliteral{22}{\isachardoublequote}}theory\ {\isaliteral{5C3C72696768746172726F773E}{\isasymrightarrow}}\ theory{\isaliteral{22}{\isachardoublequote}}} \\
\end{matharray}
\begin{rail}
('case_tac' | 'induct_tac') goalspec? name
;
indcases (prop +)
;
inductivecases (thmdecl? (prop +) + 'and')
;
\end{rail}%
\end{isamarkuptext}%
\isamarkuptrue%
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\isadelimtheory
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\end{isabellebody}%
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