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products/Sources/formale Sprachen/Isabelle/Doc/Logics_ZF/ (Isabelle Prover Version 2025-1^©) Datei vom 16.11.2025 mit Größe 4 kB

Quelle ZF_Isar.thy

Sprache: Isabelle

theory ZF_Isar
imports ZF
begin

(*<*)
ML_file ‹../antiquote_setup.ML›
(*>*)

chapter ‹Some Isar language elements›

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"}‹^*› & : & ‹context →› \\
@{method_def (ZF) typecheck} & : & ‹method› \\
@{attribute_def (ZF) TC} & : & ‹attribute› \\
\end{matharray}

🚫‹
@@{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"} & : & ‹theory → theory› \\
@{command_def (ZF) "coinductive"} & : & ‹theory → theory› \\
@{command_def (ZF) "datatype"} & : & ‹theory → theory› \\
@{command_def (ZF) "codatatype"} & : & ‹theory → theory› \\
\end{matharray}

🚫‹
(@@{command (ZF) inductive} | @@{command (ZF) coinductive}) domains intros hints
;

domains: @'domains' (@{syntax term} + '+') ('<=' | '⊆') @{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 thms}
;
condefs: @'con_defs' @{syntax thms}
;
@{syntax_def (ZF) typeintros}: @'type_intros' @{syntax thms}
;
@{syntax_def (ZF) typeelims}: @'type_elims' @{syntax thms}
›

In the following syntax specification ‹monos›, ‹typeintros›, and ‹typeelims› are the same as above.

🚫‹
(@@{command (ZF) datatype} | @@{command (ZF) codatatype}) domain? (dtspec + @'and') hints
;

domain: ('<=' | '⊆') @{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"} & : & ‹theory → theory› \\
\end{matharray}

🚫‹
@@{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}‹^*› & : & ‹method› \\
@{method_def (ZF) induct_tac}‹^*› & : & ‹method› \\
@{method_def (ZF) ind_cases}‹^*› & : & ‹method› \\
@{command_def (ZF) "inductive_cases"} & : & ‹theory → theory› \\
\end{matharray}

🚫‹
(@@{method (ZF) case_tac} | @@{method (ZF) induct_tac}) @{syntax goal_spec}? @{syntax name}
;
@@{method (ZF) ind_cases} (@{syntax prop} +)
;
@@{command (ZF) inductive_cases} (@{syntax thmdecl}? (@{syntax prop} +) + @'and')
›
›

end

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