Matching¶

On most types, a notion of subtyping is defined that is applicable in validation rules, during module instantiation when checking the types of imports, or during execution, when performing casts.

Number Types¶

The number type \({\href{../syntax/types.html#syntax-numtype}{\mathit{numtype}}}\) matches itself.

Todo

below is the official specification

A number type \({\href{../syntax/types.html#syntax-numtype}{\mathit{numtype}}}_1\) matches a number type \({\href{../syntax/types.html#syntax-numtype}{\mathit{numtype}}}_2\) if and only if:

Both \(\href{../syntax/types.html#syntax-numtype}{\mathit{numtype}}_1\) and \(\href{../syntax/types.html#syntax-numtype}{\mathit{numtype}}_2\) are the same.

\[\begin{array}{@{}c@{}}\displaystyle \frac{ }{ C \href{../valid/matching.html#match-numtype}{\vdash} {\href{../syntax/types.html#syntax-numtype}{\mathit{numtype}}} \href{../valid/matching.html#match-numtype}{\leq} {\href{../syntax/types.html#syntax-numtype}{\mathit{numtype}}} } \qquad \end{array}\]

Vector Types¶

The vector type \({\href{../syntax/types.html#syntax-vectype}{\mathit{vectype}}}\) matches itself.

Todo

below is the official specification

A vector type \({\href{../syntax/types.html#syntax-vectype}{\mathit{vectype}}}_1\) matches a vector type \({\href{../syntax/types.html#syntax-vectype}{\mathit{vectype}}}_2\) if and only if:

Both \(\href{../syntax/types.html#syntax-vectype}{\mathit{vectype}}_1\) and \(\href{../syntax/types.html#syntax-vectype}{\mathit{vectype}}_2\) are the same.

\[\begin{array}{@{}c@{}}\displaystyle \frac{ }{ C \href{../valid/matching.html#match-vectype}{\vdash} {\href{../syntax/types.html#syntax-vectype}{\mathit{vectype}}} \href{../valid/matching.html#match-vectype}{\leq} {\href{../syntax/types.html#syntax-vectype}{\mathit{vectype}}} } \qquad \end{array}\]

Heap Types¶

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) matches the heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) if:

Either:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_1\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_2\).

The heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}'}\) is valid.

The heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_1\) matches the heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}'}\).

The heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}'}\) matches the heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_2\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{any}}\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{i\scriptstyle31}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}}\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{struct}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}}\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{array}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}}\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{struct}}\).

The expansion of the defined type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}\) is the composite type \((\href{../syntax/types.html#syntax-comptype}{\mathsf{struct}}~{{\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}}^\ast})\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{array}}\).

The expansion of the defined type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}\) is the composite type \((\href{../syntax/types.html#syntax-comptype}{\mathsf{array}}~{\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}})\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{func}}\).

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

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2\).

The defined type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1\) matches the defined type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\).

The defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[{\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}}]\) exists.

The defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[{\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}}]\) matches the heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}}\).

The defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[{\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}}]\) exists.

The heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\) matches the defined type \(C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[{\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}}]\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \((\href{../syntax/types.html#syntax-heaptype}{\mathsf{rec}}~i)\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({{\href{../syntax/types.html#syntax-typeuse}{\mathit{typeuse}}}^\ast}{}[j]\).

\({|{{\href{../syntax/types.html#syntax-typeuse}{\mathit{typeuse}}}^\ast}|}\) is greater than \(j\).

The sub type \(C{.}\href{../appendix/properties.html#context-ext}{\mathsf{recs}}{}[i]\) exists.

The sub type \(C{.}\href{../appendix/properties.html#context-ext}{\mathsf{recs}}{}[i]\) is equal to \((\href{../syntax/types.html#syntax-subtype}{\mathsf{sub}}~{\href{../syntax/types.html#syntax-subtype}{\mathsf{final}}^?}~{{\href{../syntax/types.html#syntax-typeuse}{\mathit{typeuse}}}^\ast}~{\mathit{ct}})\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{none}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\).

The heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\) matches the heap type \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{any}}\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{nofunc}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\).

The heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\) matches the heap type \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{func}}\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{noextern}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\).

The heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\) matches the heap type \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{extern}}\).

Or:

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \(\href{../valid/conventions.html#syntax-valtype-ext}{\mathsf{bot}}\).

The heap type \({\mathit{ht}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}\).

Todo

below is the official specification

A heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_1\) matches a heap type \({\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_2\) if and only if:

Either both \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) and \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\) are the same.
Or there exists a valid heap type \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}'\), such that \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) matches \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}'\) and \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}'\) matches \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\).
Or \(heaptype_1\) is \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}}\) and \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\) is \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{any}}\).
Or \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) is one of \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{i\scriptstyle31}}\), \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{struct}}\), or \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{array}}\) and \(heaptype_2\) is \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}}\).
Or \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) is a defined type which expands to a structure type and \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\) is \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{struct}}\).
Or \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) is a defined type which expands to an array type and \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\) is \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{array}}\).
Or \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) is a defined type which expands to a function type and \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\) is \(\href{../syntax/modules.html#syntax-func}{\mathsf{func}}\).
Or \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) is a defined type \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1\) and \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\) is a defined type \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2\), and \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1\) matches \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2\).
Or \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) is a type index \(x_1\), and the defined type \(C.\href{../valid/conventions.html#context}{\mathsf{types}}[x_1]\) matches \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\).
Or \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\) is a type index \(x_2\), and \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) matches the defined type \(C.\href{../valid/conventions.html#context}{\mathsf{types}}[x_2]\).
Or \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) is \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{none}}\) and \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\) matches \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{any}}\).
Or \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) is \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{nofunc}}\) and \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\) matches \(\href{../syntax/modules.html#syntax-func}{\mathsf{func}}\).
Or \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) is \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{noextern}}\) and \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\) matches \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{extern}}\).
Or \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) is \(\href{../valid/conventions.html#syntax-heaptype-ext}{\mathsf{bot}}\).

\[\begin{split}\begin{array}{@{}c@{}}\displaystyle \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} \href{../valid/matching.html#match-heaptype}{\leq} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} } \qquad \frac{ C \href{../valid/types.html#valid-heaptype}{\vdash} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}'} : \href{../valid/types.html#valid-heaptype}{\mathsf{ok}} \qquad C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_1 \href{../valid/matching.html#match-heaptype}{\leq} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}'} \qquad C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}'} \href{../valid/matching.html#match-heaptype}{\leq} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_2 }{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_1 \href{../valid/matching.html#match-heaptype}{\leq} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_2 } \\[3ex]\displaystyle \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{any}} } \qquad \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{i\scriptstyle31}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}} } \qquad \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{struct}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}} } \qquad \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{array}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}} } \\[3ex]\displaystyle \frac{ {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}} \href{../valid/conventions.html#aux-expand-deftype}{\approx} \href{../syntax/types.html#syntax-comptype}{\mathsf{struct}}~{{\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}}^\ast} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{struct}} } \qquad \frac{ {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}} \href{../valid/conventions.html#aux-expand-deftype}{\approx} \href{../syntax/types.html#syntax-comptype}{\mathsf{array}}~{\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{array}} } \qquad \frac{ {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}} \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/matching.html#match-heaptype}{\vdash} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{func}} } \\[3ex]\displaystyle \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[{\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}}] \href{../valid/matching.html#match-heaptype}{\leq} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}} \href{../valid/matching.html#match-heaptype}{\leq} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} } \qquad \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} \href{../valid/matching.html#match-heaptype}{\leq} C{.}\href{../valid/conventions.html#context}{\mathsf{types}}{}[{\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}}] }{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} \href{../valid/matching.html#match-heaptype}{\leq} {\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}} } \\[3ex]\displaystyle \frac{ C{.}\href{../appendix/properties.html#context-ext}{\mathsf{recs}}{}[i] = \href{../syntax/types.html#syntax-subtype}{\mathsf{sub}}~{\href{../syntax/types.html#syntax-subtype}{\mathsf{final}}^?}~{{\href{../syntax/types.html#syntax-typeuse}{\mathit{typeuse}}}^\ast}~{\mathit{ct}} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{rec}}~i \href{../valid/matching.html#match-heaptype}{\leq} {{\href{../syntax/types.html#syntax-typeuse}{\mathit{typeuse}}}^\ast}{}[j] } \\[3ex]\displaystyle \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{any}} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{none}} \href{../valid/matching.html#match-heaptype}{\leq} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} } \qquad \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{func}} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{nofunc}} \href{../valid/matching.html#match-heaptype}{\leq} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} } \qquad \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{extern}} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{noextern}} \href{../valid/matching.html#match-heaptype}{\leq} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} } \\[3ex]\displaystyle \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../valid/conventions.html#syntax-valtype-ext}{\mathsf{bot}} \href{../valid/matching.html#match-heaptype}{\leq} {\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}} } \qquad \end{array}\end{split}\]

Reference Types¶

The reference type \((\href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~(\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}~{{\mathit{u{\kern-0.1em\scriptstyle 1}}}^?}~{}^?)~{\mathit{ht}}_1)\) matches the reference type \((\href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~(\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}~{{\mathit{u{\kern-0.1em\scriptstyle 2}}}^?}~{}^?)~{\mathit{ht}}_2)\) if:

The heap type \({\mathit{ht}}_1\) matches the heap type \({\mathit{ht}}_2\).

Either:

\({{\mathit{u{\kern-0.1em\scriptstyle 1}}}^?}\) is absent.

\({{\mathit{u{\kern-0.1em\scriptstyle 2}}}^?}\) is absent.

Or:

\({{\mathit{u{\kern-0.1em\scriptstyle 1}}}^?}\) is equal to \({()^?}\).

\({{\mathit{u{\kern-0.1em\scriptstyle 2}}}^?}\) is equal to \(()\).

Todo

below is the official specification

A reference type \(\href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~{\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}}{{{}_{1}^?}}~{\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_1\) matches a reference type \(\href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~{\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}}{{{}_{2}^?}}~{\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}}_2\) if and only if:

The heap type \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1\) matches \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2\).
\(\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}_1\) is absent or \(\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}_2\) is present.

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\mathit{ht}}_1 \href{../valid/matching.html#match-heaptype}{\leq} {\mathit{ht}}_2 }{ C \href{../valid/matching.html#match-reftype}{\vdash} \href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~{\mathit{ht}}_1 \href{../valid/matching.html#match-reftype}{\leq} \href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~{\mathit{ht}}_2 } \qquad \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} {\mathit{ht}}_1 \href{../valid/matching.html#match-heaptype}{\leq} {\mathit{ht}}_2 }{ C \href{../valid/matching.html#match-reftype}{\vdash} \href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~{\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}^?}~{\mathit{ht}}_1 \href{../valid/matching.html#match-reftype}{\leq} \href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}~{\mathit{ht}}_2 } \qquad \end{array}\]

Value Types¶

The value type \(t_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) matches the value type \(t_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) if:

Either:

The value type \(t_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../syntax/types.html#syntax-numtype}{\mathit{numtype}}}_1\).

The value type \(t_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-numtype}{\mathit{numtype}}}_2\).

The number type \({\href{../syntax/types.html#syntax-numtype}{\mathit{numtype}}}_1\) matches the number type \({\href{../syntax/types.html#syntax-numtype}{\mathit{numtype}}}_2\).

Or:

The value type \(t_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../syntax/types.html#syntax-vectype}{\mathit{vectype}}}_1\).

The value type \(t_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-vectype}{\mathit{vectype}}}_2\).

The vector type \({\href{../syntax/types.html#syntax-vectype}{\mathit{vectype}}}_1\) matches the vector type \({\href{../syntax/types.html#syntax-vectype}{\mathit{vectype}}}_2\).

Or:

The value type \(t_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_1\).

The value type \(t_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_2\).

The reference type \({\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_1\) matches the reference type \({\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_2\).

Or:

The value type \(t_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \(\href{../valid/conventions.html#syntax-valtype-ext}{\mathsf{bot}}\).

The value type \(t_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}\).

Todo

below is the official specification

A value type \({\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1\) matches a value type \({\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2\) if and only if:

Either both \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_1\) and \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_2\) are number types and \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_1\) matches \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_2\).
Or both \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_1\) and \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_2\) are reference types and \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_1\) matches \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_2\).
Or \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_1\) is \(\href{../valid/conventions.html#syntax-valtype-ext}{\mathsf{bot}}\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ }{ C \href{../valid/matching.html#match-valtype}{\vdash} \href{../valid/conventions.html#syntax-valtype-ext}{\mathsf{bot}} \href{../valid/matching.html#match-valtype}{\leq} {\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}} } \qquad \end{array}\]

Result Types¶

The value type sequence \({t_1^\ast}\) matches the value type sequence \({t_2^\ast}\) if:

\({|t_{\mathit{{\scriptstyle 2}*}}|}\) is equal to \({|t_{\mathit{{\scriptstyle 1}*}}|}\).

For all \(t_1\) in \({t_1^\ast}\) and \(t_2\) in \({t_2^\ast}\):

The value type \(t_1\) matches the value type \(t_2\).

Todo

below is the official specification

Subtyping is lifted to result types in a pointwise manner. That is, a result type \({t_1^\ast}\) matches a result type \({t_2^\ast}\) if and only if:

Every value type \(t_1\) in \([t_1^\ast]\) matches the corresponding value type \(t_2\) in \([t_2^\ast]\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ (C \href{../valid/matching.html#match-valtype}{\vdash} t_1 \href{../valid/matching.html#match-valtype}{\leq} t_2)^\ast }{ C \href{../valid/matching.html#match-resulttype}{\vdash} {t_1^\ast} \href{../valid/matching.html#match-resulttype}{\leq} {t_2^\ast} } \qquad \end{array}\]

Instruction Types¶

The instruction type \({t_{11}^\ast}~{\mathrel{\href{../valid/conventions.html#syntax-instrtype}{\rightarrow}}}_{{x_1^\ast}}\,{t_{12}^\ast}\) matches the instruction type \({t_{21}^\ast}~{\mathrel{\href{../valid/conventions.html#syntax-instrtype}{\rightarrow}}}_{{x_2^\ast}}\,{t_{22}^\ast}\) if:

The value type sequence \({t_{21}^\ast}\) matches the value type sequence \({t_{11}^\ast}\).

The value type sequence \({t_{12}^\ast}\) matches the value type sequence \({t_{22}^\ast}\).

The index sequence \({x^\ast}\) is equal to \({x_2^\ast} \setminus {x_1^\ast}\).

\({|{\mathit{x*}}|}\) is equal to \({|{\mathit{t*}}|}\).

For all \(x\) in \({x^\ast}\):

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

For all \(t\) in \({t^\ast}\) and \(x\) in \({x^\ast}\):

The local type \(C{.}\href{../valid/conventions.html#context}{\mathsf{locals}}{}[x]\) is equal to \((\href{../valid/conventions.html#syntax-init}{\mathsf{set}}, t)\).

Todo

below is the official specification

Subtyping is further lifted to instruction types. An instruction type \({t_{11}^\ast} \mathrel{\href{../valid/conventions.html#syntax-instrtype}{\rightarrow}}_{{x_1^\ast}} {t_{12}^\ast}\) matches a type \({t_{21}^\ast} \mathrel{\href{../valid/conventions.html#syntax-instrtype}{\rightarrow}}_{{x_2^\ast}} {t_{22}^\ast}\) if and only if:

There is a common sequence of value types \(t^\ast\) such that \(t_{21}^\ast\) equals \(t^\ast~{t'_{21}}^\ast\) and \(t_{22}^\ast\) equals \(t^\ast~{t'_{22}}^\ast\).
The result type \([{t'_{21}}^\ast]\) matches \([t_{11}^\ast]\).
The result type \([t_{12}^\ast]\) matches \([{t'_{22}}^\ast]\).
For every local index \(x\) that is in \(x_2^\ast\) but not in \(x_1^\ast\), the local type \(C.\href{../valid/conventions.html#context}{\mathsf{locals}}[x]\) is \(\href{../valid/conventions.html#syntax-init}{\mathsf{set}}~t_x\) for some value type \(t_x\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-resulttype}{\vdash} {t_{21}^\ast} \href{../valid/matching.html#match-resulttype}{\leq} {t_{11}^\ast} \qquad C \href{../valid/matching.html#match-resulttype}{\vdash} {t_{12}^\ast} \href{../valid/matching.html#match-resulttype}{\leq} {t_{22}^\ast} \qquad {x^\ast} = {x_2^\ast} \setminus {x_1^\ast} \qquad (C{.}\href{../valid/conventions.html#context}{\mathsf{locals}}{}[x] = \href{../valid/conventions.html#syntax-init}{\mathsf{set}}~t)^\ast }{ C \href{../valid/matching.html#match-instrtype}{\vdash} {t_{11}^\ast} \mathrel{\href{../valid/conventions.html#syntax-instrtype}{\rightarrow}}_{{x_1^\ast}} {t_{12}^\ast} \href{../valid/matching.html#match-instrtype}{\leq} {t_{21}^\ast} \mathrel{\href{../valid/conventions.html#syntax-instrtype}{\rightarrow}}_{{x_2^\ast}} {t_{22}^\ast} } \qquad \end{array}\]

Note

Instruction types are contravariant in their input and covariant in their output. Subtyping also incorporates a sort of “frame” condition, which allows adding arbitrary invariant stack elements on both sides in the super type.

Finally, the supertype may ignore variables from the init set \({x_1^\ast}\). It may also add variables to the init set, provided these are already set in the context, i.e., are vacuously initialized.

Function Types¶

The function type \({t_{11}^\ast}~\href{../syntax/types.html#syntax-functype}{\rightarrow}~{t_{12}^\ast}\) matches the function type \({t_{21}^\ast}~\href{../syntax/types.html#syntax-functype}{\rightarrow}~{t_{22}^\ast}\) if:

The value type sequence \({t_{21}^\ast}\) matches the value type sequence \({t_{11}^\ast}\).

The value type sequence \({t_{12}^\ast}\) matches the value type sequence \({t_{22}^\ast}\).

Todo

below is the official specification

A function type \({t_{11}^\ast} \href{../syntax/types.html#syntax-functype}{\rightarrow} {t_{12}^\ast}\) matches a type \({t_{21}^\ast} \href{../syntax/types.html#syntax-functype}{\rightarrow} {t_{22}^\ast}\) if and only if:

The result type \([t_{21}^\ast]\) matches \([t_{11}^\ast]\).
The result type \([t_{12}^\ast]\) matches \([t_{22}^\ast]\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-resulttype}{\vdash} {t_{21}^\ast} \href{../valid/matching.html#match-resulttype}{\leq} {t_{11}^\ast} \qquad C \href{../valid/matching.html#match-resulttype}{\vdash} {t_{12}^\ast} \href{../valid/matching.html#match-resulttype}{\leq} {t_{22}^\ast} }{ C \href{../valid/matching.html#match-functype}{\vdash} {t_{11}^\ast} \href{../syntax/types.html#syntax-functype}{\rightarrow} {t_{12}^\ast} \href{../valid/matching.html#match-functype}{\leq} {t_{21}^\ast} \href{../syntax/types.html#syntax-functype}{\rightarrow} {t_{22}^\ast} } \qquad \end{array}\]

Composite Types¶

The composite type \({\mathit{ct}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) matches the composite type \({\mathit{ct}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) if:

Either:

The composite type \({\mathit{ct}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \((\href{../syntax/types.html#syntax-comptype}{\mathsf{struct}}~{{\mathit{yt}}_1^\ast}~{\mathit{yt}'}_1)\).

The composite type \({\mathit{ct}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \((\href{../syntax/types.html#syntax-comptype}{\mathsf{struct}}~{{\mathit{yt}}_2^\ast})\).

\({|{\mathit{yt}}_{\mathit{{\scriptstyle 2}*}}|}\) is equal to \({|{\mathit{yt}}_{\mathit{{\scriptstyle 1}*}}|}\).

For all \({\mathit{yt}}_1\) in \({{\mathit{yt}}_1^\ast}\) and \({\mathit{yt}}_2\) in \({{\mathit{yt}}_2^\ast}\):

The field type \({\mathit{yt}}_1\) matches the field type \({\mathit{yt}}_2\).

Or:

The composite type \({\mathit{ct}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \((\href{../syntax/types.html#syntax-comptype}{\mathsf{array}}~{\mathit{yt}}_1)\).

The composite type \({\mathit{ct}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \((\href{../syntax/types.html#syntax-comptype}{\mathsf{array}}~{\mathit{yt}}_2)\).

The field type \({\mathit{yt}}_1\) matches the field type \({\mathit{yt}}_2\).

Or:

The composite type \({\mathit{ct}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \((\href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~{\mathit{ft}}_1)\).

The composite type \({\mathit{ct}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \((\href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~{\mathit{ft}}_2)\).

The function type \({\mathit{ft}}_1\) matches the function type \({\mathit{ft}}_2\).

Todo

below is the official specification

A composite type \({\href{../syntax/types.html#syntax-comptype}{\mathit{comptype}}}_1\) matches a type \({\href{../syntax/types.html#syntax-comptype}{\mathit{comptype}}}_2\) if and only if:

Either the composite type \(\href{../syntax/types.html#syntax-comptype}{\mathit{comptype}}_1\) is \(\href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~\href{../syntax/types.html#syntax-functype}{\mathit{functype}}_1\) and \(\href{../syntax/types.html#syntax-comptype}{\mathit{comptype}}_2\) is \(\href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~\href{../syntax/types.html#syntax-functype}{\mathit{functype}}_2\) and:
- The function type \(\href{../syntax/types.html#syntax-functype}{\mathit{functype}}_1\) matches \(\href{../syntax/types.html#syntax-functype}{\mathit{functype}}_2\).
Or the composite type \(\href{../syntax/types.html#syntax-comptype}{\mathit{comptype}}_1\) is \(\href{../syntax/types.html#syntax-comptype}{\mathsf{struct}}~\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_1^{n_1}\) and \(\href{../syntax/types.html#syntax-comptype}{\mathit{comptype}}_2\) is \(\href{../syntax/types.html#syntax-comptype}{\mathsf{struct}}~\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_2\) and:
- The arity \(n_1\) is greater than or equal to \(n_2\).
- For every field type \(\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_{2i}\) in \(\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_2^{n_2}\) and corresponding \(\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_{1i}\) in \(\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_1^{n_1}\)
  - The field type \(\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_{1i}\) matches \(\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_{2i}\).
Or the composite type \(\href{../syntax/types.html#syntax-comptype}{\mathit{comptype}}_1\) is \(\href{../syntax/types.html#syntax-comptype}{\mathsf{array}}~\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_1\) and \(\href{../syntax/types.html#syntax-comptype}{\mathit{comptype}}_2\) is \(\href{../syntax/types.html#syntax-comptype}{\mathsf{array}}~\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_2\) and:
- The field type \(\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_1\) matches \(\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_2\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ (C \href{../valid/matching.html#match-fieldtype}{\vdash} {\mathit{yt}}_1 \href{../valid/matching.html#match-fieldtype}{\leq} {\mathit{yt}}_2)^\ast }{ C \href{../valid/matching.html#match-comptype}{\vdash} \href{../syntax/types.html#syntax-comptype}{\mathsf{struct}}~({{\mathit{yt}}_1^\ast}~{\mathit{yt}'}_1) \href{../valid/matching.html#match-comptype}{\leq} \href{../syntax/types.html#syntax-comptype}{\mathsf{struct}}~{{\mathit{yt}}_2^\ast} } \qquad \frac{ C \href{../valid/matching.html#match-fieldtype}{\vdash} {\mathit{yt}}_1 \href{../valid/matching.html#match-fieldtype}{\leq} {\mathit{yt}}_2 }{ C \href{../valid/matching.html#match-comptype}{\vdash} \href{../syntax/types.html#syntax-comptype}{\mathsf{array}}~{\mathit{yt}}_1 \href{../valid/matching.html#match-comptype}{\leq} \href{../syntax/types.html#syntax-comptype}{\mathsf{array}}~{\mathit{yt}}_2 } \qquad \frac{ C \href{../valid/matching.html#match-functype}{\vdash} {\mathit{ft}}_1 \href{../valid/matching.html#match-functype}{\leq} {\mathit{ft}}_2 }{ C \href{../valid/matching.html#match-comptype}{\vdash} \href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~{\mathit{ft}}_1 \href{../valid/matching.html#match-comptype}{\leq} \href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~{\mathit{ft}}_2 } \qquad \end{array}\]

Field Types¶

The field type \(((\href{../syntax/types.html#syntax-mut}{\mathsf{mut}}~{{\mathit{u{\kern-0.1em\scriptstyle 1}}}^?}~{}^?), {\mathit{zt}}_1)\) matches the field type \(((\href{../syntax/types.html#syntax-mut}{\mathsf{mut}}~{{\mathit{u{\kern-0.1em\scriptstyle 2}}}^?}~{}^?), {\mathit{zt}}_2)\) if:

The storage type \({\mathit{zt}}_1\) matches the storage type \({\mathit{zt}}_2\).

Either:

\({{\mathit{u{\kern-0.1em\scriptstyle 1}}}^?}\) is absent.

\({{\mathit{u{\kern-0.1em\scriptstyle 2}}}^?}\) is absent.

Or:

\({{\mathit{u{\kern-0.1em\scriptstyle 1}}}^?}\) is equal to \(()\).

\({{\mathit{u{\kern-0.1em\scriptstyle 2}}}^?}\) is equal to \(()\).

The storage type \({\mathit{zt}}_2\) matches the storage type \({\mathit{zt}}_1\).

Todo

below is the official specification

A field type \(({\href{../syntax/types.html#syntax-mut}{\mathsf{mut}}}{{{}_{1}^?}}~{\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}}_1)\) matches a type \(({\href{../syntax/types.html#syntax-mut}{\mathsf{mut}}}{{{}_{2}^?}}~{\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}}_2)\) if and only if:

Storage type \(\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_1\) matches \(\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2\).
Either both \(\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_1\) and \(\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_2\) are \(\href{../syntax/types.html#syntax-mut}{\mathsf{const}}\).
Or both \(\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_1\) and \(\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_2\) are \(\href{../syntax/types.html#syntax-mut}{\mathsf{var}}\) and \(\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2\) matches \(\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_1\) as well.

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-storagetype}{\vdash} {\mathit{zt}}_1 \href{../valid/matching.html#match-storagetype}{\leq} {\mathit{zt}}_2 }{ C \href{../valid/matching.html#match-fieldtype}{\vdash} {\mathit{zt}}_1 \href{../valid/matching.html#match-fieldtype}{\leq} {\mathit{zt}}_2 } \qquad \frac{ C \href{../valid/matching.html#match-storagetype}{\vdash} {\mathit{zt}}_1 \href{../valid/matching.html#match-storagetype}{\leq} {\mathit{zt}}_2 \qquad C \href{../valid/matching.html#match-storagetype}{\vdash} {\mathit{zt}}_2 \href{../valid/matching.html#match-storagetype}{\leq} {\mathit{zt}}_1 }{ C \href{../valid/matching.html#match-fieldtype}{\vdash} \href{../syntax/types.html#syntax-mut}{\mathsf{mut}}~{\mathit{zt}}_1 \href{../valid/matching.html#match-fieldtype}{\leq} \href{../syntax/types.html#syntax-mut}{\mathsf{mut}}~{\mathit{zt}}_2 } \qquad \end{array}\]

The storage type \({\mathit{zt}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) matches the storage type \({\mathit{zt}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) if:

Either:

The storage type \({\mathit{zt}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1\).

The storage type \({\mathit{zt}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2\).

The value type \({\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1\) matches the value type \({\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2\).

Or:

The storage type \({\mathit{zt}}_{\mathit{u{\kern-0.1em\scriptstyle 1}}}\) is equal to \({\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}}_1\).

The storage type \({\mathit{zt}}_{\mathit{u{\kern-0.1em\scriptstyle 2}}}\) is equal to \({\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}}_2\).

The packed type \({\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}}_1\) matches the packed type \({\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}}_2\).

Todo

below is the official specification

A storage type \(\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_1\) matches a type \(\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2\) if and only if:

Either \(\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_1\) is a value type \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_1\) and \(\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2\) is a value type \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_2\) and \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_1\) matches \(\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}_2\).
Or \(\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_1\) is a packed type \(\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}_1\) and \(\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2\) is a packed type \(\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}_2\) and \(\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}_1\) matches \(\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}_2\).

The packed type \({\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}}\) matches itself.

Todo

below is the official specification

A packed type \({\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}}_1\) matches a type \({\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}}_2\) if and only if:

The packed type \(\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}_1\) is the same as \(\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}_2\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ }{ C \href{../valid/matching.html#match-packtype}{\vdash} {\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}} \href{../valid/matching.html#match-packtype}{\leq} {\href{../syntax/types.html#syntax-packtype}{\mathit{packtype}}} } \qquad \end{array}\]

Defined Types¶

The defined type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1\) matches the defined type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2\) if:

Either:

The defined type \({{\href{../valid/conventions.html#aux-clostype}{\mathsf{clos}}}}_{C}({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1)\) is equal to \({{\href{../valid/conventions.html#aux-clostype}{\mathsf{clos}}}}_{C}({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2)\).

Or:

The sub type \({\href{../valid/conventions.html#aux-unroll-deftype}{\mathrm{unroll}}}({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1)\) is equal to \((\href{../syntax/types.html#syntax-subtype}{\mathsf{sub}}~{\href{../syntax/types.html#syntax-subtype}{\mathsf{final}}^?}~{{\href{../syntax/types.html#syntax-typeuse}{\mathit{typeuse}}}^\ast}~{\mathit{ct}})\).

\({|{{\href{../syntax/types.html#syntax-typeuse}{\mathit{typeuse}}}^\ast}|}\) is greater than \(i\).

The type use \({{\href{../syntax/types.html#syntax-typeuse}{\mathit{typeuse}}}^\ast}{}[i]\) matches the defined type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2\).

Todo

below is the official specification

A defined type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1\) matches a type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2\) if and only if:

Either \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1\) and \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2\) are equal when closed under context \(C\).
Or:
- Let the sub type \(\href{../syntax/types.html#syntax-subtype}{\mathsf{sub}}~\href{../syntax/types.html#syntax-subtype}{\mathsf{final}}^?~\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}^\ast~\href{../syntax/types.html#syntax-comptype}{\mathit{comptype}}\) be the result of unrolling \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1\).
- Then there must exist a heap type \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_i\) in \(\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}^\ast\) that matches \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2\).

\[\begin{split}\begin{array}{@{}c@{}}\displaystyle \frac{ {{\href{../valid/conventions.html#aux-clostype}{\mathsf{clos}}}}_{C}({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1) = {{\href{../valid/conventions.html#aux-clostype}{\mathsf{clos}}}}_{C}({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2) }{ C \href{../valid/matching.html#match-deftype}{\vdash} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1 \href{../valid/matching.html#match-deftype}{\leq} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2 } \\[3ex]\displaystyle \frac{ {\href{../valid/conventions.html#aux-unroll-deftype}{\mathrm{unroll}}}({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1) = \href{../syntax/types.html#syntax-subtype}{\mathsf{sub}}~{\href{../syntax/types.html#syntax-subtype}{\mathsf{final}}^?}~{{\href{../syntax/types.html#syntax-typeuse}{\mathit{typeuse}}}^\ast}~{\mathit{ct}} \qquad C \href{../valid/matching.html#match-heaptype}{\vdash} {{\href{../syntax/types.html#syntax-typeuse}{\mathit{typeuse}}}^\ast}{}[i] \href{../valid/matching.html#match-heaptype}{\leq} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2 }{ C \href{../valid/matching.html#match-deftype}{\vdash} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1 \href{../valid/matching.html#match-deftype}{\leq} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2 } \qquad \end{array}\end{split}\]

Note

Note that there is no explicit definition of type equivalence, since it coincides with syntactic equality, as used in the premise of the former rule above.

Limits¶

The limits \(({}[~n_1~\href{../syntax/types.html#syntax-limits}{\,{..}\,}~m_1~])\) matches the limits \(({}[~n_2~\href{../syntax/types.html#syntax-limits}{\,{..}\,}~m_2~])\) if:

\(n_1\) is greater than or equal to \(n_2\).

\(m_1\) is less than or equal to \(m_2\).

Todo

below is the official specification

Limits \({}[ n_1 \href{../syntax/types.html#syntax-limits}{\,{..}\,} m_1 ]\) match limits \({}[ n_2 \href{../syntax/types.html#syntax-limits}{\,{..}\,} m_2 ]\) if and only if:

\(n_1\) is larger than or equal to \(n_2\).
Either:
- \(m_2^?\) is empty.
Or:
- Both \(m_1^?\) and \(m_2^?\) are non-empty.
- \(m_1\) is smaller than or equal to \(m_2\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ n_1 \geq n_2 \qquad m_1 \leq m_2 }{ C \href{../valid/matching.html#match-limits}{\vdash} {}[ n_1 \href{../syntax/types.html#syntax-limits}{\,{..}\,} m_1 ] \href{../valid/matching.html#match-limits}{\leq} {}[ n_2 \href{../syntax/types.html#syntax-limits}{\,{..}\,} m_2 ] } \qquad \end{array}\]

Table Types¶

The table type \(({\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}, {\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_1, {\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_1)\) matches the table type \(({\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}, {\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_2, {\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_2)\) if:

The limits \({\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_1\) matches the limits \({\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_2\).

The reference type \({\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_1\) matches the reference type \({\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_2\).

The reference type \({\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_2\) matches the reference type \({\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_1\).

Todo

below is the official specification

A table type \(({\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}_1~{\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_1~{\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_1)\) matches \(({\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}_2~{\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_1~{\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_1)\) if and only if:

Address types \(\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}_1\) and \(\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}_2\) are the same.
Limits \(\href{../syntax/types.html#syntax-limits}{\mathit{limits}}_1\) match \(\href{../syntax/types.html#syntax-limits}{\mathit{limits}}_2\).
The reference type \(\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}_1\) matches \(\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}_2\), and vice versa.

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-limits}{\vdash} {\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_1 \href{../valid/matching.html#match-limits}{\leq} {\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_2 \qquad C \href{../valid/matching.html#match-reftype}{\vdash} {\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_1 \href{../valid/matching.html#match-reftype}{\leq} {\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_2 \qquad C \href{../valid/matching.html#match-reftype}{\vdash} {\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_2 \href{../valid/matching.html#match-reftype}{\leq} {\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_1 }{ C \href{../valid/matching.html#match-tabletype}{\vdash} {\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}~{\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_1~{\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_1 \href{../valid/matching.html#match-tabletype}{\leq} {\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}~{\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_2~{\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}}_2 } \qquad \end{array}\]

Memory Types¶

The memory type \(({\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}~{\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_1~\href{../syntax/types.html#syntax-memtype}{\mathsf{page}})\) matches the memory type \(({\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}~{\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_2~\href{../syntax/types.html#syntax-memtype}{\mathsf{page}})\) if:

The limits \({\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_1\) matches the limits \({\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_2\).

Todo

below is the official specification

A memory type \(({\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}_1~{\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_1~\href{../syntax/types.html#syntax-memtype}{\mathsf{page}})\) matches \(({\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}_2~{\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_2~\href{../syntax/types.html#syntax-memtype}{\mathsf{page}})\) if and only if:

Address types \(\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}_1\) and \(\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}_2\) are the same.
Limits \(\href{../syntax/types.html#syntax-limits}{\mathit{limits}}_1\) match \(\href{../syntax/types.html#syntax-limits}{\mathit{limits}}_2\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-limits}{\vdash} {\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_1 \href{../valid/matching.html#match-limits}{\leq} {\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_2 }{ C \href{../valid/matching.html#match-memtype}{\vdash} {\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}~{\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_1~\href{../syntax/types.html#syntax-memtype}{\mathsf{page}} \href{../valid/matching.html#match-memtype}{\leq} {\href{../syntax/types.html#syntax-addrtype}{\mathit{addrtype}}}~{\href{../syntax/types.html#syntax-limits}{\mathit{limits}}}_2~\href{../syntax/types.html#syntax-memtype}{\mathsf{page}} } \qquad \end{array}\]

Global Types¶

The global type \(((\href{../syntax/types.html#syntax-mut}{\mathsf{mut}}~{{\mathit{u{\kern-0.1em\scriptstyle 1}}}^?}~{}^?), {\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1)\) matches the global type \(((\href{../syntax/types.html#syntax-mut}{\mathsf{mut}}~{{\mathit{u{\kern-0.1em\scriptstyle 2}}}^?}~{}^?), {\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2)\) if:

The value type \({\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1\) matches the value type \({\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2\).

Either:

\({{\mathit{u{\kern-0.1em\scriptstyle 1}}}^?}\) is absent.

\({{\mathit{u{\kern-0.1em\scriptstyle 2}}}^?}\) is absent.

Or:

\({{\mathit{u{\kern-0.1em\scriptstyle 1}}}^?}\) is equal to \(()\).

\({{\mathit{u{\kern-0.1em\scriptstyle 2}}}^?}\) is equal to \(()\).

The value type \({\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2\) matches the value type \({\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1\).

Todo

below is the official specification

A global type \(({\href{../syntax/types.html#syntax-mut}{\mathsf{mut}}}{{{}_{1}^?}}~{\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1)\) matches \(({\href{../syntax/types.html#syntax-mut}{\mathsf{mut}}}{{{}_{2}^?}}~{\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2)\) if and only if:

Either both \(\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_1\) and \(\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_2\) are \(\href{../syntax/types.html#syntax-mut}{\mathsf{var}}\) and \(t_1\) matches \(t_2\) and vice versa.
Or both \(\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_1\) and \(\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_2\) are \(\href{../syntax/types.html#syntax-mut}{\mathsf{const}}\) and \(t_1\) matches \(t_2\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-valtype}{\vdash} {\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1 \href{../valid/matching.html#match-valtype}{\leq} {\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2 }{ C \href{../valid/matching.html#match-globaltype}{\vdash} {\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1 \href{../valid/matching.html#match-globaltype}{\leq} {\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2 } \qquad \frac{ C \href{../valid/matching.html#match-valtype}{\vdash} {\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1 \href{../valid/matching.html#match-valtype}{\leq} {\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2 \qquad C \href{../valid/matching.html#match-valtype}{\vdash} {\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2 \href{../valid/matching.html#match-valtype}{\leq} {\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1 }{ C \href{../valid/matching.html#match-globaltype}{\vdash} \href{../syntax/types.html#syntax-mut}{\mathsf{mut}}~{\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_1 \href{../valid/matching.html#match-globaltype}{\leq} \href{../syntax/types.html#syntax-mut}{\mathsf{mut}}~{\href{../syntax/types.html#syntax-valtype}{\mathit{valtype}}}_2 } \qquad \end{array}\]

Tag Types¶

The tag type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1\) matches the tag type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2\) if:

The tag type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1\) matches the tag type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2\).

The tag type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2\) matches the tag type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1\).

Todo

below is the official specification

A tag type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1\) matches \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2\) if and only if:

Defined type \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1\) matches \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2\) and vice versa.

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-deftype}{\vdash} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1 \href{../valid/matching.html#match-deftype}{\leq} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2 \qquad C \href{../valid/matching.html#match-deftype}{\vdash} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2 \href{../valid/matching.html#match-deftype}{\leq} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1 }{ C \href{../valid/matching.html#match-tagtype}{\vdash} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1 \href{../valid/matching.html#match-tagtype}{\leq} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2 } \qquad \end{array}\]

External Types¶

Functions¶

The external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~{\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1)\) matches the external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~{\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2)\) if:

The defined type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1\) matches the defined type \({\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2\).

Todo

below is the official specification

An external type \(\href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~{\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1\) matches \(\href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~{\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2\) if and only if:

The defined type \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1\) matches \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-deftype}{\vdash} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1 \href{../valid/matching.html#match-deftype}{\leq} {\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2 }{ C \href{../valid/matching.html#match-externtype}{\vdash} \href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~{\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_1 \href{../valid/matching.html#match-externtype}{\leq} \href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~{\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}}_2 } \qquad \end{array}\]

Tables¶

The external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~{\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}_1)\) matches the external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~{\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}_2)\) if:

The table type \({\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}_1\) matches the table type \({\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}_2\).

Todo

below is the official specification

An external type \(\href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~{\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}_1\) matches \(\href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~{\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}_2\) if and only if:

Table type \(\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}_1\) matches \(\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}_2\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-tabletype}{\vdash} {\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}_1 \href{../valid/matching.html#match-tabletype}{\leq} {\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}_2 }{ C \href{../valid/matching.html#match-externtype}{\vdash} \href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~{\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}_1 \href{../valid/matching.html#match-externtype}{\leq} \href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~{\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}}_2 } \qquad \end{array}\]

Memories¶

The external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~{\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}_1)\) matches the external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~{\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}_2)\) if:

The memory type \({\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}_1\) matches the memory type \({\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}_2\).

Todo

below is the official specification

An external type \(\href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~{\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}_1\) matches \(\href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~{\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}_2\) if and only if:

Memory type \(\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}_1\) matches \(\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}_2\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-memtype}{\vdash} {\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}_1 \href{../valid/matching.html#match-memtype}{\leq} {\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}_2 }{ C \href{../valid/matching.html#match-externtype}{\vdash} \href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~{\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}_1 \href{../valid/matching.html#match-externtype}{\leq} \href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~{\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}}_2 } \qquad \end{array}\]

Globals¶

The external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~{\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}_1)\) matches the external type \((\href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~{\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}_2)\) if:

The global type \({\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}_1\) matches the global type \({\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}_2\).

Todo

below is the official specification

An external type \(\href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~{\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}_1\) matches \(\href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~{\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}_2\) if and only if:

Global type \(\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}_1\) matches \(\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}_2\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-globaltype}{\vdash} {\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}_1 \href{../valid/matching.html#match-globaltype}{\leq} {\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}_2 }{ C \href{../valid/matching.html#match-externtype}{\vdash} \href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~{\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}_1 \href{../valid/matching.html#match-externtype}{\leq} \href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~{\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}}_2 } \qquad \end{array}\]

Tags¶

The external type \((\href{../syntax/types.html#syntax-tagtype}{\mathsf{tag}}~{\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}}_1)\) matches the external type \((\href{../syntax/types.html#syntax-tagtype}{\mathsf{tag}}~{\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}}_2)\) if:

The tag type \({\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}}_1\) matches the tag type \({\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}}_2\).

Todo

below is the official specification

An external type \(\href{../syntax/types.html#syntax-tagtype}{\mathsf{tag}}~{\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}}_1\) matches \(\href{../syntax/types.html#syntax-tagtype}{\mathsf{tag}}~{\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}}_2\) if and only if:

Tag type \(\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}_1\) matches \(\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}_2\).

\[\begin{array}{@{}c@{}}\displaystyle \frac{ C \href{../valid/matching.html#match-tagtype}{\vdash} {\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}}_1 \href{../valid/matching.html#match-tagtype}{\leq} {\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}}_2 }{ C \href{../valid/matching.html#match-externtype}{\vdash} \href{../syntax/types.html#syntax-tagtype}{\mathsf{tag}}~{\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}}_1 \href{../valid/matching.html#match-externtype}{\leq} \href{../syntax/types.html#syntax-tagtype}{\mathsf{tag}}~{\href{../syntax/types.html#syntax-tagtype}{\mathit{tagtype}}}_2 } \qquad \end{array}\]