theory ZF_Specific
imports Base Main
begin
chapter {* Isabelle/ZF \label{ch:zf} *}
section {* Type checking *}
text {*
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}
@{command_def (ZF) "print_tcset"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
@{method_def (ZF) typecheck} & : & @{text method} \\
@{attribute_def (ZF) TC} & : & @{text attribute} \\
\end{matharray}
@{rail "
@@{attribute (ZF) TC} (() | 'add' | 'del')
"}
\begin{description}
\item @{command (ZF) "print_tcset"} prints the collection of
typechecking rules of the current context.
\item @{method (ZF) typecheck} attempts to solve any pending
type-checking problems in subgoals.
\item @{attribute (ZF) TC} adds or deletes type-checking rules from
the context.
\end{description}
*}
section {* (Co)Inductive sets and datatypes *}
subsection {* Set definitions *}
text {*
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}
@{command_def (ZF) "inductive"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def (ZF) "coinductive"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def (ZF) "datatype"} & : & @{text "theory \<rightarrow> theory"} \\
@{command_def (ZF) "codatatype"} & : & @{text "theory \<rightarrow> theory"} \\
\end{matharray}
@{rail "
(@@{command (ZF) inductive} | @@{command (ZF) coinductive}) domains intros hints
;
domains: @'domains' (@{syntax term} + '+') ('<=' | '\<subseteq>') @{syntax term}
;
intros: @'intros' (@{syntax thmdecl}? @{syntax prop} +)
;
hints: @{syntax (ZF) \"monos\"}? condefs? @{syntax (ZF) typeintros}? @{syntax (ZF) typeelims}?
;
@{syntax_def (ZF) \"monos\"}: @'monos' @{syntax thmrefs}
;
condefs: @'con_defs' @{syntax thmrefs}
;
@{syntax_def (ZF) typeintros}: @'type_intros' @{syntax thmrefs}
;
@{syntax_def (ZF) typeelims}: @'type_elims' @{syntax thmrefs}
"}
In the following syntax specification @{text "monos"}, @{text
typeintros}, and @{text typeelims} are the same as above.
@{rail "
(@@{command (ZF) datatype} | @@{command (ZF) codatatype}) domain? (dtspec + @'and') hints
;
domain: ('<=' | '\<subseteq>') @{syntax term}
;
dtspec: @{syntax term} '=' (con + '|')
;
con: @{syntax name} ('(' (@{syntax term} ',' +) ')')?
;
hints: @{syntax (ZF) \"monos\"}? @{syntax (ZF) typeintros}? @{syntax (ZF) typeelims}?
"}
See \cite{isabelle-ZF} for further information on inductive
definitions in ZF, but note that this covers the old-style theory
format.
*}
subsection {* Primitive recursive functions *}
text {*
\begin{matharray}{rcl}
@{command_def (ZF) "primrec"} & : & @{text "theory \<rightarrow> theory"} \\
\end{matharray}
@{rail "
@@{command (ZF) primrec} (@{syntax thmdecl}? @{syntax prop} +)
"}
*}
subsection {* Cases and induction: emulating tactic scripts *}
text {*
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}
@{method_def (ZF) case_tac}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def (ZF) induct_tac}@{text "\<^sup>*"} & : & @{text method} \\
@{method_def (ZF) ind_cases}@{text "\<^sup>*"} & : & @{text method} \\
@{command_def (ZF) "inductive_cases"} & : & @{text "theory \<rightarrow> theory"} \\
\end{matharray}
@{rail "
(@@{method (ZF) case_tac} | @@{method (ZF) induct_tac}) @{syntax goalspec}? @{syntax name}
;
@@{method (ZF) ind_cases} (@{syntax prop} +)
;
@@{command (ZF) inductive_cases} (@{syntax thmdecl}? (@{syntax prop} +) + @'and')
;
"}
*}
end