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Side by Side Diff: docs/language/dartLangSpec.tex

Issue 1179113010: Add missing clause for LUB(Bottom, T) (Closed) Base URL: git@github.com:dart-lang/sdk.git@master
Patch Set: Created 5 years, 6 months ago
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1 \documentclass{article} 1 \documentclass{article}
2 \usepackage{epsfig} 2 \usepackage{epsfig}
3 \usepackage{color} 3 \usepackage{color}
4 \usepackage{dart} 4 \usepackage{dart}
5 \usepackage{bnf} 5 \usepackage{bnf}
6 \usepackage{hyperref} 6 \usepackage{hyperref}
7 \usepackage{lmodern} 7 \usepackage{lmodern}
8 \newcommand{\code}[1]{{\sf #1}} 8 \newcommand{\code}[1]{{\sf #1}}
9 \title{Dart Programming Language Specification \\ 9 \title{Dart Programming Language Specification \\
10 {\large Version 1.10}} 10 {\large Version 1.10}}
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7503 \LMLabel{leastUpperBounds} 7503 \LMLabel{leastUpperBounds}
7504 7504
7505 \LMHash{} 7505 \LMHash{}
7506 % does this diverge in some cases? 7506 % does this diverge in some cases?
7507 Given two interfaces $I$ and $J$, let $S_I$ be the set of superinterfaces of $I$ , let $S_J$ be the set of superinterfaces of $J$ and let $S = (I \cup S_I) \ca p (J \cup S_J)$. Furthermore, we define $S_n = \{T | T \in S \wedge depth(T) = n\}$ for any finite $n$ %, and $k=max(depth(T_1), \ldots, depth(T_m)), T_i \in S , i \in 1..m$, 7507 Given two interfaces $I$ and $J$, let $S_I$ be the set of superinterfaces of $I$ , let $S_J$ be the set of superinterfaces of $J$ and let $S = (I \cup S_I) \ca p (J \cup S_J)$. Furthermore, we define $S_n = \{T | T \in S \wedge depth(T) = n\}$ for any finite $n$ %, and $k=max(depth(T_1), \ldots, depth(T_m)), T_i \in S , i \in 1..m$,
7508 where $depth(T)$ is the number of steps in the longest inheritance path from $T$ to \code{Object}. Let $q$ be the largest number such that $S_q$ has cardinality one. The least upper bound of $I$ and $J$ is the sole element of $S_q$. 7508 where $depth(T)$ is the number of steps in the longest inheritance path from $T$ to \code{Object}. Let $q$ be the largest number such that $S_q$ has cardinality one. The least upper bound of $I$ and $J$ is the sole element of $S_q$.
7509 7509
7510 \LMHash{} 7510 \LMHash{}
7511 The least upper bound of \DYNAMIC{} and any type $T$ is \DYNAMIC{}. 7511 The least upper bound of \DYNAMIC{} and any type $T$ is \DYNAMIC{}.
7512 The least upper bound of \VOID{} and any type $T \ne \DYNAMIC{}$ is \VOID{}. 7512 The least upper bound of \VOID{} and any type $T \ne \DYNAMIC{}$ is \VOID{}.
7513 Let $U$ be a type variable with upper bound $B$. The least upper bound of $U$ an d a type $T$ is the least upper bound of $B$ and $T$. 7513 The least upper bound of $\bot$ and any type $T$ is $T$.
7514 Let $U$ be a type variable with upper bound $B$. The least upper bound of $U$ an d a type $T \ne \bot$ is the least upper bound of $B$ and $T$.
7514 7515
7515 \LMHash{} 7516 \LMHash{}
7516 The least upper bound relation is symmetric and reflexive. 7517 The least upper bound relation is symmetric and reflexive.
7517 7518
7518 % Function types 7519 % Function types
7519 7520
7520 \LMHash{} 7521 \LMHash{}
7521 The least upper bound of a function type and an interface type $T$ is the least upper bound of \cd{Function} and $T$. 7522 The least upper bound of a function type and an interface type $T$ is the least upper bound of \cd{Function} and $T$.
7522 Let $F$ and $G$ be function types. If $F$ and $G$ differ in their number of requ ired parameters, then the least upper bound of $F$ and $G$ is \cd{Function}. Ot herwise: 7523 Let $F$ and $G$ be function types. If $F$ and $G$ differ in their number of requ ired parameters, then the least upper bound of $F$ and $G$ is \cd{Function}. Ot herwise:
7523 \begin{itemize} 7524 \begin{itemize}
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7845 7846
7846 The invariant that each normative paragraph is associated with a line 7847 The invariant that each normative paragraph is associated with a line
7847 containing the text \LMHash{} should be maintained. Extra occurrences 7848 containing the text \LMHash{} should be maintained. Extra occurrences
7848 of \LMHash{} can be added if needed, e.g., in order to make 7849 of \LMHash{} can be added if needed, e.g., in order to make
7849 individual \item{}s in itemized lists addressable. Each \LM.. command 7850 individual \item{}s in itemized lists addressable. Each \LM.. command
7850 must occur on a separate line. \LMHash{} must occur immediately 7851 must occur on a separate line. \LMHash{} must occur immediately
7851 before the associated paragraph, and \LMLabel must occur immediately 7852 before the associated paragraph, and \LMLabel must occur immediately
7852 after the associated \section{}, \subsection{} etc. 7853 after the associated \section{}, \subsection{} etc.
7853 7854
7854 ---------------------------------------------------------------------- 7855 ----------------------------------------------------------------------
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