\documentclass[../generics]{subfiles}

\begin{document}

\chapter{Derived Requirements}\label{derived summary}

\index{$\vdash$}

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:
\begin{center}
\begin{tabular}{ll}
\toprule
\textbf{Symbol}&\textbf{Description}\\
\midrule
\tT, \tU, and \tV&\index{type parameter!summary}type parameters\\
\texttt{Self.U} and \texttt{Self.V}&type parameters rooted in the \Index{protocol Self type@protocol \tSelf\ type!summary}protocol \tSelf\ type\\
\texttt{X}&a concrete type\\
\texttt{C}&a concrete \index{class type!summary}class type\\
$\Xprime$ and $\Cprime$&obtained from \texttt{X} and \texttt{C} by replacing \tSelf\ with \tT\\
\tP\ and \tQ&protocols\\
\nA&the name of an \index{associated type declaration!summary}associated type of \tP\\
\texttt{[P]A}&an associated type declaration of \tP\\
\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\\
$\TP$&a \index{conformance requirement!summary}conformance requirement\\
$\TU$&a \index{same-type requirement!summary}same-type requirement between type parameters\\
$\TX$&a concrete same-type requirement\\
$\TC$&a \index{superclass requirement!summary}superclass requirement\\
$\TAnyObject$&a \index{layout requirement!summary}layout requirement\\
$\AssocConfReq{Self.U}{Q}{P}$&an \index{associated requirement!summary}associated requirement of protocol \tP\\
\bottomrule
\end{tabular}
\end{center}
See \index{derived requirement!summary}\index{valid type parameter!summary}\SecRef{derived req} and \SecRef{type params} for details.

\index{elementary derivation step!summary}\paragraph{Elementary statements.}
For \IndexStepTwo{Generic}{summary}each generic parameter \ttgp{d}{i} of~$G$:
\begin{gather*}
\GenericStepDef
\end{gather*}
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:
\begin{gather*}
\ConfStepDef\\
\SameStepDef\\
\ConcreteStepDef\\
\SuperStepDef\\
\LayoutStepDef
\end{gather*}

\index{requirement signature!summary}
\paragraph{Requirement signatures.}
Assume $G\vdash\TP$. For \IndexStepTwo{AssocName}{summary}\IndexStepTwo{AssocDecl}{summary}\IndexStepTwo{AssocBind}{summary}each associated type~\nA\ of~\tP:
\begin{gather*}
\AssocNameStepDef\\
\AssocDeclStepDef\\
\AssocBindStepDef
\end{gather*}
For each \IndexStepTwo{AssocConf}{summary}\IndexStepTwo{AssocSame}{summary}\IndexStepTwo{AssocConcrete}{summary}\IndexStepTwo{AssocSuper}{summary}\IndexStepTwo{AssocLayout}{summary}associated requirement of~\tP\ by kind:
\begin{gather*}
\AssocConfStepDef\\
\AssocSameStepDef\\
\AssocConcreteStepDef\\
\AssocSuperStepDef\\
\AssocLayoutStepDef
\end{gather*}

\paragraph{Equivalence.}
Same-type requirements \IndexStepTwo{Reflex}{summary}\IndexStepTwo{Sym}{summary}\IndexStepTwo{Trans}{summary}generate an equivalence relation:
\begin{gather*}
\ReflexStepDef\\
\SymStepDef\\
\TransStepDef
\end{gather*}

\paragraph{Compatibility.}
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:
\begin{gather*}
\SameConfStepDef\\
\SameConcreteStepDef\\
\SameSuperStepDef\\
\SameLayoutStepDef
\end{gather*}
If two type parameters are equivalent, so are \IndexStepTwo{SameName}{summary}\IndexStepTwo{SameDecl}{summary}all corresponding member types:
\begin{gather*}
\SameNameStepDef\\
\SameDeclStepDef
\end{gather*}

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