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

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.

\[\begin{split}~\\[-1ex] \frac{ }{ C \href{../valid/matching.html#match-numtype}{\vdash} \href{../syntax/types.html#syntax-numtype}{\mathit{numtype}} \href{../valid/matching.html#match-numtype}{\leq} \href{../syntax/types.html#syntax-numtype}{\mathit{numtype}} }\end{split}\]

Vector Types

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.

\[\begin{split}~\\[-1ex] \frac{ }{ C \href{../valid/matching.html#match-vectype}{\vdash} \href{../syntax/types.html#syntax-vectype}{\mathit{vectype}} \href{../valid/matching.html#match-vectype}{\leq} \href{../syntax/types.html#syntax-vectype}{\mathit{vectype}} }\end{split}\]

Heap Types

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 \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{i31}}\), \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{struct}}\), 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 \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{func}}\).

  • 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 \(\href{../syntax/types.html#syntax-heaptype}{\mathsf{func}}\).

  • 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 \(\href{../valid/conventions.html#syntax-heaptype-ext}{\mathsf{bot}}\).

\[\begin{split}~\\[-1ex] \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}} } \qquad \frac{ C \href{../valid/types.html#valid-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}' \mathrel{\mbox{ok}} \qquad C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1 \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}' \qquad C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}' \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2 }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1 \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2 }\end{split}\]
\[\begin{split}~\\[-1ex] \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{any}} } \qquad \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{i31}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}} } \qquad \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{struct}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}} } \qquad \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{array}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{eq}} }\end{split}\]
\[\begin{split}~\\[-1ex] \frac{ \href{../valid/conventions.html#aux-expand-deftype}{\mathrm{expand}}(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}) = \href{../syntax/types.html#syntax-comptype}{\mathsf{struct}}~\mathit{st} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{struct}} } \qquad \frac{ \href{../valid/conventions.html#aux-expand-deftype}{\mathrm{expand}}(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}) = \href{../syntax/types.html#syntax-comptype}{\mathsf{array}}~\mathit{at} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{array}} } \qquad \frac{ \href{../valid/conventions.html#aux-expand-deftype}{\mathrm{expand}}(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}) = \href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~\mathit{ft} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{func}} }\end{split}\]
\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} C.\href{../valid/conventions.html#context}{\mathsf{types}}[\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}_1] \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2 }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}_1 \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2 } \qquad \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1 \href{../valid/matching.html#match-heaptype}{\leq} C.\href{../valid/conventions.html#context}{\mathsf{types}}[\href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}_2] }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1 \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/modules.html#syntax-typeidx}{\mathit{typeidx}}_2 }\end{split}\]
\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} \mathit{ht} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{any}} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{none}} \href{../valid/matching.html#match-heaptype}{\leq} \mathit{ht} } \qquad \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} \mathit{ht} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{func}} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{nofunc}} \href{../valid/matching.html#match-heaptype}{\leq} \mathit{ht} } \qquad \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} \mathit{ht} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathsf{extern}} }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathsf{noextern}} \href{../valid/matching.html#match-heaptype}{\leq} \mathit{ht} }\end{split}\]
\[\begin{split}~\\[-1ex] \frac{ }{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../valid/conventions.html#syntax-heaptype-ext}{\mathsf{bot}} \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}} }\end{split}\]

Reference Types

A reference type \(\href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}_1^?~heaptype_1\) matches a reference type \(\href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}_2^?~heaptype_2\) 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.

\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1 \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2 }{ C \href{../valid/matching.html#match-reftype}{\vdash} \href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1 \href{../valid/matching.html#match-reftype}{\leq} \href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2 } \qquad \frac{ C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1 \href{../valid/matching.html#match-heaptype}{\leq} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2 }{ C \href{../valid/matching.html#match-reftype}{\vdash} \href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}^?~\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_1 \href{../valid/matching.html#match-reftype}{\leq} \href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}~\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}_2 }\end{split}\]

Value Types

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 \(\href{../valid/conventions.html#syntax-valtype-ext}{\mathsf{bot}}\).

\[\begin{split}~\\[-1ex] \frac{ }{ C \href{../valid/matching.html#match-valtype}{\vdash} \href{../valid/conventions.html#syntax-valtype-ext}{\mathsf{bot}} \href{../valid/matching.html#match-valtype}{\leq} \href{../syntax/types.html#syntax-valtype}{\mathit{valtype}} }\end{split}\]

Result Types

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:

\[\begin{split}~\\[-1ex] \frac{ (C \href{../valid/matching.html#match-valtype}{\vdash} t_1 \href{../valid/matching.html#match-valtype}{\leq} t_2)^\ast }{ C \href{../valid/matching.html#match-resulttype}{\vdash} [t_1^\ast] \href{../valid/matching.html#match-resulttype}{\leq} [t_2^\ast] }\end{split}\]

Instruction Types

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 \(t_x\).

\[\begin{split}~\\[-1ex] \frac{ \begin{array}{@{}c@{\qquad}l@{}} C \href{../valid/matching.html#match-resulttype}{\vdash} [t_{21}^\ast] \href{../valid/matching.html#match-resulttype}{\leq} [t_{11}^\ast] & \{ x^\ast \} = \{ x_2^\ast \} \setminus \{ x_1^\ast \} \\ C \href{../valid/matching.html#match-resulttype}{\vdash} [t_{12}^\ast] \href{../valid/matching.html#match-resulttype}{\leq} [t_{22}^\ast] & (C.\href{../valid/conventions.html#context}{\mathsf{locals}}[x] = \href{../valid/conventions.html#syntax-init}{\mathsf{set}}~t_x)^\ast \end{array} }{ C \href{../valid/matching.html#match-instrtype}{\vdash} [t_{11}^\ast] \mathrel{\href{../valid/conventions.html#syntax-instrtype}{\rightarrow}}_{x_1^\ast} [t_{12}^\ast] \href{../valid/matching.html#match-instrtype}{\leq} [t^\ast~t_{21}^\ast] \mathrel{\href{../valid/conventions.html#syntax-instrtype}{\rightarrow}}_{x_2^\ast} [t^\ast~t_{22}^\ast] }\end{split}\]

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

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:

\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-resulttype}{\vdash} [t_{21}^\ast] \href{../valid/matching.html#match-resulttype}{\leq} [t_{11}^\ast] \qquad C \href{../valid/matching.html#match-resulttype}{\vdash} [t_{12}^\ast] \href{../valid/matching.html#match-resulttype}{\leq} [t_{22}^\ast] }{ C \href{../valid/matching.html#match-functype}{\vdash} [t_{11}^\ast] \href{../syntax/types.html#syntax-functype}{\rightarrow} [t_{12}^\ast] \href{../valid/matching.html#match-functype}{\leq} [t_{21}^\ast] \href{../syntax/types.html#syntax-functype}{\rightarrow} [t_{22}^\ast] }\end{split}\]

Composite Types

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 \(\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_2\).

\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-functype}{\vdash} \href{../syntax/types.html#syntax-functype}{\mathit{functype}}_1 \href{../valid/matching.html#match-functype}{\leq} \href{../syntax/types.html#syntax-functype}{\mathit{functype}}_2 }{ C \href{../valid/matching.html#match-comptype}{\vdash} \href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~\href{../syntax/types.html#syntax-functype}{\mathit{functype}}_1 \href{../valid/matching.html#match-comptype}{\leq} \href{../syntax/types.html#syntax-comptype}{\mathsf{func}}~\href{../syntax/types.html#syntax-functype}{\mathit{functype}}_2 }\end{split}\]
\[\begin{split}~\\[-1ex] \frac{ (C \href{../valid/matching.html#match-fieldtype}{\vdash} \href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_1 \href{../valid/matching.html#match-fieldtype}{\leq} \href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_2)^\ast }{ C \href{../valid/matching.html#match-comptype}{\vdash} \href{../syntax/types.html#syntax-comptype}{\mathsf{struct}}~\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_1^\ast~{\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}'}_1^\ast \href{../valid/matching.html#match-comptype}{\leq} \href{../syntax/types.html#syntax-comptype}{\mathsf{struct}}~\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_2^\ast }\end{split}\]
\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-fieldtype}{\vdash} \href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_1 \href{../valid/matching.html#match-fieldtype}{\leq} \href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_2 }{ C \href{../valid/matching.html#match-comptype}{\vdash} \href{../syntax/types.html#syntax-comptype}{\mathsf{array}}~\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_1 \href{../valid/matching.html#match-comptype}{\leq} \href{../syntax/types.html#syntax-comptype}{\mathsf{array}}~\href{../syntax/types.html#syntax-fieldtype}{\mathit{fieldtype}}_2 }\end{split}\]

Field Types

A field type \(\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_1~\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_1\) matches a type \(\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_2~\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2\) 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.

\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-storagetype}{\vdash} \href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_1 \href{../valid/matching.html#match-storagetype}{\leq} \href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2 }{ C \href{../valid/matching.html#match-fieldtype}{\vdash} \href{../syntax/types.html#syntax-mut}{\mathsf{const}}~\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_1 \href{../valid/matching.html#match-comptype}{\leq} \href{../syntax/types.html#syntax-mut}{\mathsf{const}}~\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2 } \qquad \frac{ \begin{array}[b]{@{}c@{}} C \href{../valid/matching.html#match-storagetype}{\vdash} \href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_1 \href{../valid/matching.html#match-storagetype}{\leq} \href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2 \\ C \href{../valid/matching.html#match-storagetype}{\vdash} \href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2 \href{../valid/matching.html#match-storagetype}{\leq} \href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_1 \end{array} }{ C \href{../valid/matching.html#match-fieldtype}{\vdash} \href{../syntax/types.html#syntax-mut}{\mathsf{var}}~\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_1 \href{../valid/matching.html#match-comptype}{\leq} \href{../syntax/types.html#syntax-mut}{\mathsf{var}}~\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2 }\end{split}\]

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 \(\href{../syntax/types.html#syntax-packedtype}{\mathit{packedtype}}_1\) and \(\href{../syntax/types.html#syntax-storagetype}{\mathit{storagetype}}_2\) is a packed type \(\href{../syntax/types.html#syntax-packedtype}{\mathit{packedtype}}_2\) and \(\href{../syntax/types.html#syntax-packedtype}{\mathit{packedtype}}_1\) matches \(\href{../syntax/types.html#syntax-packedtype}{\mathit{packedtype}}_2\).

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

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

\[\begin{split}~\\[-1ex] \frac{ }{ C \href{../valid/matching.html#match-packedtype}{\vdash} \href{../syntax/types.html#syntax-packedtype}{\mathit{packedtype}} \href{../valid/matching.html#match-packedtype}{\leq} \href{../syntax/types.html#syntax-packedtype}{\mathit{packedtype}} }\end{split}\]

Defined Types

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 \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2\).

\[\begin{split}~\\[-1ex] \frac{ \href{../valid/conventions.html#aux-clostype}{\mathsf{clos}}_C(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1) = \href{../valid/conventions.html#aux-clostype}{\mathsf{clos}}_C(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2) }{ C \href{../valid/matching.html#match-deftype}{\vdash} \href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1 \href{../valid/matching.html#match-deftype}{\leq} \href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2 }\end{split}\]
\[\begin{split}~\\[-1ex] \frac{ \href{../valid/conventions.html#aux-unroll-deftype}{\mathrm{unroll}}(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1) = \href{../syntax/types.html#syntax-subtype}{\mathsf{sub}}~\href{../syntax/types.html#syntax-subtype}{\mathsf{final}}^?~\href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}^\ast~\href{../syntax/types.html#syntax-comptype}{\mathit{comptype}} \qquad C \href{../valid/matching.html#match-heaptype}{\vdash} \href{../syntax/types.html#syntax-heaptype}{\mathit{heaptype}}^\ast[i] \href{../valid/matching.html#match-heaptype}{\leq} \href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2 }{ C \href{../valid/matching.html#match-deftype}{\vdash} \href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1 \href{../valid/matching.html#match-deftype}{\leq} \href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2 }\end{split}\]

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

Limits \(\{ \href{../syntax/types.html#syntax-limits}{\mathsf{min}}~n_1, \href{../syntax/types.html#syntax-limits}{\mathsf{max}}~m_1^? \}\) match limits \(\{ \href{../syntax/types.html#syntax-limits}{\mathsf{min}}~n_2, \href{../syntax/types.html#syntax-limits}{\mathsf{max}}~m_2^? \}\) 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 \(m_2\).

\[\begin{split}~\\[-1ex] \frac{ n_1 \geq n_2 }{ C \href{../valid/matching.html#match-limits}{\vdash} \{ \href{../syntax/types.html#syntax-limits}{\mathsf{min}}~n_1, \href{../syntax/types.html#syntax-limits}{\mathsf{max}}~m_1^? \} \href{../valid/matching.html#match-limits}{\leq} \{ \href{../syntax/types.html#syntax-limits}{\mathsf{min}}~n_2, \href{../syntax/types.html#syntax-limits}{\mathsf{max}}~\epsilon \} } \quad \frac{ n_1 \geq n_2 \qquad m_1 \leq m_2 }{ C \href{../valid/matching.html#match-limits}{\vdash} \{ \href{../syntax/types.html#syntax-limits}{\mathsf{min}}~n_1, \href{../syntax/types.html#syntax-limits}{\mathsf{max}}~m_1 \} \href{../valid/matching.html#match-limits}{\leq} \{ \href{../syntax/types.html#syntax-limits}{\mathsf{min}}~n_2, \href{../syntax/types.html#syntax-limits}{\mathsf{max}}~m_2 \} }\end{split}\]

Table Types

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

  • 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.

\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-limits}{\vdash} \href{../syntax/types.html#syntax-limits}{\mathit{limits}}_1 \href{../valid/matching.html#match-limits}{\leq} \href{../syntax/types.html#syntax-limits}{\mathit{limits}}_2 \qquad C \href{../valid/matching.html#match-reftype}{\vdash} \href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}_1 \href{../valid/matching.html#match-reftype}{\leq} \href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}_2 \qquad C \href{../valid/matching.html#match-reftype}{\vdash} \href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}_2 \href{../valid/matching.html#match-reftype}{\leq} \href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}_1 }{ C \href{../valid/matching.html#match-tabletype}{\vdash} \href{../syntax/types.html#syntax-limits}{\mathit{limits}}_1~\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}_1 \href{../valid/matching.html#match-tabletype}{\leq} \href{../syntax/types.html#syntax-limits}{\mathit{limits}}_2~\href{../syntax/types.html#syntax-reftype}{\mathit{reftype}}_2 }\end{split}\]

Memory Types

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

  • Limits \(\href{../syntax/types.html#syntax-limits}{\mathit{limits}}_1\) match \(\href{../syntax/types.html#syntax-limits}{\mathit{limits}}_2\).

\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-limits}{\vdash} \href{../syntax/types.html#syntax-limits}{\mathit{limits}}_1 \href{../valid/matching.html#match-limits}{\leq} \href{../syntax/types.html#syntax-limits}{\mathit{limits}}_2 }{ C \href{../valid/matching.html#match-memtype}{\vdash} \href{../syntax/types.html#syntax-limits}{\mathit{limits}}_1 \href{../valid/matching.html#match-memtype}{\leq} \href{../syntax/types.html#syntax-limits}{\mathit{limits}}_2 }\end{split}\]

Global Types

A global type \((\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_1~t_1)\) matches \((\href{../syntax/types.html#syntax-mut}{\mathit{mut}}_2~t_2)\) 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 \(t_2\).

\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-valtype}{\vdash} t_1 \href{../valid/matching.html#match-valtype}{\leq} t_2 \qquad C \href{../valid/matching.html#match-valtype}{\vdash} t_2 \href{../valid/matching.html#match-valtype}{\leq} t_1 }{ C \href{../valid/matching.html#match-globaltype}{\vdash} \href{../syntax/types.html#syntax-mut}{\mathsf{var}}~t_1 \href{../valid/matching.html#match-globaltype}{\leq} \href{../syntax/types.html#syntax-mut}{\mathsf{var}}~t_2 } \qquad \frac{ C \href{../valid/matching.html#match-valtype}{\vdash} t_1 \href{../valid/matching.html#match-valtype}{\leq} t_2 }{ C \href{../valid/matching.html#match-globaltype}{\vdash} \href{../syntax/types.html#syntax-mut}{\mathsf{const}}~t_1 \href{../valid/matching.html#match-globaltype}{\leq} \href{../syntax/types.html#syntax-mut}{\mathsf{const}}~t_2 }\end{split}\]

External Types

Functions

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 \(\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2\).

\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-deftype}{\vdash} \href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1 \href{../valid/matching.html#match-functype}{\leq} \href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2 }{ C \href{../valid/matching.html#match-externtype}{\vdash} \href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_1 \href{../valid/matching.html#match-externtype}{\leq} \href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~\href{../valid/conventions.html#syntax-deftype}{\mathit{deftype}}_2 }\end{split}\]

Tables

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

\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-tabletype}{\vdash} \href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}_1 \href{../valid/matching.html#match-tabletype}{\leq} \href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}_2 }{ C \href{../valid/matching.html#match-externtype}{\vdash} \href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}_1 \href{../valid/matching.html#match-externtype}{\leq} \href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~\href{../syntax/types.html#syntax-tabletype}{\mathit{tabletype}}_2 }\end{split}\]

Memories

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

\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-memtype}{\vdash} \href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}_1 \href{../valid/matching.html#match-memtype}{\leq} \href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}_2 }{ C \href{../valid/matching.html#match-externtype}{\vdash} \href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}_1 \href{../valid/matching.html#match-externtype}{\leq} \href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~\href{../syntax/types.html#syntax-memtype}{\mathit{memtype}}_2 }\end{split}\]

Globals

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

\[\begin{split}~\\[-1ex] \frac{ C \href{../valid/matching.html#match-globaltype}{\vdash} \href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}_1 \href{../valid/matching.html#match-globaltype}{\leq} \href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}_2 }{ C \href{../valid/matching.html#match-externtype}{\vdash} \href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}_1 \href{../valid/matching.html#match-externtype}{\leq} \href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~\href{../syntax/types.html#syntax-globaltype}{\mathit{globaltype}}_2 }\end{split}\]