Subtyping¶

Subtyping defines when a value of one type may be used when a value of another type is expected.

TODO: This is not complete, pending further discussion, especially in re the special treatment that may or may not be required or specialized value types.

Value types¶

Reflexivity¶

Any value type is a subtype of itself

\[\frac{ }{ \href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e} \preccurlyeq \href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e} }\]

Numeric types¶

\(\href{../valid/types.html#syntax-evaltype}{\mathsf{s8}}\) is a subtype of \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s16}}\), \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s32}}\), and \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s64}}\).
\(\href{../valid/types.html#syntax-evaltype}{\mathsf{s16}}\) is a subtype of \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s32}}\) and \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s64}}\).
\(\href{../valid/types.html#syntax-evaltype}{\mathsf{s32}}\) is a subtype of \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s64}}\).
\(\href{../valid/types.html#syntax-evaltype}{\mathsf{u8}}\) is a subtype of \(\href{../valid/types.html#syntax-evaltype}{\mathsf{u16}}\), \(\href{../valid/types.html#syntax-evaltype}{\mathsf{u32}}\), \(\href{../valid/types.html#syntax-evaltype}{\mathsf{u64}}\), \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s16}}\), \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s32}}\), and \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s64}}\).
\(\href{../valid/types.html#syntax-evaltype}{\mathsf{u16}}\) is a subtype of \(\href{../valid/types.html#syntax-evaltype}{\mathsf{u32}}\), \(\href{../valid/types.html#syntax-evaltype}{\mathsf{u64}}\), \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s32}}\), and \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s64}}\).
\(\href{../valid/types.html#syntax-evaltype}{\mathsf{u32}}\) is a subtype of \(\href{../valid/types.html#syntax-evaltype}{\mathsf{u64}}\) and \(\href{../valid/types.html#syntax-evaltype}{\mathsf{s64}}\).
\(\href{../valid/types.html#syntax-evaltype}{\mathsf{float32}}\) is a subtype of \(\href{../valid/types.html#syntax-evaltype}{\mathsf{float64}}\).

\[\frac{ m > n }{ \mathsf{s}n \preccurlyeq \mathsf{s}m }\]

\[\frac{ m > n }{ \mathsf{u}n \preccurlyeq \mathsf{u}m }\]

\[\frac{ m > n }{ \mathsf{u}n \preccurlyeq \mathsf{s}m }\]

\[\frac{ }{ \href{../valid/types.html#syntax-evaltype}{\mathsf{float32}} \preccurlyeq \href{../valid/types.html#syntax-evaltype}{\mathsf{float64}} }\]

Records¶

A type \(\href{../valid/types.html#syntax-evaltype2}{\mathsf{record}}~\overline{{\href{../valid/types.html#syntax-erecordfield}{\mathit{record\_field}_e}}_i}\) is a subtype of a type \(\href{../valid/types.html#syntax-evaltype2}{\mathsf{record}}~\overline{{\href{../valid/types.html#syntax-erecordfield}{\mathit{record\_field}_e}'}_j}\) if, for each named field of the latter type, a field with the same name is present in the former, and the type of the field in the former is a subtype of the type of the field in the latter.

Todo: We may need to move despecialization later because of subtyping?

\[\begin{split}\frac{ \begin{aligned} \forall j, \exists i,&{\href{../valid/types.html#syntax-erecordfield}{\mathit{record\_field}_e}}_i.\href{../valid/types.html#syntax-erecordfield}{\mathsf{name}} = {\href{../valid/types.html#syntax-erecordfield}{\mathit{record\_field}_e}'}_j.\href{../valid/types.html#syntax-erecordfield}{\mathsf{name}}\\ \land{}& {\href{../valid/types.html#syntax-erecordfield}{\mathit{record\_field}_e}}_i.\href{../valid/types.html#syntax-erecordfield}{\mathsf{type}} \preccurlyeq {\href{../valid/types.html#syntax-erecordfield}{\mathit{record\_field}_e}}_j.\href{../valid/types.html#syntax-erecordfield}{\mathsf{type}} \end{aligned} }{ \href{../valid/types.html#syntax-evaltype2}{\mathsf{record}}~\overline{{\href{../valid/types.html#syntax-erecordfield}{\mathit{record\_field}_e}}_i} \preccurlyeq \href{../valid/types.html#syntax-evaltype2}{\mathsf{record}}~\overline{{\href{../valid/types.html#syntax-erecordfield}{\mathit{record\_field}_e}'}_j} }\end{split}\]

Variants¶

A type \(\href{../valid/types.html#syntax-evaltype2}{\mathsf{variant}}~\overline{{\href{../valid/types.html#syntax-evariantcase}{\mathit{variant\_case}_e}}_i}\) is a subtype of a type \(\href{../valid/types.html#syntax-evaltype2}{\mathsf{variant}}~\overline{{\href{../valid/types.html#syntax-evariantcase}{\mathit{variant\_case}_e}'}_j}\) if, or each named case of the former type, either:
- A case of the same name exists in the latter type, such that the type of the field in the former is a subtype of the type of the field in the latter; or
- No case of the same name exists in the latter type, and the case in the former contains a \(\href{../valid/types.html#syntax-evariantcase}{\mathsf{refines}}\).

\[\begin{split}\frac{ \begin{alignedat}{2} \forall i,&&(\exists j,&{\href{../valid/types.html#syntax-evariantcase}{\mathit{variant\_case}_e}'}_j.\href{../valid/types.html#syntax-evariantcase}{\mathsf{name}} = {\href{../valid/types.html#syntax-evariantcase}{\mathit{variant\_case}_e}}_i.\href{../valid/types.html#syntax-evariantcase}{\mathsf{name}}\\&& \land{}& {\href{../valid/types.html#syntax-evariantcase}{\mathit{variant\_case}_e}}_i \preccurlyeq {\href{../valid/types.html#syntax-evariantcase}{\mathit{variant\_case}_e}'}_j)\\ \lor{}&&(\forall j,&{\href{../valid/types.html#syntax-evariantcase}{\mathit{variant\_case}_e}'}_j.\href{../valid/types.html#syntax-evariantcase}{\mathsf{name}} \neq {\href{../valid/types.html#syntax-evariantcase}{\mathit{variant\_case}_e}}_i.\href{../valid/types.html#syntax-evariantcase}{\mathsf{name}}\\&& \land{}&\exists \href{../syntax/values.html#syntax-name}{\mathit{name}}, {\href{../valid/types.html#syntax-evariantcase}{\mathit{variant\_case}_e}}_i.\href{../valid/types.html#syntax-evariantcase}{\mathsf{refines}} = \href{../syntax/values.html#syntax-name}{\mathit{name}}) \end{alignedat} }{ \href{../valid/types.html#syntax-evaltype2}{\mathsf{variant}}~\overline{{\href{../valid/types.html#syntax-evariantcase}{\mathit{variant\_case}_e}}_i} \preccurlyeq \href{../valid/types.html#syntax-evaltype2}{\mathsf{variant}}~\overline{{\href{../valid/types.html#syntax-evariantcase}{\mathit{variant\_case}_e}'}_j} }\end{split}\]

Lists¶

A type \(\href{../valid/types.html#syntax-evaltype}{\mathsf{list}}~\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}\) is a subtype of a type \(\href{../valid/types.html#syntax-evaltype}{\mathsf{list}}~\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}'\) if \(\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}'\)

\[\frac{ \href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e} \preccurlyeq \href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}' }{ \href{../valid/types.html#syntax-evaltype}{\mathsf{list}}~\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e} \preccurlyeq \href{../valid/types.html#syntax-evaltype}{\mathsf{list}}~\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}' }\]

Result types¶

A result type of the form \(\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}\) is a subtype of a result type of te form \(\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}'\) if \(\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}'\).

\[\frac{ \href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e} \preccurlyeq \href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}' }{ \href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e} \preccurlyeq \href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}' }\]
A result type of the form \(\overline{\{ \href{../valid/types.html#syntax-eresulttype}{\mathsf{name}}~\href{../syntax/values.html#syntax-name}{\mathit{name}}_i, \href{../valid/types.html#syntax-eresulttype}{\mathsf{type}}~{\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}}_i \}}\) is a subtype of a result type of the form \(\overline{\{ \href{../valid/types.html#syntax-eresulttype}{\mathsf{name}}~\href{../syntax/values.html#syntax-name}{\mathit{name}}'_j, \href{../valid/types.html#syntax-eresulttype}{\mathsf{type}}~{\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}'}_j \}}\) when:
- For each \(\href{../syntax/values.html#syntax-name}{\mathit{name}}'_j\), there is some \(i\) such that \(\href{../syntax/values.html#syntax-name}{\mathit{name}}'_j = \href{../syntax/values.html#syntax-name}{\mathit{name}}_i\) and \({\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}}_i \preccurlyeq {\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}}'_j\).
\[\frac{ \forall j, \exists i, \href{../syntax/values.html#syntax-name}{\mathit{name}}_i = \href{../syntax/values.html#syntax-name}{\mathit{name}}'_j \land {\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}}_i \preccurlyeq {\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}}'_j }{ \overline{\{ \href{../valid/types.html#syntax-eresulttype}{\mathsf{name}}~\href{../syntax/values.html#syntax-name}{\mathit{name}}_i, \href{../valid/types.html#syntax-eresulttype}{\mathsf{type}}~{\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}}_i \}} \preccurlyeq \overline{\{ \href{../valid/types.html#syntax-eresulttype}{\mathsf{name}}~\href{../syntax/values.html#syntax-name}{\mathit{name}}'_j, \href{../valid/types.html#syntax-eresulttype}{\mathsf{type}}~{\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}'}_j \}} }\]

Function types¶

A function type \({\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}}_1 \to {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}}_2\) is a subtype of a function \({\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}'}_1 \to {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}'}_2\) if \({\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}'}_1 \preccurlyeq {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}}_1\) and \({\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}}_2 \preccurlyeq {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}'}_2\).

\[\begin{split}\frac{ \begin{array}{@{}c@{}} {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}'}_1 \preccurlyeq {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}}_1\\ {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}}_2 \preccurlyeq {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}'}_2 \end{array} }{ {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}}_1 \to {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}}_2 \preccurlyeq {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}'}_1 \to {\href{../valid/types.html#syntax-eresulttype}{\mathit{resulttype}_e}'}_2 }\end{split}\]

Type bound¶

\(\href{../valid/types.html#syntax-etypebound}{\mathsf{eq}}~\href{../valid/types.html#syntax-edeftype}{\mathit{deftype}_e}\)¶

A type bound \(\href{../valid/types.html#syntax-etypebound}{\mathsf{eq}}~\href{../valid/types.html#syntax-edeftype}{\mathit{deftype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-etypebound}{\mathsf{eq}}~\href{../valid/types.html#syntax-edeftype}{\mathit{deftype}_e}'\) if \(\href{../valid/types.html#syntax-edeftype}{\mathit{deftype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-edeftype}{\mathit{deftype}_e}'\).

\[\frac{ \href{../valid/types.html#syntax-edeftype}{\mathit{deftype}_e} \preccurlyeq \href{../valid/types.html#syntax-edeftype}{\mathit{deftype}_e}' }{ \href{../valid/types.html#syntax-etypebound}{\mathsf{eq}}~\href{../valid/types.html#syntax-edeftype}{\mathit{deftype}_e} \preccurlyeq \href{../valid/types.html#syntax-etypebound}{\mathsf{eq}}~\href{../valid/types.html#syntax-edeftype}{\mathit{deftype}_e}' }\]

Extern descriptors¶

\(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{core\_module}}~\href{../valid/types.html#syntax-ecoremoduletype}{\mathit{core{:}moduletype}_e}\)¶

A extern descriptor \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{core\_module}}~\href{../valid/types.html#syntax-ecoremoduletype}{\mathit{core{:}moduletype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{core\_module}}~\href{../valid/types.html#syntax-ecoremoduletype}{\mathit{core{:}moduletype}_e}'\) if \(\href{../valid/types.html#syntax-ecoremoduletype}{\mathit{core{:}moduletype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-ecoremoduletype}{\mathit{core{:}moduletype}_e}'\).

\[\frac{ \href{../valid/types.html#syntax-ecoremoduletype}{\mathit{core{:}moduletype}_e}' \preccurlyeq \href{../syntax/types.html#syntax-coremoduletype}{\mathit{core{:}moduletype}}' }{ \href{../valid/types.html#syntax-eexterndesc}{\mathsf{core\_module}}~\href{../valid/types.html#syntax-ecoremoduletype}{\mathit{core{:}moduletype}_e} \preccurlyeq \href{../valid/types.html#syntax-eexterndesc}{\mathsf{core\_module}}~\href{../valid/types.html#syntax-ecoremoduletype}{\mathit{core{:}moduletype}_e}' }\]

\(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{func}}~\href{../valid/types.html#syntax-efunctype}{\mathit{functype}_e}\)¶

An extern descriptor \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{func}}~\href{../valid/types.html#syntax-efunctype}{\mathit{functype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{func}}~\href{../valid/types.html#syntax-efunctype}{\mathit{functype}_e}'\) if \(\href{../valid/types.html#syntax-efunctype}{\mathit{functype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-efunctype}{\mathit{functype}_e}'\).

\[\frac{ \href{../valid/types.html#syntax-efunctype}{\mathit{functype}_e} \preccurlyeq \href{../valid/types.html#syntax-efunctype}{\mathit{functype}_e}' }{ \href{../valid/types.html#syntax-eexterndesc}{\mathsf{func}}~\href{../valid/types.html#syntax-efunctype}{\mathit{functype}_e} \preccurlyeq \href{../valid/types.html#syntax-eexterndesc}{\mathsf{func}}~\href{../valid/types.html#syntax-efunctype}{\mathit{functype}_e}' }\]

\(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{value}}~\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}\)¶

An extern descriptor \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{value}}~\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{value}}~\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}'\) if \(\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}'\).

\[\frac{ \href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e} \preccurlyeq \href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}' }{ \href{../valid/types.html#syntax-eexterndesc}{\mathsf{value}}~\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e} \preccurlyeq \href{../valid/types.html#syntax-eexterndesc}{\mathsf{value}}~\href{../valid/types.html#syntax-evaltype}{\mathit{valtype}_e}' }\]

\(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{type}}~\href{../valid/types.html#syntax-etypebound}{\mathit{typebound}_e}\)¶

An extern descriptor \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{type}}~\href{../valid/types.html#syntax-etypebound}{\mathit{typebound}_e}\) is a subtype of \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{type}}~\href{../valid/types.html#syntax-etypebound}{\mathit{typebound}_e}'\) if \(\href{../valid/types.html#syntax-etypebound}{\mathit{typebound}_e}\) is a subtype of \(\href{../valid/types.html#syntax-etypebound}{\mathit{typebound}_e}'\).

\[\frac{ \href{../valid/types.html#syntax-etypebound}{\mathit{typebound}_e} \preccurlyeq \href{../valid/types.html#syntax-etypebound}{\mathit{typebound}_e}' }{ \href{../valid/types.html#syntax-eexterndesc}{\mathsf{type}}~\href{../valid/types.html#syntax-etypebound}{\mathit{typebound}_e} \preccurlyeq \href{../valid/types.html#syntax-eexterndesc}{\mathsf{type}}~\href{../valid/types.html#syntax-etypebound}{\mathit{typebound}_e}' }\]

\(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{instance}}~\href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}\)¶

An extern descriptor \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{instance}}~\href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{instance}}~\href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}'\) if \(\href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}'\).

\[\frac{ \href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e} \preccurlyeq \href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}' }{ \href{../valid/types.html#syntax-eexterndesc}{\mathsf{instance}}~\href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e} \preccurlyeq \href{../valid/types.html#syntax-eexterndesc}{\mathsf{instance}}~\href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}' }\]

\(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{component}}~\href{../valid/types.html#syntax-ecomponenttype}{\mathit{componenttype}_e}\)¶

An extern descriptor \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{component}}~\href{../valid/types.html#syntax-ecomponenttype}{\mathit{componenttype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-eexterndesc}{\mathsf{component}}~\href{../valid/types.html#syntax-ecomponenttype}{\mathit{componenttype}_e}'\) if \(\href{../valid/types.html#syntax-ecomponenttype}{\mathit{componenttype}_e}\) is a subtype of \(\href{../valid/types.html#syntax-ecomponenttype}{\mathit{componenttype}_e}'\).

\[\frac{ \href{../valid/types.html#syntax-ecomponenttype}{\mathit{componenttype}_e} \preccurlyeq \href{../valid/types.html#syntax-ecomponenttype}{\mathit{componenttype}_e}' }{ \href{../valid/types.html#syntax-eexterndesc}{\mathsf{component}}~\href{../valid/types.html#syntax-ecomponenttype}{\mathit{componenttype}_e} \preccurlyeq \href{../valid/types.html#syntax-eexterndesc}{\mathsf{component}}~\href{../valid/types.html#syntax-ecomponenttype}{\mathit{componenttype}_e}' }\]

Instance types¶

An instance type \(\overline{{\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i}\) is a subtype of an instance type \(\overline{{\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j}\) if:
- For each \(j\), there exists some \(i\) such that \({\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{name}} = {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{name}}\) and \({\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{desc}} \preccurlyeq {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{desc}}\).

\[\frac{ \forall j, \exists i, {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{name}} = {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{name}} \land {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{desc}} \preccurlyeq {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{desc}}. }{ \overline{{\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i} \preccurlyeq \overline{{\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j} }\]

Component types¶

A component type \(\overline{{\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i} \to \href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}\) is a subtype of a \(\overline{{\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j} \to \href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}'\) if:
- For each \(i\), there exists some \(j\), such that \({\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{name}} = {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{name}}\) and \({\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{desc}} \preccurlyeq {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{desc}}\); and
- \(\href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e} \preccurlyeq \href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}'\)

\[\begin{split}\frac{ \begin{array}{@{}c@{}} \forall i, \exists j, {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{name}} = {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{name}} \land {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{desc}} \preccurlyeq {\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i.\href{../valid/types.html#syntax-eexportdecl}{\mathsf{desc}}\\ \href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e} \preccurlyeq \href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}' \end{array} }{ \overline{{\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}}_i} \to \href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e} \preccurlyeq \overline{{\href{../valid/types.html#syntax-eexportdecl}{\mathit{externdecl}_e}'}_j} \to \href{../valid/types.html#syntax-einstancetype}{\mathit{instancetype}_e}' }\end{split}\]

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