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89 lines
4.1 KiB
TeX
89 lines
4.1 KiB
TeX
\documentclass[../generics]{subfiles}
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\begin{document}
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\chapter{Derived Requirements Summary}\label{derived summary}
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\index{$\vdash$}
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Let $G$ be a \index{generic signature!summary}generic signature. We generate the theory of~$G$ by repeated application of inference rules, starting from a finite set of elementary statements. A derivation proves that some element belongs to this set by giving a list of derivation steps where the assumptions in each step are conclusions of previous steps. Nomenclature:
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\begin{center}
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\begin{tabular}{ll}
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\toprule
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\textbf{Symbol}&\textbf{Description}\\
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\midrule
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\texttt{T}, \texttt{U}, and \texttt{V}&\index{type parameter!summary}type parameters\\
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\texttt{Self.U} and \texttt{Self.V}&type parameters rooted in the \index{protocol Self type!summary}protocol \texttt{Self} type\\
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\texttt{X}&a concrete type\\
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\texttt{C}&a concrete \index{class type!summary}class type\\
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$\Xprime$ and $\Cprime$&obtained from \texttt{X} and \texttt{C} by replacing \texttt{Self} with \texttt{T}\\
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\texttt{P} and \texttt{Q}&protocols\\
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\texttt{A}&the name of an \index{associated type declaration!summary}associated type of \texttt{P}\\
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\texttt{[P]A}&an associated type declaration of \texttt{P}\\
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\texttt{T.[P]A} and \texttt{T.A}&\index{bound dependent member type!summary}bound and \index{unbound dependent member type!summary}unbound dependent member type\\
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$\ConfReq{T}{P}$&a \index{conformance requirement!summary}conformance requirement\\
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$\SameReq{T}{U}$&a \index{same-type requirement!summary}same-type requirement between type parameters\\
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$\SameReq{T}{X}$&a concrete same-type requirement\\
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$\ConfReq{T}{C}$&a \index{superclass requirement!summary}superclass requirement\\
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$\ConfReq{T}{AnyObject}$&a \index{layout requirement!summary}layout requirement\\
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$\ConfReq{Self.U}{Q}_\texttt{P}$&an \index{associated requirement!summary}associated requirement of protocol \texttt{P}\\
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\bottomrule
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\end{tabular}
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\end{center}
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See \index{derived requirement!summary}\index{valid type parameter!summary}\SecRef{derived req} and \SecRef{type params} for details.
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\index{elementary derivation step!summary}\paragraph{Elementary statements.}
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For \IndexStepTwo{Generic}{summary}each generic parameter \ttgp{d}{i} of~$G$:
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\begin{gather*}
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\GenericStepDef
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\end{gather*}
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For \IndexStepTwo{Conf}{summary}\IndexStepTwo{Same}{summary}\IndexStepTwo{Concrete}{summary}\IndexStepTwo{Super}{summary}\IndexStepTwo{Layout}{summary}each explicit requirement of~$G$ by \index{requirement kind!summary}kind:
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\begin{gather*}
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\ConfStepDef\\
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\SameStepDef\\
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\ConcreteStepDef\\
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\SuperStepDef\\
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\LayoutStepDef
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\end{gather*}
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\index{requirement signature!summary}
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\paragraph{Requirement signatures.}
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Assume $G\vdash\ConfReq{T}{P}$. For \IndexStepTwo{AssocName}{summary}\IndexStepTwo{AssocDecl}{summary}\IndexStepTwo{AssocBind}{summary}each associated type~\texttt{A} of~\texttt{P}:
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\begin{gather*}
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\AssocNameStepDef\\
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\AssocDeclStepDef\\
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\AssocBindStepDef
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\end{gather*}
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For each \IndexStepTwo{AssocConf}{summary}\IndexStepTwo{AssocSame}{summary}\IndexStepTwo{AssocConcrete}{summary}\IndexStepTwo{AssocSuper}{summary}\IndexStepTwo{AssocLayout}{summary}associated requirement of~\texttt{P} by kind:
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\begin{gather*}
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\AssocConfStepDef\\
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\AssocSameStepDef\\
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\AssocConcreteStepDef\\
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\AssocSuperStepDef\\
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\AssocLayoutStepDef
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\end{gather*}
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\paragraph{Equivalence.}
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Same-type requirements \IndexStepTwo{Reflex}{summary}\IndexStepTwo{Sym}{summary}\IndexStepTwo{Trans}{summary}generate an equivalence relation:
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\begin{gather*}
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\ReflexStepDef\\
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\SymStepDef\\
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\TransStepDef
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\end{gather*}
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\paragraph{Compatibility.}
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A derived conformance, superclass or layout requirement \IndexStepTwo{SameConf}{summary}\IndexStepTwo{SameSuper}{summary}\IndexStepTwo{SameLayout}{summary}\IndexStepTwo{SameName}{summary}\IndexStepTwo{SameDecl}{summary}applies to all type parameters in an equivalence class:
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\begin{gather*}
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\SameConfStepDef\\
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\SameConcreteStepDef\\
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\SameSuperStepDef\\
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\SameLayoutStepDef
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\end{gather*}
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If two type parameters are equivalent, so are \IndexStepTwo{SameName}{summary}\IndexStepTwo{SameDecl}{summary}all corresponding member types:
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\begin{gather*}
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\SameNameStepDef\\
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\SameDeclStepDef
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\end{gather*}
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\end{document}
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