Modules

Modules are valid when all the components they contain are valid. To verify this, most definitions are themselves classified with a suitable type.

Types

The sequence of types defined in a module is validated incrementally, yielding a sequence of defined types representing them individually.

The type definition \((\href{../syntax/modules.html#syntax-type}{\mathsf{type}}~{\href{../syntax/types.html#syntax-rectype}{\mathit{rectype}}})\) is valid with the defined type sequence \({{\mathit{dt}}^\ast}\) if:

  • The length of \(C{.}\href{../valid/conventions.html#context}{\mathsf{types}}\) is equal to \(x\).

  • The defined type sequence \({{\mathit{dt}}^\ast}\) is of the form \({{{{\href{../valid/conventions.html#aux-roll-deftype}{\mathrm{roll}}}}_{x}^\ast}}{({\href{../syntax/types.html#syntax-rectype}{\mathit{rectype}}})}\).

  • Let \({C'}\) be the same context as \(C\), but with the defined type sequence \({{\mathit{dt}}^\ast}\) appended to the field \(\href{../valid/conventions.html#context}{\mathsf{types}}\).

  • Under the context \({C'}\), the recursive type \({\href{../syntax/types.html#syntax-rectype}{\mathit{rectype}}}\) is valid for \(({\href{../valid/types.html#valid-subtype}{\mathsf{ok}}}{(x)})\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ x = {|C{.}\href{../valid/conventions.html#context}{\mathsf{types}}|} \qquad {{\mathit{dt}}^\ast} = {{{{\href{../valid/conventions.html#aux-roll-deftype}{\mathrm{roll}}}}_{x}^\ast}}{({\href{../syntax/types.html#syntax-rectype}{\mathit{rectype}}})} \qquad C \oplus \{ \begin{array}[t]{@{}l@{}} \href{../valid/conventions.html#context}{\mathsf{types}}~{{\mathit{dt}}^\ast} \}\end{array} \href{../valid/types.html#valid-rectype}{\vdash} {\href{../syntax/types.html#syntax-rectype}{\mathit{rectype}}} : {\href{../valid/types.html#valid-subtype}{\mathsf{ok}}}{((x))} }{ C \href{../valid/modules.html#valid-type}{\vdash} \href{../syntax/modules.html#syntax-type}{\mathsf{type}}~{\href{../syntax/types.html#syntax-rectype}{\mathit{rectype}}} : {{\mathit{dt}}^\ast} } \qquad \end{array}\]

The type definition sequence \({{\href{../syntax/types.html#syntax-rectype}{\mathit{type}}'}^\ast}\) is valid with the defined type sequence \({{\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}^\ast}\) if:

  • Either:

    • The type definition sequence \({{\href{../syntax/types.html#syntax-rectype}{\mathit{type}}'}^\ast}\) is empty.

    • The defined type sequence \({{\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}^\ast}\) is empty.

  • Or:

    • The type definition sequence \({{\href{../syntax/types.html#syntax-rectype}{\mathit{type}}'}^\ast}\) is of the form \({\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}_1~{{\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}^\ast}\).

    • The defined type sequence \({{\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}^\ast}\) is of the form \({{\mathit{dt}}_1^\ast}~{{\mathit{dt}}^\ast}\).

    • The type definition \({\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}_1\) is valid with the defined type sequence \({{\mathit{dt}}_1^\ast}\).

    • Let \({C'}\) be the same context as \(C\), but with the defined type sequence \({{\mathit{dt}}_1^\ast}\) appended to the field \(\href{../valid/conventions.html#context}{\mathsf{types}}\).

    • Under the context \({C'}\), the type definition sequence \({{\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}^\ast}\) is valid with the defined type sequence \({{\mathit{dt}}^\ast}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ }{ C \href{../valid/modules.html#valid-types}{\vdash} \epsilon : \epsilon } \qquad \frac{ C \href{../valid/modules.html#valid-type}{\vdash} {\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}_1 : {{\mathit{dt}}_1^\ast} \qquad C \oplus \{ \begin{array}[t]{@{}l@{}} \href{../valid/conventions.html#context}{\mathsf{types}}~{{\mathit{dt}}_1^\ast} \}\end{array} \href{../valid/modules.html#valid-types}{\vdash} {{\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}^\ast} : {{\mathit{dt}}^\ast} }{ C \href{../valid/modules.html#valid-types}{\vdash} {\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}_1~{{\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}^\ast} : {{\mathit{dt}}_1^\ast}~{{\mathit{dt}}^\ast} } \qquad \end{array}\]

Functions

Functions \({\href{../syntax/modules.html#syntax-func}{\mathit{func}}}\) are classified by defined types that expand to function types of the form \(\href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~({t_1^\ast} \href{../syntax/types.html#syntax-functype}{\rightarrow} {t_2^\ast})\).

The function \((\href{../syntax/modules.html#syntax-func}{\mathsf{func}}~x~{{\href{../syntax/modules.html#syntax-local}{\mathit{local}}}^\ast}~{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}})\) is valid with the defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[x]\) if:

  • The defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[x]\) exists.

  • The expansion of the defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[x]\) is the composite type \((\href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~{t_1^\ast}~\href{../syntax/types.html#syntax-functype}{\rightarrow}~{t_2^\ast})\).

  • The length of \({{{\mathit{lt}}}^\ast}\) is equal to the length of \({{\href{../syntax/modules.html#syntax-local}{\mathit{local}}}^\ast}\).

  • For all \({{\mathit{lt}}}\) in \({{{\mathit{lt}}}^\ast}\), and corresponding \({\href{../syntax/modules.html#syntax-local}{\mathit{local}}}\) in \({{\href{../syntax/modules.html#syntax-local}{\mathit{local}}}^\ast}\):

    • The local \({\href{../syntax/modules.html#syntax-local}{\mathit{local}}}\) is valid with the local type \({{\mathit{lt}}}\).

  • Under the context \(C{}[{.}\href{../valid/conventions.html#context}{\mathsf{locals}} \mathrel{{=}{\oplus}} {(\href{../valid/conventions.html#syntax-init}{\mathsf{set}}~t_1)^\ast}~{{{\mathit{lt}}}^\ast}]{}[{.}\href{../valid/conventions.html#context}{\mathsf{labels}} \mathrel{{=}{\oplus}} {t_2^\ast}]{}[{.}\href{../valid/conventions.html#context}{\mathsf{return}} \mathrel{{=}{\oplus}} {t_2^\ast}]\), the expression \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) is valid with the result type \({t_2^\ast}\).

Todo

below is the official specification

  • The defined type \(C.\href{../valid/conventions.html#context}{\mathsf{types}}[x]\) must be a function type.

  • Let \(\href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~[t_1^\ast] \href{../syntax/types.html#syntax-functype}{\rightarrow} [t_2^\ast]\) be the expansion of the defined type \(C.\href{../valid/conventions.html#context}{\mathsf{types}}[x]\).

  • For each local declared by a value type \(t\) in \(t^\ast\):

    • The local for type \(t\) must be valid with local type \(\href{../valid/conventions.html#syntax-localtype}{\mathit{localtype}}_i\).

  • Let \(\href{../valid/conventions.html#syntax-localtype}{\mathit{localtype}}^\ast\) be the concatenation of all \(\href{../valid/conventions.html#syntax-localtype}{\mathit{localtype}}_i\).

  • Let \(C'\) be the same context as \(C\), but with:

    • \(\href{../valid/conventions.html#context}{\mathsf{locals}}\) set to the sequence of value types \((\href{../valid/conventions.html#syntax-init}{\mathsf{set}}~t_1)^\ast~\href{../valid/conventions.html#syntax-localtype}{\mathit{localtype}}^\ast\), concatenating parameters and locals,

    • \(\href{../valid/conventions.html#context}{\mathsf{labels}}\) set to the singular sequence containing only result type \([t_2^\ast]\).

    • \(\href{../valid/conventions.html#context}{\mathsf{return}}\) set to the result type \([t_2^\ast]\).

  • Under the context \(C'\), the expression \(\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}\) must be valid with type \([t_2^\ast]\).

  • Then the function definition is valid with type \(C.\href{../valid/conventions.html#context}{\mathsf{types}}[x]\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[x] \href{../valid/conventions.html#aux-expand-deftype}{\approx} \href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~({t_1^\ast} \href{../syntax/types.html#syntax-functype}{\rightarrow} {t_2^\ast}) \qquad (C \href{../valid/modules.html#valid-local}{\vdash} {\href{../syntax/modules.html#syntax-local}{\mathit{local}}} : {{\mathit{lt}}})^\ast \qquad C \oplus \{ \begin{array}[t]{@{}l@{}} \href{../valid/conventions.html#context}{\mathsf{locals}}~{(\href{../valid/conventions.html#syntax-init}{\mathsf{set}}~t_1)^\ast}~{{{\mathit{lt}}}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{labels}}~({t_2^\ast}),\; \href{../valid/conventions.html#context}{\mathsf{return}}~({t_2^\ast}) \}\end{array} \href{../valid/instructions.html#valid-expr}{\vdash} {\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}} : {t_2^\ast} }{ C \href{../valid/modules.html#valid-func}{\vdash} \href{../syntax/modules.html#syntax-func}{\mathsf{func}}~x~{{\href{../syntax/modules.html#syntax-local}{\mathit{local}}}^\ast}~{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}} : C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[x] } \qquad \end{array}\]

Locals

Locals \({\href{../syntax/modules.html#syntax-local}{\mathit{local}}}\) are classified with local types.

The local \((\href{../syntax/modules.html#syntax-local}{\mathsf{local}}~t)\) is valid with the local type \(({\href{../valid/conventions.html#syntax-init}{\mathit{init}}}~t)\) if:

  • Either:

    • The initialization status \({\href{../valid/conventions.html#syntax-init}{\mathit{init}}}\) is of the form \(\href{../valid/conventions.html#syntax-init}{\mathsf{set}}\).

    • A default value for the value type \(t\) is defined.

  • Or:

    • The initialization status \({\href{../valid/conventions.html#syntax-init}{\mathit{init}}}\) is of the form \(\href{../valid/conventions.html#syntax-init}{\mathsf{unset}}\).

    • A default value for the value type \(t\) is not defined.

\[\begin{array}{@{}c@{}}\displaystyle \frac{ {{\href{../exec/runtime.html#aux-default}{\mathrm{default}}}}_{t} \neq \epsilon }{ C \href{../valid/modules.html#valid-local}{\vdash} \href{../syntax/modules.html#syntax-local}{\mathsf{local}}~t : \href{../valid/conventions.html#syntax-init}{\mathsf{set}}~t } \qquad \frac{ {{\href{../exec/runtime.html#aux-default}{\mathrm{default}}}}_{t} = \epsilon }{ C \href{../valid/modules.html#valid-local}{\vdash} \href{../syntax/modules.html#syntax-local}{\mathsf{local}}~t : \href{../valid/conventions.html#syntax-init}{\mathsf{unset}}~t } \qquad \end{array}\]

Note

For cases where both rules are applicable, the former yields the more permissable type.

Tables

Tables \({\href{../syntax/modules.html#syntax-table}{\mathit{table}}}\) are classified by table types.

The table \((\href{../syntax/modules.html#syntax-table}{\mathsf{table}}~{\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}~{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}})\) is valid with the table type \({\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}\) if:

  • The table type \({\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}\) is valid.

  • The table type \({\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}\) is of the form \(({\mathit{at}}~{\mathit{lim}}~{\mathit{rt}})\).

  • The expression \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) is valid with the value type \({\mathit{rt}}\).

  • \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) is constant.

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/types.html#valid-tabletype}{\vdash} {\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}} : \href{../valid/types.html#valid-tabletype}{\mathsf{ok}} \qquad {\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}} = {\mathit{at}}~{\mathit{lim}}~{\mathit{rt}} \qquad C \href{../valid/instructions.html#valid-const}{\vdash} {\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}} : {\mathit{rt}}~\href{../valid/instructions.html#valid-const}{\mathsf{const}} }{ C \href{../valid/modules.html#valid-table}{\vdash} \href{../syntax/modules.html#syntax-table}{\mathsf{table}}~{\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}~{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}} : {\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}} } \qquad \end{array}\]

Memories

Memories \({\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}}\) are classified by memory types.

The memory \((\href{../syntax/modules.html#syntax-mem}{\mathsf{memory}}~{\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}})\) is valid with the memory type \({\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}\) if:

  • The memory type \({\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}\) is valid.

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/types.html#valid-memtype}{\vdash} {\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}} : \href{../valid/types.html#valid-memtype}{\mathsf{ok}} }{ C \href{../valid/modules.html#valid-mem}{\vdash} \href{../syntax/modules.html#syntax-mem}{\mathsf{memory}}~{\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}} : {\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}} } \qquad \end{array}\]

Globals

Globals \({\href{../syntax/modules.html#syntax-global}{\mathit{global}}}\) are classified by global types.

The global \((\href{../syntax/modules.html#syntax-global}{\mathsf{global}}~{\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}~{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}})\) is valid with the global type \({\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}\) if:

  • The global type \({\mathit{gt}}\) is valid.

  • The global type \({\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}\) is of the form \(({\href{../syntax/types.html#syntax-mut}{\mathsf{mut}}^?}~t)\).

  • The expression \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) is valid with the value type \(t\).

  • \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) is constant.

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/types.html#valid-globaltype}{\vdash} {\mathit{gt}} : \href{../valid/types.html#valid-globaltype}{\mathsf{ok}} \qquad {\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}} = {\href{../syntax/types.html#syntax-mut}{\mathsf{mut}}^?}~t \qquad C \href{../valid/instructions.html#valid-const}{\vdash} {\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}} : t~\href{../valid/instructions.html#valid-const}{\mathsf{const}} }{ C \href{../valid/modules.html#valid-global}{\vdash} \href{../syntax/modules.html#syntax-global}{\mathsf{global}}~{\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}~{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}} : {\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}} } \qquad \end{array}\]

Sequences of globals are handled incrementally, such that each definition has access to previous definitions.

The global sequence \({{\href{../syntax/modules.html#syntax-global}{\mathit{global}}'}^\ast}\) is valid with the global type sequence \({{\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}^\ast}\) if:

  • Either:

    • The global sequence \({{\href{../syntax/modules.html#syntax-global}{\mathit{global}}'}^\ast}\) is empty.

    • The global type sequence \({{\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}^\ast}\) is empty.

  • Or:

    • The global sequence \({{\href{../syntax/modules.html#syntax-global}{\mathit{global}}'}^\ast}\) is of the form \({\href{../syntax/modules.html#syntax-global}{\mathit{global}}}_1~{{\href{../syntax/modules.html#syntax-global}{\mathit{global}}}^\ast}\).

    • The global type sequence \({{\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}^\ast}\) is of the form \({\mathit{gt}}_1~{{\mathit{gt}}^\ast}\).

    • The global \({\href{../syntax/modules.html#syntax-global}{\mathit{global}}}_1\) is valid with the global type \({\mathit{gt}}_1\).

    • Let \({C'}\) be the same context as \(C\), but with the global type sequence \({\mathit{gt}}_1\) appended to the field \(\href{../valid/conventions.html#context}{\mathsf{globals}}\).

    • Under the context \({C'}\), the global sequence \({{\href{../syntax/modules.html#syntax-global}{\mathit{global}}}^\ast}\) is valid with the global type sequence \({{\mathit{gt}}^\ast}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ }{ C \href{../valid/modules.html#valid-globalseq}{\vdash} \epsilon : \epsilon } \qquad \frac{ C \href{../valid/modules.html#valid-global}{\vdash} {\href{../syntax/modules.html#syntax-global}{\mathit{global}}}_1 : {\mathit{gt}}_1 \qquad C \oplus \{ \begin{array}[t]{@{}l@{}} \href{../valid/conventions.html#context}{\mathsf{globals}}~{\mathit{gt}}_1 \}\end{array} \href{../valid/modules.html#valid-globalseq}{\vdash} {{\href{../syntax/modules.html#syntax-global}{\mathit{global}}}^\ast} : {{\mathit{gt}}^\ast} }{ C \href{../valid/modules.html#valid-globalseq}{\vdash} {\href{../syntax/modules.html#syntax-global}{\mathit{global}}}_1~{{\href{../syntax/modules.html#syntax-global}{\mathit{global}}}^\ast} : {\mathit{gt}}_1~{{\mathit{gt}}^\ast} } \qquad \end{array}\]

Tags

Tags \(\href{../syntax/modules.html#syntax-tag}{\mathit{tag}}\) are classified by their tag types, which are defined types expanding to function types.

The tag \((\href{../syntax/modules.html#syntax-tag}{\mathsf{tag}}~x)\) is valid with the tag type \(C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[x]\) if:

  • The defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[x]\) exists.

  • The expansion of the defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[x]\) is the composite type \((\href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~{\href{../syntax/types.html#syntax-functype}{\mathit{functype}}})\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[x] \href{../valid/conventions.html#aux-expand-deftype}{\approx} \href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~{\href{../syntax/types.html#syntax-functype}{\mathit{functype}}} }{ C \href{../valid/modules.html#valid-tag}{\vdash} \href{../syntax/modules.html#syntax-tag}{\mathsf{tag}}~x : C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[x] } \qquad \end{array}\]

Element Segments

Element segments \({\href{../syntax/modules.html#syntax-elem}{\mathit{elem}}}\) are classified by their element type.

The table segment \((\href{../syntax/modules.html#syntax-elem}{\mathsf{elem}}~{\href{../syntax/types.html#syntax-elemtype}{\mathit{elemtype}}}~{{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}^\ast}~{\href{../syntax/modules.html#syntax-elemmode}{\mathit{elemmode}}})\) is valid with the element type \({\href{../syntax/types.html#syntax-elemtype}{\mathit{elemtype}}}\) if:

  • The reference type \({\href{../syntax/types.html#syntax-elemtype}{\mathit{elemtype}}}\) is valid.

  • For all \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) in \({{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}^\ast}\):

    • The expression \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) is valid with the value type \({\href{../syntax/types.html#syntax-elemtype}{\mathit{elemtype}}}\).

    • \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) is constant.

  • The element mode \({\href{../syntax/modules.html#syntax-elemmode}{\mathit{elemmode}}}\) is valid with the element type \({\href{../syntax/types.html#syntax-elemtype}{\mathit{elemtype}}}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/types.html#valid-reftype}{\vdash} {\href{../syntax/types.html#syntax-elemtype}{\mathit{elemtype}}} : \href{../valid/types.html#valid-reftype}{\mathsf{ok}} \qquad (C \href{../valid/instructions.html#valid-const}{\vdash} {\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}} : {\href{../syntax/types.html#syntax-elemtype}{\mathit{elemtype}}}~\href{../valid/instructions.html#valid-const}{\mathsf{const}})^\ast \qquad C \href{../valid/modules.html#valid-elemmode}{\vdash} {\href{../syntax/modules.html#syntax-elemmode}{\mathit{elemmode}}} : {\href{../syntax/types.html#syntax-elemtype}{\mathit{elemtype}}} }{ C \href{../valid/modules.html#valid-elem}{\vdash} \href{../syntax/modules.html#syntax-elem}{\mathsf{elem}}~{\href{../syntax/types.html#syntax-elemtype}{\mathit{elemtype}}}~{{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}^\ast}~{\href{../syntax/modules.html#syntax-elemmode}{\mathit{elemmode}}} : {\href{../syntax/types.html#syntax-elemtype}{\mathit{elemtype}}} } \qquad \end{array}\]

The element mode \({\href{../syntax/modules.html#syntax-elemmode}{\mathit{elemmode}}}\) is valid with the element type \({\mathit{rt}}\) if:

  • Either:

    • The element mode \({\href{../syntax/modules.html#syntax-elemmode}{\mathit{elemmode}}}\) is of the form \((\href{../syntax/modules.html#syntax-elemmode}{\mathsf{active}}~x~{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}})\).

    • The table type \(C{.}\href{../valid/conventions.html#context}{\mathsf{tables}}{}[x]\) exists.

    • The table type \(C{.}\href{../valid/conventions.html#context}{\mathsf{tables}}{}[x]\) is of the form \(({\mathit{at}}~{\mathit{lim}}~{\mathit{rt}'})\).

    • The reference type \({\mathit{rt}}\) matches the reference type \({\mathit{rt}'}\).

    • The expression \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) is valid with the value type \(\href{../syntax/types.html#syntax-numtype}{\mathsf{i\scriptstyle32}}\).

    • \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) is constant.

  • Or:

    • The element mode \({\href{../syntax/modules.html#syntax-elemmode}{\mathit{elemmode}}}\) is of the form \(\href{../syntax/modules.html#syntax-elemmode}{\mathsf{passive}}\).

  • Or:

    • The element mode \({\href{../syntax/modules.html#syntax-elemmode}{\mathit{elemmode}}}\) is of the form \(\href{../syntax/modules.html#syntax-elemmode}{\mathsf{declare}}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C{.}\href{../valid/conventions.html#context}{\mathsf{tables}}{}[x] = {\mathit{at}}~{\mathit{lim}}~{\mathit{rt}'} \qquad C \href{../valid/matching.html#match-reftype}{\vdash} {\mathit{rt}} \href{../valid/matching.html#match-reftype}{\leq} {\mathit{rt}'} \qquad C \href{../valid/instructions.html#valid-const}{\vdash} {\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}} : \href{../syntax/types.html#syntax-numtype}{\mathsf{i\scriptstyle32}}~\href{../valid/instructions.html#valid-const}{\mathsf{const}} }{ C \href{../valid/modules.html#valid-elemmode}{\vdash} \href{../syntax/modules.html#syntax-elemmode}{\mathsf{active}}~x~{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}} : {\mathit{rt}} } \qquad \frac{ }{ C \href{../valid/modules.html#valid-elemmode}{\vdash} \href{../syntax/modules.html#syntax-elemmode}{\mathsf{passive}} : {\mathit{rt}} } \qquad \frac{ }{ C \href{../valid/modules.html#valid-elemmode}{\vdash} \href{../syntax/modules.html#syntax-elemmode}{\mathsf{declare}} : {\mathit{rt}} } \qquad \end{array}\]

Data Segments

Data segments \({\href{../syntax/modules.html#syntax-data}{\mathit{data}}}\) are classified by the singleton data type, which merely expresses well-formedness.

The memory segment \((\href{../syntax/modules.html#syntax-data}{\mathsf{data}}~{b^\ast}~{\href{../syntax/modules.html#syntax-datamode}{\mathit{datamode}}})\) is valid with the data type \(\href{../valid/modules.html#valid-data}{\mathsf{ok}}\) if:

  • The data mode \({\href{../syntax/modules.html#syntax-datamode}{\mathit{datamode}}}\) is valid with the data type \(\href{../valid/modules.html#valid-data}{\mathsf{ok}}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/modules.html#valid-datamode}{\vdash} {\href{../syntax/modules.html#syntax-datamode}{\mathit{datamode}}} : \href{../valid/modules.html#valid-data}{\mathsf{ok}} }{ C \href{../valid/modules.html#valid-data}{\vdash} \href{../syntax/modules.html#syntax-data}{\mathsf{data}}~{b^\ast}~{\href{../syntax/modules.html#syntax-datamode}{\mathit{datamode}}} : \href{../valid/modules.html#valid-data}{\mathsf{ok}} } \qquad \end{array}\]

The data mode \({\href{../syntax/modules.html#syntax-datamode}{\mathit{datamode}}}\) is valid with the data type \(\href{../valid/modules.html#valid-data}{\mathsf{ok}}\) if:

  • Either:

    • The data mode \({\href{../syntax/modules.html#syntax-datamode}{\mathit{datamode}}}\) is of the form \((\href{../syntax/modules.html#syntax-datamode}{\mathsf{active}}~x~{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}})\).

    • The memory type \(C{.}\href{../valid/conventions.html#context}{\mathsf{mems}}{}[x]\) exists.

    • The memory type \(C{.}\href{../valid/conventions.html#context}{\mathsf{mems}}{}[x]\) is of the form \({\mathit{mt}}\).

    • The expression \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) is valid with the value type \(\href{../syntax/types.html#syntax-numtype}{\mathsf{i\scriptstyle32}}\).

    • \({\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}}\) is constant.

  • Or:

    • The data mode \({\href{../syntax/modules.html#syntax-datamode}{\mathit{datamode}}}\) is of the form \(\href{../syntax/modules.html#syntax-datamode}{\mathsf{passive}}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C{.}\href{../valid/conventions.html#context}{\mathsf{mems}}{}[x] = {\mathit{mt}} \qquad C \href{../valid/instructions.html#valid-const}{\vdash} {\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}} : \href{../syntax/types.html#syntax-numtype}{\mathsf{i\scriptstyle32}}~\href{../valid/instructions.html#valid-const}{\mathsf{const}} }{ C \href{../valid/modules.html#valid-datamode}{\vdash} \href{../syntax/modules.html#syntax-datamode}{\mathsf{active}}~x~{\href{../syntax/instructions.html#syntax-expr}{\mathit{expr}}} : \href{../valid/modules.html#valid-data}{\mathsf{ok}} } \qquad \frac{ }{ C \href{../valid/modules.html#valid-datamode}{\vdash} \href{../syntax/modules.html#syntax-datamode}{\mathsf{passive}} : \href{../valid/modules.html#valid-data}{\mathsf{ok}} } \qquad \end{array}\]

Start Function

The start function \((\href{../syntax/modules.html#syntax-start}{\mathsf{start}}~x)\) is valid if:

  • The defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{funcs}}{}[x]\) exists.

  • The expansion of the defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{funcs}}{}[x]\) is the composite type \((\href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~\epsilon~\href{../syntax/types.html#syntax-functype}{\rightarrow}~\epsilon)\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C{.}\href{../valid/conventions.html#context}{\mathsf{funcs}}{}[x] \href{../valid/conventions.html#aux-expand-deftype}{\approx} \href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~(\epsilon \href{../syntax/types.html#syntax-functype}{\rightarrow} \epsilon) }{ C \href{../valid/modules.html#valid-start}{\vdash} \href{../syntax/modules.html#syntax-start}{\mathsf{start}}~x : \href{../valid/modules.html#valid-start}{\mathsf{ok}} } \qquad \end{array}\]

Exports

Exports \({\href{../syntax/modules.html#syntax-export}{\mathit{export}}}\) are classified by their external type.

The export \((\href{../syntax/modules.html#syntax-export}{\mathsf{export}}~{\href{../syntax/values.html#syntax-name}{\mathit{name}}}~{\href{../syntax/modules.html#syntax-externidx}{\mathit{externidx}}})\) is valid with the name \({\href{../syntax/values.html#syntax-name}{\mathit{name}}}\) and the external type \({\mathit{xt}}\) if:

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/modules.html#valid-externidx}{\vdash} {\href{../syntax/modules.html#syntax-externidx}{\mathit{externidx}}} : {\mathit{xt}} }{ C \href{../valid/modules.html#valid-export}{\vdash} \href{../syntax/modules.html#syntax-export}{\mathsf{export}}~{\href{../syntax/values.html#syntax-name}{\mathit{name}}}~{\href{../syntax/modules.html#syntax-externidx}{\mathit{externidx}}} : {\href{../syntax/values.html#syntax-name}{\mathit{name}}}~{\mathit{xt}} } \qquad \end{array}\]

\(\href{../syntax/modules.html#syntax-exportdesc}{\mathsf{func}}~x\)

The external index \((\href{../syntax/modules.html#syntax-externidx}{\mathsf{func}}~x)\) is valid with the external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~{\mathit{dt}})\) if:

  • The defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{funcs}}{}[x]\) exists.

  • The defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{funcs}}{}[x]\) is of the form \({\mathit{dt}}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C{.}\href{../valid/conventions.html#context}{\mathsf{funcs}}{}[x] = {\mathit{dt}} }{ C \href{../valid/modules.html#valid-externidx}{\vdash} \href{../syntax/modules.html#syntax-externidx}{\mathsf{func}}~x : \href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~{\mathit{dt}} } \qquad \end{array}\]

\(\href{../syntax/modules.html#syntax-exportdesc}{\mathsf{table}}~x\)

The external index \((\href{../syntax/modules.html#syntax-externidx}{\mathsf{table}}~x)\) is valid with the external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~{\mathit{tt}})\) if:

  • The table type \(C{.}\href{../valid/conventions.html#context}{\mathsf{tables}}{}[x]\) exists.

  • The table type \(C{.}\href{../valid/conventions.html#context}{\mathsf{tables}}{}[x]\) is of the form \({\mathit{tt}}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C{.}\href{../valid/conventions.html#context}{\mathsf{tables}}{}[x] = {\mathit{tt}} }{ C \href{../valid/modules.html#valid-externidx}{\vdash} \href{../syntax/modules.html#syntax-externidx}{\mathsf{table}}~x : \href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~{\mathit{tt}} } \qquad \end{array}\]

\(\href{../syntax/modules.html#syntax-exportdesc}{\mathsf{mem}}~x\)

The external index \((\href{../syntax/modules.html#syntax-externidx}{\mathsf{memory}}~x)\) is valid with the external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~{\mathit{mt}})\) if:

  • The memory type \(C{.}\href{../valid/conventions.html#context}{\mathsf{mems}}{}[x]\) exists.

  • The memory type \(C{.}\href{../valid/conventions.html#context}{\mathsf{mems}}{}[x]\) is of the form \({\mathit{mt}}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C{.}\href{../valid/conventions.html#context}{\mathsf{mems}}{}[x] = {\mathit{mt}} }{ C \href{../valid/modules.html#valid-externidx}{\vdash} \href{../syntax/modules.html#syntax-externidx}{\mathsf{memory}}~x : \href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~{\mathit{mt}} } \qquad \end{array}\]

\(\href{../syntax/modules.html#syntax-exportdesc}{\mathsf{global}}~x\)

The external index \((\href{../syntax/modules.html#syntax-externidx}{\mathsf{global}}~x)\) is valid with the external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~{\mathit{gt}})\) if:

  • The global type \(C{.}\href{../valid/conventions.html#context}{\mathsf{globals}}{}[x]\) exists.

  • The global type \(C{.}\href{../valid/conventions.html#context}{\mathsf{globals}}{}[x]\) is of the form \({\mathit{gt}}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C{.}\href{../valid/conventions.html#context}{\mathsf{globals}}{}[x] = {\mathit{gt}} }{ C \href{../valid/modules.html#valid-externidx}{\vdash} \href{../syntax/modules.html#syntax-externidx}{\mathsf{global}}~x : \href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~{\mathit{gt}} } \qquad \end{array}\]

\(\href{../syntax/modules.html#syntax-exportdesc}{\mathsf{tag}}~x\)

The external index \((\href{../syntax/modules.html#syntax-externidx}{\mathsf{tag}}~x)\) is valid with the external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{tag}}~{\mathit{jt}})\) if:

  • The tag type \(C{.}\href{../valid/conventions.html#context}{\mathsf{tags}}{}[x]\) exists.

  • The tag type \(C{.}\href{../valid/conventions.html#context}{\mathsf{tags}}{}[x]\) is of the form \({\mathit{jt}}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C{.}\href{../valid/conventions.html#context}{\mathsf{tags}}{}[x] = {\mathit{jt}} }{ C \href{../valid/modules.html#valid-externidx}{\vdash} \href{../syntax/modules.html#syntax-externidx}{\mathsf{tag}}~x : \href{../syntax/types.html#syntax-externtype}{\mathsf{tag}}~{\mathit{jt}} } \qquad \end{array}\]

Imports

Imports \({\href{../syntax/modules.html#syntax-import}{\mathit{import}}}\) are classified by external types.

The import \((\href{../syntax/modules.html#syntax-import}{\mathsf{import}}~{\href{../syntax/values.html#syntax-name}{\mathit{name}}}_1~{\href{../syntax/values.html#syntax-name}{\mathit{name}}}_2~{\mathit{xt}})\) is valid with the external type \({\mathit{xt}}\) if:

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/types.html#valid-externtype}{\vdash} {\mathit{xt}} : \href{../valid/types.html#valid-externtype}{\mathsf{ok}} }{ C \href{../valid/modules.html#valid-import}{\vdash} \href{../syntax/modules.html#syntax-import}{\mathsf{import}}~{\href{../syntax/values.html#syntax-name}{\mathit{name}}}_1~{\href{../syntax/values.html#syntax-name}{\mathit{name}}}_2~{\mathit{xt}} : {\mathit{xt}} } \qquad \end{array}\]

Modules

Modules are classified by their mapping from the external types of their imports to those of their exports.

A module is entirely closed, that is, its components can only refer to definitions that appear in the module itself. Consequently, no initial context is required. Instead, the context \(C\) for validation of the module’s content is constructed from the definitions in the module.

The module \((\href{../syntax/modules.html#syntax-module}{\mathsf{module}}~{{\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}^\ast}~{{\href{../syntax/modules.html#syntax-import}{\mathit{import}}}^\ast}~{{\href{../syntax/modules.html#syntax-func}{\mathit{func}}}^\ast}~{{\href{../syntax/modules.html#syntax-global}{\mathit{global}}}^\ast}~{{\href{../syntax/modules.html#syntax-table}{\mathit{table}}}^\ast}~{{\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}}^\ast}~{{\href{../syntax/modules.html#syntax-tag}{\mathit{tag}}}^\ast}~{{\href{../syntax/modules.html#syntax-elem}{\mathit{elem}}}^\ast}~{{\href{../syntax/modules.html#syntax-data}{\mathit{data}}}^\ast}~{{\href{../syntax/modules.html#syntax-start}{\mathit{start}}}^?}~{{\href{../syntax/modules.html#syntax-export}{\mathit{export}}}^\ast})\) is valid with the module type \(t\) if:

  • Under the context \(\{ \begin{array}[t]{@{}l@{}}\href{../valid/conventions.html#context}{\mathsf{return}}~\epsilon \}\end{array}\), the type definition sequence \({{\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}^\ast}\) is valid with the defined type sequence \({{\mathit{dt}'}^\ast}\).

  • The length of \({{\href{../syntax/modules.html#syntax-import}{\mathit{import}}}^\ast}\) is equal to the length of \({{\mathit{xt}}_{\mathsf{i}}^\ast}\).

  • For all \({\href{../syntax/modules.html#syntax-import}{\mathit{import}}}\) in \({{\href{../syntax/modules.html#syntax-import}{\mathit{import}}}^\ast}\), and corresponding \({\mathit{xt}}_{\mathsf{i}}\) in \({{\mathit{xt}}_{\mathsf{i}}^\ast}\):

    • Under the context \(\{ \begin{array}[t]{@{}l@{}}\href{../valid/conventions.html#context}{\mathsf{types}}~{{\mathit{dt}'}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{return}}~\epsilon \}\end{array}\), the import \({\href{../syntax/modules.html#syntax-import}{\mathit{import}}}\) is valid with the external type \({\mathit{xt}}_{\mathsf{i}}\).

  • Under the context \({C'}\), the global sequence \({{\href{../syntax/modules.html#syntax-global}{\mathit{global}}}^\ast}\) is valid with the global type sequence \({{\mathit{gt}}^\ast}\).

  • The length of \({{\href{../syntax/modules.html#syntax-table}{\mathit{table}}}^\ast}\) is equal to the length of \({{\mathit{tt}}^\ast}\).

  • For all \({\href{../syntax/modules.html#syntax-table}{\mathit{table}}}\) in \({{\href{../syntax/modules.html#syntax-table}{\mathit{table}}}^\ast}\), and corresponding \({\mathit{tt}}\) in \({{\mathit{tt}}^\ast}\):

    • Under the context \({C'}\), the table \({\href{../syntax/modules.html#syntax-table}{\mathit{table}}}\) is valid with the table type \({\mathit{tt}}\).

  • The length of \({{\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}}^\ast}\) is equal to the length of \({{\mathit{mt}}^\ast}\).

  • For all \({\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}}\) in \({{\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}}^\ast}\), and corresponding \({\mathit{mt}}\) in \({{\mathit{mt}}^\ast}\):

    • Under the context \({C'}\), the memory \({\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}}\) is valid with the memory type \({\mathit{mt}}\).

  • The length of \({{\mathit{jt}}^\ast}\) is equal to the length of \({{\href{../syntax/modules.html#syntax-tag}{\mathit{tag}}}^\ast}\).

  • For all \({\mathit{jt}}\) in \({{\mathit{jt}}^\ast}\), and corresponding \({\href{../syntax/modules.html#syntax-tag}{\mathit{tag}}}\) in \({{\href{../syntax/modules.html#syntax-tag}{\mathit{tag}}}^\ast}\):

    • Under the context \({C'}\), the tag \({\href{../syntax/modules.html#syntax-tag}{\mathit{tag}}}\) is valid with the tag type \({\mathit{jt}}\).

  • The length of \({{\mathit{dt}}^\ast}\) is equal to the length of \({{\href{../syntax/modules.html#syntax-func}{\mathit{func}}}^\ast}\).

  • For all \({\mathit{dt}}\) in \({{\mathit{dt}}^\ast}\), and corresponding \({\href{../syntax/modules.html#syntax-func}{\mathit{func}}}\) in \({{\href{../syntax/modules.html#syntax-func}{\mathit{func}}}^\ast}\):

    • The function \({\href{../syntax/modules.html#syntax-func}{\mathit{func}}}\) is valid with the defined type \({\mathit{dt}}\).

  • The length of \({{\href{../syntax/modules.html#syntax-elem}{\mathit{elem}}}^\ast}\) is equal to the length of \({{\mathit{rt}}^\ast}\).

  • For all \({\href{../syntax/modules.html#syntax-elem}{\mathit{elem}}}\) in \({{\href{../syntax/modules.html#syntax-elem}{\mathit{elem}}}^\ast}\), and corresponding \({\mathit{rt}}\) in \({{\mathit{rt}}^\ast}\):

  • The length of \({{\href{../syntax/modules.html#syntax-data}{\mathit{data}}}^\ast}\) is equal to the length of \({{\mathit{ok}}^\ast}\).

  • For all \({\href{../syntax/modules.html#syntax-data}{\mathit{data}}}\) in \({{\href{../syntax/modules.html#syntax-data}{\mathit{data}}}^\ast}\), and corresponding \({\mathit{ok}}\) in \({{\mathit{ok}}^\ast}\):

    • The memory segment \({\href{../syntax/modules.html#syntax-data}{\mathit{data}}}\) is valid.

  • If \({\href{../syntax/modules.html#syntax-start}{\mathit{start}}}\) is defined, then:

    • The start function \({\href{../syntax/modules.html#syntax-start}{\mathit{start}}}\) is valid.

  • The length of \({{\href{../syntax/modules.html#syntax-export}{\mathit{export}}}^\ast}\) is equal to the length of \({{\mathit{nm}}^\ast}\).

  • The length of \({{\href{../syntax/modules.html#syntax-export}{\mathit{export}}}^\ast}\) is equal to the length of \({{\mathit{xt}}_{\mathsf{e}}^\ast}\).

  • For all \({\href{../syntax/modules.html#syntax-export}{\mathit{export}}}\) in \({{\href{../syntax/modules.html#syntax-export}{\mathit{export}}}^\ast}\), and corresponding \({\mathit{nm}}\) in \({{\mathit{nm}}^\ast}\), and corresponding \({\mathit{xt}}_{\mathsf{e}}\) in \({{\mathit{xt}}_{\mathsf{e}}^\ast}\):

    • The export \({\href{../syntax/modules.html#syntax-export}{\mathit{export}}}\) is valid with the name \({\mathit{nm}}\) and the external type \({\mathit{xt}}_{\mathsf{e}}\).

  • \({{\mathit{nm}}^\ast}~{\mathrm{disjoint}}\) is of the form true.

  • The context \(C\) is of the form \({C'}{}[{.}\href{../valid/conventions.html#context}{\mathsf{globals}} \mathrel{{=}{\oplus}} {{\mathit{gt}}^\ast}]{}[{.}\href{../valid/conventions.html#context}{\mathsf{tables}} \mathrel{{=}{\oplus}} {{\mathit{tt}}_{\mathsf{i}}^\ast}~{{\mathit{tt}}^\ast}]{}[{.}\href{../valid/conventions.html#context}{\mathsf{mems}} \mathrel{{=}{\oplus}} {{\mathit{mt}}_{\mathsf{i}}^\ast}~{{\mathit{mt}}^\ast}]{}[{.}\href{../valid/conventions.html#context}{\mathsf{tags}} \mathrel{{=}{\oplus}} {{\mathit{jt}}_{\mathsf{i}}^\ast}~{{\mathit{jt}}^\ast}]{}[{.}\href{../valid/conventions.html#context}{\mathsf{elems}} \mathrel{{=}{\oplus}} {{\mathit{rt}}^\ast}]{}[{.}\href{../valid/conventions.html#context}{\mathsf{datas}} \mathrel{{=}{\oplus}} {{\mathit{ok}}^\ast}]\).

  • The context \({C'}\) is of the form \(\{ \begin{array}[t]{@{}l@{}}\href{../valid/conventions.html#context}{\mathsf{types}}~{{\mathit{dt}'}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{funcs}}~{{\mathit{dt}}_{\mathsf{i}}^\ast}~{{\mathit{dt}}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{globals}}~{{\mathit{gt}}_{\mathsf{i}}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{return}}~\epsilon,\; \href{../valid/conventions.html#context}{\mathsf{refs}}~{x^\ast} \}\end{array}\).

  • The function index sequence \({x^\ast}\) is of the form \({\href{../syntax/modules.html#syntax-funcidx}{\mathrm{funcidx}}}({{\href{../syntax/modules.html#syntax-global}{\mathit{global}}}^\ast}~{{\href{../syntax/modules.html#syntax-table}{\mathit{table}}}^\ast}~{{\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}}^\ast}~{{\href{../syntax/modules.html#syntax-elem}{\mathit{elem}}}^\ast}~{{\href{../syntax/modules.html#syntax-data}{\mathit{data}}}^\ast})\).

  • The defined type sequence \({{\mathit{dt}}_{\mathsf{i}}^\ast}\) is of the form \({\href{../syntax/types.html#syntax-externtype}{\mathrm{funcs}}}({{\mathit{xt}}_{\mathsf{i}}^\ast})\).

  • The global type sequence \({{\mathit{gt}}_{\mathsf{i}}^\ast}\) is of the form \({\href{../syntax/types.html#syntax-externtype}{\mathrm{globals}}}({{\mathit{xt}}_{\mathsf{i}}^\ast})\).

  • The table type sequence \({{\mathit{tt}}_{\mathsf{i}}^\ast}\) is of the form \({\href{../syntax/types.html#syntax-externtype}{\mathrm{tables}}}({{\mathit{xt}}_{\mathsf{i}}^\ast})\).

  • The memory type sequence \({{\mathit{mt}}_{\mathsf{i}}^\ast}\) is of the form \({\href{../syntax/types.html#syntax-externtype}{\mathrm{mems}}}({{\mathit{xt}}_{\mathsf{i}}^\ast})\).

  • The tag type sequence \({{\mathit{jt}}_{\mathsf{i}}^\ast}\) is of the form \({\href{../syntax/types.html#syntax-externtype}{\mathrm{tags}}}({{\mathit{xt}}_{\mathsf{i}}^\ast})\).

  • Let \(t\) be the module type \({{\href{../valid/conventions.html#aux-clostype}{\mathsf{clos}}}}_{C}({{\mathit{xt}}_{\mathsf{i}}^\ast}~\mathrel{\href{../valid/modules.html#syntax-moduletype}{\rightarrow}}~{{\mathit{xt}}_{\mathsf{e}}^\ast})\).

Todo

below is the official specification

  • Let \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}\) be the module to validate.

  • The types \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{types}}\) must be valid yielding a context \(C_0\).

  • Let \(C\) be a context where:

    • \(C.\href{../valid/conventions.html#context}{\mathsf{types}}\) is \(C_0.\href{../valid/conventions.html#context}{\mathsf{types}}\),

    • \(C.\href{../valid/conventions.html#context}{\mathsf{funcs}}\) is \(\href{../syntax/types.html#syntax-externtype}{\mathrm{funcs}}(\mathit{it}^\ast)\) concatenated with \(\mathit{dt}^\ast\), with the import’s external types \(\mathit{it}^\ast\) and the internal defined types \(\mathit{dt}^\ast\) as determined below,

    • \(C.\href{../valid/conventions.html#context}{\mathsf{tables}}\) is \(\href{../syntax/types.html#syntax-externtype}{\mathrm{tables}}(\mathit{it}^\ast)\) concatenated with \(\mathit{tt}^\ast\), with the import’s external types \(\mathit{it}^\ast\) and the internal table types \(\mathit{tt}^\ast\) as determined below,

    • \(C.\href{../valid/conventions.html#context}{\mathsf{mems}}\) is \(\href{../syntax/types.html#syntax-externtype}{\mathrm{mems}}(\mathit{it}^\ast)\) concatenated with \(\mathit{mt}^\ast\), with the import’s external types \(\mathit{it}^\ast\) and the internal memory types \(\mathit{mt}^\ast\) as determined below,

    • \(C.\href{../valid/conventions.html#context}{\mathsf{globals}}\) is \(\href{../syntax/types.html#syntax-externtype}{\mathrm{globals}}(\mathit{it}^\ast)\) concatenated with \(\mathit{gt}^\ast\), with the import’s external types \(\mathit{it}^\ast\) and the internal global types \(\mathit{gt}^\ast\) as determined below,

    • \(C.\href{../valid/conventions.html#context}{\mathsf{tags}}\) is \(\href{../syntax/types.html#syntax-externtype}{\mathrm{tags}}(\mathit{it}^\ast)\) concatenated with \(\mathit{ht}^\ast\), with the import’s external types \(\mathit{it}^\ast\) and the internal tag types \(\mathit{ht}^\ast\) as determined below,

    • \(C.\href{../valid/conventions.html#context}{\mathsf{elems}}\) is \({\mathit{rt}}^\ast\) as determined below,

    • \(C.\href{../valid/conventions.html#context}{\mathsf{datas}}\) is \({\mathit{ok}}^\ast\) as determined below,

    • \(C.\href{../valid/conventions.html#context}{\mathsf{locals}}\) is empty,

    • \(C.\href{../valid/conventions.html#context}{\mathsf{labels}}\) is empty,

    • \(C.\href{../valid/conventions.html#context}{\mathsf{return}}\) is empty.

    • \(C.\href{../valid/conventions.html#context}{\mathsf{refs}}\) is the set \(\href{../syntax/modules.html#syntax-funcidx}{\mathrm{funcidx}}(\href{../syntax/modules.html#syntax-module}{\mathit{module}} \href{../syntax/conventions.html#notation-replace}{\mathrel{\mbox{with}}} \href{../syntax/modules.html#syntax-module}{\mathsf{funcs}} = \epsilon \href{../syntax/conventions.html#notation-replace}{\mathrel{\mbox{with}}} \href{../syntax/modules.html#syntax-module}{\mathsf{start}} = \epsilon)\), i.e., the set of function indices occurring in the module, except in its functions or start function.

  • Let \(C'\) be the context where:

    • \(C'.\href{../valid/conventions.html#context}{\mathsf{globals}}\) is the sequence \(\href{../syntax/types.html#syntax-externtype}{\mathrm{globals}}(\mathit{it}^\ast)\),

    • \(C'.\href{../valid/conventions.html#context}{\mathsf{types}}\) is the same as \(C.\href{../valid/conventions.html#context}{\mathsf{types}}\),

    • \(C'.\href{../valid/conventions.html#context}{\mathsf{funcs}}\) is the same as \(C.\href{../valid/conventions.html#context}{\mathsf{funcs}}\),

    • \(C'.\href{../valid/conventions.html#context}{\mathsf{tables}}\) is the same as \(C.\href{../valid/conventions.html#context}{\mathsf{tables}}\),

    • \(C'.\href{../valid/conventions.html#context}{\mathsf{mems}}\) is the same as \(C.\href{../valid/conventions.html#context}{\mathsf{mems}}\),

    • \(C'.\href{../valid/conventions.html#context}{\mathsf{refs}}\) is the same as \(C.\href{../valid/conventions.html#context}{\mathsf{refs}}\),

    • all other fields are empty.

  • Under the context \(C'\):

    • The sequence \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{globals}}\) of globals must be valid with a sequence \(\mathit{gt}^\ast\) of global types.

    • For each \(\href{../syntax/modules.html#syntax-table}{\mathit{table}}_i\) in \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{tables}}\), the definition \(\href{../syntax/modules.html#syntax-table}{\mathit{table}}_i\) must be valid with a table type \(\mathit{tt}_i\).

    • For each \(\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}_i\) in \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{mems}}\), the definition \(\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}_i\) must be valid with a memory type \(\mathit{mt}_i\).

  • Under the context \(C\):

    • For each \(\href{../syntax/modules.html#syntax-func}{\mathit{func}}_i\) in \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{funcs}}\), the definition \(\href{../syntax/modules.html#syntax-func}{\mathit{func}}_i\) must be valid with a defined type \(\mathit{dt}_i\).

    • For each \(\href{../syntax/modules.html#syntax-tag}{\mathit{tag}}_i\) in \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{tags}}\), the definition \(\href{../syntax/modules.html#syntax-tag}{\mathit{tag}}_i\) must be valid with a tag type \(\mathit{ht}_i\).

    • For each \(\href{../syntax/modules.html#syntax-elem}{\mathit{elem}}_i\) in \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{elems}}\), the segment \(\href{../syntax/modules.html#syntax-elem}{\mathit{elem}}_i\) must be valid with reference type \(\mathit{rt}_i\).

    • For each \(\href{../syntax/modules.html#syntax-data}{\mathit{data}}_i\) in \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{datas}}\), the segment \(\href{../syntax/modules.html#syntax-data}{\mathit{data}}_i\) must be valid with data type \(\mathit{ok}_i\).

    • If \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{start}}\) is non-empty, then \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{start}}\) must be valid.

    • For each \(\href{../syntax/modules.html#syntax-import}{\mathit{import}}_i\) in \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{imports}}\), the segment \(\href{../syntax/modules.html#syntax-import}{\mathit{import}}_i\) must be valid with an external type \(\mathit{it}_i\).

    • For each \(\href{../syntax/modules.html#syntax-export}{\mathit{export}}_i\) in \(\href{../syntax/modules.html#syntax-module}{\mathit{module}}.\href{../syntax/modules.html#syntax-module}{\mathsf{exports}}\), the segment \(\href{../syntax/modules.html#syntax-export}{\mathit{export}}_i\) must be valid with external type \(\mathit{et}_i\).

  • Let \(\mathit{dt}^\ast\) be the concatenation of the internal function types \(\mathit{dt}_i\), in index order.

  • Let \(\mathit{tt}^\ast\) be the concatenation of the internal table types \(\mathit{tt}_i\), in index order.

  • Let \(\mathit{mt}^\ast\) be the concatenation of the internal memory types \(\mathit{mt}_i\), in index order.

  • Let \(\mathit{gt}^\ast\) be the concatenation of the internal global types \(\mathit{gt}_i\), in index order.

  • Let \(\mathit{ht}^\ast\) be the concatenation of the internal tag types \(\mathit{ht}_i\), in index order.

  • Let \(\mathit{rt}^\ast\) be the concatenation of the reference types \(\mathit{rt}_i\), in index order.

  • Let \(\mathit{ok}^\ast\) be the concatenation of the data types \(\mathit{ok}_i\), in index order.

  • Let \(\mathit{it}^\ast\) be the concatenation of external types \(\mathit{it}_i\) of the imports, in index order.

  • Let \(\mathit{et}^\ast\) be the concatenation of external types \(\mathit{et}_i\) of the exports, in index order.

  • All export names \(\href{../syntax/modules.html#syntax-export}{\mathit{export}}_i.\href{../syntax/modules.html#syntax-export}{\mathsf{name}}\) must be different.

  • Then the module is valid with external types \(\mathit{it}^\ast \mathrel{\href{../valid/conventions.html#syntax-instrtype}{\rightarrow}} \mathit{et}^\ast\).

\[\begin{split}\begin{array}{@{}c@{}}\displaystyle \frac{ \begin{array}{@{}c@{}} \{ \begin{array}[t]{@{}l@{}} \}\end{array} \href{../valid/modules.html#valid-types}{\vdash} {{\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}^\ast} : {{\mathit{dt}'}^\ast} \qquad (\{ \begin{array}[t]{@{}l@{}} \href{../valid/conventions.html#context}{\mathsf{types}}~{{\mathit{dt}'}^\ast} \}\end{array} \href{../valid/modules.html#valid-import}{\vdash} {\href{../syntax/modules.html#syntax-import}{\mathit{import}}} : {\mathit{xt}}_{\mathsf{i}})^\ast \\ {C'} \href{../valid/modules.html#valid-globalseq}{\vdash} {{\href{../syntax/modules.html#syntax-global}{\mathit{global}}}^\ast} : {{\mathit{gt}}^\ast} \qquad ({C'} \href{../valid/modules.html#valid-table}{\vdash} {\href{../syntax/modules.html#syntax-table}{\mathit{table}}} : {\mathit{tt}})^\ast \qquad ({C'} \href{../valid/modules.html#valid-mem}{\vdash} {\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}} : {\mathit{mt}})^\ast \qquad ({C'} \href{../valid/modules.html#valid-tag}{\vdash} {\href{../syntax/modules.html#syntax-tag}{\mathit{tag}}} : {\mathit{jt}})^\ast \qquad (C \href{../valid/modules.html#valid-func}{\vdash} {\href{../syntax/modules.html#syntax-func}{\mathit{func}}} : {\mathit{dt}})^\ast \\ (C \href{../valid/modules.html#valid-elem}{\vdash} {\href{../syntax/modules.html#syntax-elem}{\mathit{elem}}} : {\mathit{rt}})^\ast \qquad (C \href{../valid/modules.html#valid-data}{\vdash} {\href{../syntax/modules.html#syntax-data}{\mathit{data}}} : {\mathit{ok}})^\ast \qquad (C \href{../valid/modules.html#valid-start}{\vdash} {\href{../syntax/modules.html#syntax-start}{\mathit{start}}} : \href{../valid/modules.html#valid-start}{\mathsf{ok}})^? \qquad (C \href{../valid/modules.html#valid-export}{\vdash} {\href{../syntax/modules.html#syntax-export}{\mathit{export}}} : {\mathit{nm}}~{\mathit{xt}}_{\mathsf{e}})^\ast \qquad {{\mathit{nm}}^\ast}~{\mathrm{disjoint}} \\ C = {C'} \oplus \{ \begin{array}[t]{@{}l@{}} \href{../valid/conventions.html#context}{\mathsf{globals}}~{{\mathit{gt}}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{tables}}~{{\mathit{tt}}_{\mathsf{i}}^\ast}~{{\mathit{tt}}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{mems}}~{{\mathit{mt}}_{\mathsf{i}}^\ast}~{{\mathit{mt}}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{tags}}~{{\mathit{jt}}_{\mathsf{i}}^\ast}~{{\mathit{jt}}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{elems}}~{{\mathit{rt}}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{datas}}~{{\mathit{ok}}^\ast} \}\end{array} \\ {C'} = \{ \begin{array}[t]{@{}l@{}} \href{../valid/conventions.html#context}{\mathsf{types}}~{{\mathit{dt}'}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{funcs}}~{{\mathit{dt}}_{\mathsf{i}}^\ast}~{{\mathit{dt}}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{globals}}~{{\mathit{gt}}_{\mathsf{i}}^\ast},\; \href{../valid/conventions.html#context}{\mathsf{refs}}~{x^\ast} \}\end{array} \qquad {x^\ast} = {\href{../syntax/modules.html#syntax-funcidx}{\mathrm{funcidx}}}({{\href{../syntax/modules.html#syntax-global}{\mathit{global}}}^\ast}~{{\href{../syntax/modules.html#syntax-table}{\mathit{table}}}^\ast}~{{\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}}^\ast}~{{\href{../syntax/modules.html#syntax-elem}{\mathit{elem}}}^\ast}~{{\href{../syntax/modules.html#syntax-data}{\mathit{data}}}^\ast}) \\ {{\mathit{dt}}_{\mathsf{i}}^\ast} = {\href{../syntax/types.html#syntax-externtype}{\mathrm{funcs}}}({{\mathit{xt}}_{\mathsf{i}}^\ast}) \qquad {{\mathit{gt}}_{\mathsf{i}}^\ast} = {\href{../syntax/types.html#syntax-externtype}{\mathrm{globals}}}({{\mathit{xt}}_{\mathsf{i}}^\ast}) \qquad {{\mathit{tt}}_{\mathsf{i}}^\ast} = {\href{../syntax/types.html#syntax-externtype}{\mathrm{tables}}}({{\mathit{xt}}_{\mathsf{i}}^\ast}) \qquad {{\mathit{mt}}_{\mathsf{i}}^\ast} = {\href{../syntax/types.html#syntax-externtype}{\mathrm{mems}}}({{\mathit{xt}}_{\mathsf{i}}^\ast}) \qquad {{\mathit{jt}}_{\mathsf{i}}^\ast} = {\href{../syntax/types.html#syntax-externtype}{\mathrm{tags}}}({{\mathit{xt}}_{\mathsf{i}}^\ast}) \end{array} }{ {\href{../valid/modules.html#valid-module}{\vdash}}\, \href{../syntax/modules.html#syntax-module}{\mathsf{module}}~{{\href{../syntax/types.html#syntax-rectype}{\mathit{type}}}^\ast}~{{\href{../syntax/modules.html#syntax-import}{\mathit{import}}}^\ast}~{{\href{../syntax/modules.html#syntax-func}{\mathit{func}}}^\ast}~{{\href{../syntax/modules.html#syntax-global}{\mathit{global}}}^\ast}~{{\href{../syntax/modules.html#syntax-table}{\mathit{table}}}^\ast}~{{\href{../syntax/modules.html#syntax-mem}{\mathit{mem}}}^\ast}~{{\href{../syntax/modules.html#syntax-tag}{\mathit{tag}}}^\ast}~{{\href{../syntax/modules.html#syntax-elem}{\mathit{elem}}}^\ast}~{{\href{../syntax/modules.html#syntax-data}{\mathit{data}}}^\ast}~{{\href{../syntax/modules.html#syntax-start}{\mathit{start}}}^?}~{{\href{../syntax/modules.html#syntax-export}{\mathit{export}}}^\ast} : {{\href{../valid/conventions.html#aux-clostype}{\mathsf{clos}}}}_{C}({{\mathit{xt}}_{\mathsf{i}}^\ast} \mathrel{\href{../valid/modules.html#syntax-moduletype}{\rightarrow}} {{\mathit{xt}}_{\mathsf{e}}^\ast}) } \qquad \end{array}\end{split}\]

Todo

Check refs; check export names

Note

All functions in a module are mutually recursive. Consequently, the definition of the context \(C\) in this rule is recursive: it depends on the outcome of validation of the function, table, memory, and global definitions contained in the module, which itself depends on \(C\). However, this recursion is just a specification device. All types needed to construct \(C\) can easily be determined from a simple pre-pass over the module that does not perform any actual validation.

Globals, however, are not recursive but evaluated sequentially, such that each constant expressions only has access to imported or previously defined globals.