From d5c4db23bf30103cda12cc40e18ee8c19e1a2b88 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Sat, 20 Mar 2021 20:23:08 -0500 Subject: [PATCH 001/181] Initial draft Merges in content from PHI 5 (type variables) with latex judgements for bidirectional type elaboration and kind synthesis/analysis. Will follow up with content from the PHI 3 (type aliases) later. --- 9-type-aliases-redux/9-type-aliases-redux.md | 108 +++ 9-type-aliases-redux/latex/.gitignore | 2 + 9-type-aliases-redux/latex/etal.sty | 94 +++ 9-type-aliases-redux/latex/joshuadunfield.sty | 699 ++++++++++++++++++ .../latex/kind-judgements.pdf | Bin 0 -> 140515 bytes .../latex/kind-judgements.tex | 154 ++++ 6 files changed, 1057 insertions(+) create mode 100644 9-type-aliases-redux/9-type-aliases-redux.md create mode 100644 9-type-aliases-redux/latex/.gitignore create mode 100644 9-type-aliases-redux/latex/etal.sty create mode 100644 9-type-aliases-redux/latex/joshuadunfield.sty create mode 100644 9-type-aliases-redux/latex/kind-judgements.pdf create mode 100644 9-type-aliases-redux/latex/kind-judgements.tex diff --git a/9-type-aliases-redux/9-type-aliases-redux.md b/9-type-aliases-redux/9-type-aliases-redux.md new file mode 100644 index 0000000..28903f7 --- /dev/null +++ b/9-type-aliases-redux/9-type-aliases-redux.md @@ -0,0 +1,108 @@ +# Introduction + +Several future Hazel extensions will require support for type variables, a +prerequisite for type aliases including PHI 3 (8?) (Explicit Polymorphism). +This PHI adds support for type variables and type aliases to Hazel. + +TODO: Merge with Type-alias content + +# Type Language +## Type Identifiers +We define a type `TyId.t` as follows: + +``` +type TyId.t = string +``` + +The syntax of type variable identifiers is initially the same as expression +variable identifiers, but this may change in the future. We restrict type +variable identifiers from using keywords in the same way (to support +extraction). + +## Unexpanded Types + +We extend the syntax of `UHTyp.t` as follows: + +``` +type UHTyp.t = ... | TyVar(VarErrStatus.t, TyId.t) +``` + +## Expanded Types + +In expanded types, we will use a de Bruijn indexed representation. This requires +defining a type abbreviation `HTyp.idx` as follows: + +``` +type HTyp.idx = int +``` + +Then, we extend the syntax of `HTyp.t` as follows: + +``` +type HTyp.t = ... + | TyVar(idx, TyId.t) + | TyVarHole(MetaVar.t, Var.t) +``` + +The `TyVar` constructor represents a bound type variable. We retain the original +identifier for the purposes of supporting contraction back to a `UHTyp.t`. + +The `TyVarHole` constructor represents a free type variable, which is understood +as a type variable hole. Like other holes, each type variable hole has a +metavariable. + +## Kinds +We will need kinds to distinguish types from type-level holes, such as `TyVarHole`. +We define a new type, `Kind.t`, as follows: +``` +type Kind.t = KHole + | Type +``` + +# Static Semantics +## Type Variable Contexts +Type variable contexts are values of type `TyVarCtx.t`, which is defined +as follows: + +``` +type TyVarCtx.t = list((TyId.t, Kind.t)) +``` + +The position within a type variable context determines the de Bruijn index +(most recent bindings at the head). For this reason, type identifiers *can* +be duplicated. + +## Kind Consistency + +Kind consistency is defined by a judgement of the form `Kind.t ~ Kind.t` with +the rules presented in the attached +[latex document](./latex/kind-judgements.pdf). + +## Type Elaboration + +Expansion from unexpanded types to expanded types is defined by +bidirectional judgements `TyVarCtx.t |- UHTyp.t => Kind.t ~> HTyp.t -| Delta` +and `TyVarCtx.t |- UHtyp.t <= Kind.t_1 ~> HTyp.t : Kind.t_2` for consistent +`Kind.t_1` and `Kind.t_2`. These judgements are shown in the attached +[latex document](./latex/kind-judgements.pdf). + +# Action Semantics + +Adding support for type variables requires adding support for character-level +editing at the type-level. Currently, types are bound to individual uppercase +characters, e.g. `I` inserts the `Int` type immediately. This needs to be +changed to support natural text editing. We will follow the approach taken +with variables in expressions and patterns. + +We will need to change the action semantics for types to support kinds: + +Previously the type-level action semantics was defined by a judgement of +the form: + +`ZTyp.t --Action.t--> ZTyp.t` + +Now, we will need to define a bidirectional action semantics as with +expressions by judgements of the following form: + +1. `TVarCtx.t |- ZTyp.t => Kind.t --Action.t--> ZTyp.t => Kind.t` +2. `TVarCtx.t |- ZTyp.t --Action.t--> ZTyp.t <= Kind.t` diff --git a/9-type-aliases-redux/latex/.gitignore b/9-type-aliases-redux/latex/.gitignore new file mode 100644 index 0000000..0e98f99 --- /dev/null +++ b/9-type-aliases-redux/latex/.gitignore @@ -0,0 +1,2 @@ +*.log +*.aux diff --git a/9-type-aliases-redux/latex/etal.sty b/9-type-aliases-redux/latex/etal.sty new file mode 100644 index 0000000..4a31e23 --- /dev/null +++ b/9-type-aliases-redux/latex/etal.sty @@ -0,0 +1,94 @@ +\ProvidesPackage{etal}[2015/04/17 - Extract from local] + +\RequirePackage{xspace} + +% These are separate from joshua.sty, because I'd like everyone, everywhere, +% to use them, and I realize not everyone wants the baggage of hundreds of +% other macros. +% +% These macros protect you from the following scourges: +% +% 1. Misspelled/mispunctuated ``et al.'', ``cf.'', ``e.g.'', ``i.e.'', ``\'a la'', and ``vis-\'a-vis''. +% +% 2. Too much space after the ``.'' in ``cf.'', etc., because TeX likes to guess +% that a ``.'' might end a sentence, but ``i.e.'' doesn't end a sentence. +% +% +% \etal: +% +% I've lost track of the number of papers that have +% +% et. al. +% et. al +% et al +% +% ``Et'' is Latin for ``and''; it's why the ampersand has the shape it does +% (particularly the form that looks more like an E with a little T attached below). +% ``Et al.'' means ``and others''. +% +% If you don't want to remember all of that, you can still use the \etal macro. +% +% I can't think of any good reason to put a comma before \etal, and lots of +% people (including at least one person who does it frequently) think it's wrong. +% +% If you're in the ``\eg and \ie should always have a comma after'' camp, +% you presumably agree that \etal should *not* have a comma before, +% by the same analogy to what you would do if you replaced Latin with English. +% +% +% \eg and \ie: +% +% These work whether or not you put a comma after them. AIUI, there is no +% consensus on whether you should or shouldn't put a comma after them. +% I believe the reasonable positions are +% +% (1) Always put a comma, because if you wrote English instead of Latin, +% you'd have to use a comma (``(for example a rainbow)'' isn't grammatical; +% ``(for example, a rainbow)'' is grammatical). +% +% (2) Put a comma only if it precedes something long: ``(\eg a rainbow)'', but +% ``(\eg, the absurd kerfuffle over whether or not REDACTED should, like +% essentially all conferences nowadays, have a code of conduct)''. +% +% (3) Never put a comma; it's a Latin abbreviation, so it need not behave +% the same as its English replacement. +% If what follows is long, use English (\eg ``for example'') instead. +% +% (4) People care about this? Seriously? +% +% +% \cf: +% +% ``Cf.'' means ``compare''; it comes from one Latin word, so it's not ``c.f.''. +% Note that if you just want the reader to refer to something, +% ``see'' is about the same length as ``cf.'' but doesn't require +% knowing a somewhat obscure abbreviation. +% (Which suggests that, even when you mean ``compare'' and not ``see'', +% you should probably write ``compare''...) +% +% I can't think of any reason that you'd want to put a comma after \cf, +% so I deliberately defeated that possibility by defining it as ``cf.\ '' +% (which will produce a jarring space before the following comma). +% If you really want to, you can redefine it to ``cf.\@\xspace''. +% +% \ala, \visavis: +% +% I *really* can't think of any way these could ever make sense with +% a comma, or any punctuation, following them, so they also end with ``\ ''. + + +\newcommand{\etal}{et al.\@\xspace} +\newcommand{\eg}{e.g.\@\xspace} +\newcommand{\ie}{i.e.\@\xspace} + +\newcommand{\cf}{cf.\ } + +\newcommand{\ala}{\`a la\ } +\newcommand{\visavis}{vis-\`a-vis\ } + + +% Postscript: +% +% I rarely use these macros; I usually write ``for example'' instead of \eg, +% and ``that is'' instead of \ie. I think that, in general, when you're trying +% to write technical English, you should write English, not Latin. diff --git a/9-type-aliases-redux/latex/joshuadunfield.sty b/9-type-aliases-redux/latex/joshuadunfield.sty new file mode 100644 index 0000000..9b2b354 --- /dev/null +++ b/9-type-aliases-redux/latex/joshuadunfield.sty @@ -0,0 +1,699 @@ +\ProvidesClass{joshuadunfield}[2015/06/20 - Minor revision] + +% \RequirePackage{etex} +\RequirePackage{amsmath} + +% +% Bring in \etal, \eg, etc. +% +\RequirePackage{etal} + +% +% The mathpar package. +% +\RequirePackage{mathpartir} + +% +% Colours +% +\RequirePackage{color} +\RequirePackage{xcolor} + +% Yellow suitable for highlighting +% +\definecolor{dHilite}{rgb}{0.9, 0.9, 0.6} + +% Darker versions (dRed, dBlue, etc.) of common colours; +% pure red/blue/green are violently garish, and too light in monochrome. +% +% dDkRed, etc. are darker still. +% dLight[er]{Blue|Purple} are lighter colours suitable for backgrounds. +% +% dFaint, dGray, dDark, dAlmostBlack are shades of gray. +% +\definecolor{dRed}{rgb}{0.65, 0.0, 0.0} +\definecolor{dDkRed}{rgb}{0.35, 0.0, 0.0} +\definecolor{dGreen}{rgb}{0.0, 0.65, 0.0} +\definecolor{dDkGreen}{rgb}{0.0, 0.35, 0.0} +\definecolor{dBlue}{rgb}{0.0, 0.0, 0.65} +\definecolor{dLightBlue}{rgb}{0.4, 0.4, 0.9} +\definecolor{dLighterBlue}{rgb}{0.8, 0.8, 1.0} +\definecolor{dDkBlue}{rgb}{0.0, 0.0, 0.45} +\definecolor{dLightPurple}{rgb}{0.9, 0.5, 0.9} +\definecolor{dLighterPurple}{rgb}{1.0, 0.7, 1.0} +\definecolor{dPurple}{rgb}{0.65, 0.0, 0.65} +\definecolor{dDigPurple}{rgb}{0.5, 0.0, 0.5} +% \definecolor{DDIGPURPLE}{rgb}{0.5, 0.0, 0.5} % laughable +\definecolor{dFaint}{rgb}{0.7, 0.7, 0.7} +\definecolor{dGray}{rgb}{0.5, 0.5, 0.5} +\definecolor{dDark}{rgb}{0.2, 0.2, 0.2} +\definecolor{dAlmostBlack}{rgb}{0.1, 0.1, 0.1} + +\definecolor{lred}{rgb}{1.0, 0.3, 0.3} + + +\newcommand{\mathcolor}[2]{{\textcolor{#1}{\ensuremath{#2}}}} + +\newcommand{\textgraybox}[1]{\boxed{#1}} +\newcommand{\graybox}[1]{\textgraybox{\ensuremath{#1}}} + +\newcommand{\tightcolorbox}[2]{\setlength{\fboxsep}{1pt}\colorbox{#1}{#2}} + +\definecolor{grayboxgray}{rgb}{0.85, 0.85, 0.85} +\newcommand{\textcolorbox}[2]{\tightcolorbox{#1}{#2}} +\newcommand{\textshadebox}[1]{\textcolorbox{grayboxgray}{#1}} +\newcommand{\shadebox}[1]{\text{\textshadebox{\ensuremath{#1}}}} + +\newcommand{\mathcolorbox}[2]{\text{\tightcolorbox{#1}{$\displaystyle {#2}$}}} + +\newcommand{\hilite}[2]{\mathcolorbox{#1}{{#2}\mathstrut}} +\newcommand{\redhi}[1]{\hilite{red!40}{#1}} +\newcommand{\yellhi}[1]{\hilite{yellow}{#1}} +\newcommand{\fighi}[1]{\hilite{yellow!40}{#1}} +\newcommand{\purplehi}[1]{\hilite{purple!40}{#1}} +\newcommand{\purpleop}[1]{\mathrel{\hilite{purple!40}{{#1}}}} + + +% +% Font handling. +% +% First, some generally useful macros: +% +% +% \textvtt{...}: Like \texttt, but the *variable*-width CM typewriter font. +% +% \textbi{...}: Bold italic +% +% + + +%\newcommand{\textvtt}[1]{{\normalfont\fontfamily{cmvtt}\selectfont {#1}}} +\newcommand{\textvtt}[1]{\text{\normalfont\fontfamily{cmvtt}\selectfont {#1}}} +%\newcommand{\textvtt}[1]{\text{\normalfont\fontfamily{cmvtt}\selectfont\fontsz{11.0pt}{#1}}} +\newcommand{\textbi}[1]{\textit{\bfseries#1}} + +% +% Non-``run-on'' font macros that also set the line spacing to 1.2*the font size. +% Examples: +% +% \fontsz{18pt}{Eighteen-point text, in text mode} +% \mathsz{18pt}{18~pt~\in~\text{math} + \text{mode}} + +\newdimen\zzfontsz +\newcommand{\fontsz}[2]{\zzfontsz=#1% +{\fontsize{\zzfontsz}{1.2\zzfontsz}\selectfont{#2}}} + +\newcommand{\mathsz}[2]{\text{\fontsz{#1}{$#2$}}} + + +% +% Second, macros for particular animals (keywords, function names, etc.): +% +% \keyword{...}: Keyword (= \textvtt) +% (synonym: \textkw{...}) +% +% \textid{...}: Identifier (= \textit) +% +% \textfn{...}: Function name (italic variable-width typewriter) +% +% \datacon{...}: Data constructor (= \mathsf) +% +% \tyname{...}: Type name (= \mathsf) +% +% \sortname{...}: Sort name (= \tyname) +% + + +%\newcommand{\textfn}[1]{\textit{#1}} +\newcommand{\textfn}[1]{\text{\itshape\fontfamily{cmvtt}\selectfont {#1}}} +\newcommand{\datacon}[1]{\ensuremath{\mathsf{#1}}} +\newcommand{\tyname}[1]{\ensuremath{\mathsf{#1}}} +\newcommand{\sortname}[1]{\tyname{#1}} +%\newcommand{\keyword}[1]{\textbf{#1}}%{\text{\usefont{T1}{phv}{b}{n}\fontsize{11pt}{13pt}\selectfont#1}} +%\newcommand{\keyword}[1]{\text{\usefont{T1}{\rmdefault}{b}{n}\fontsize{\keywordfontsize}{12pt}\selectfont#1}} +\newcommand{\keyword}[1]{\text{\textvtt{#1}}} +\newcommand{\textkw}[1]{\keyword{#1}} +\newcommand{\textid}[1]{\textit{#1}} + +% Set the font size for \textkw and \keyword +% +\newcommand{\keywordfontsize}{9.7pt} % should be about (document point size - 0.4pt) + +% Third, screaming red: +% +\newcommand{\flaming}[1]{\textcolor{red}{\fontsz{18pt}{\bf #1}}} +\newcommand{\flamingmath}[1]{\textcolor{red}{\fontsz{18pt}{\bf \ensuremath{#1}}}} +\newcommand{\semiflaming}[1]{\textcolor{dRed}{\sl #1}} + +% \normalfont capers: +% +% If your theorems, lemmas, etc. are italic or slanted, and they mention +% judgments that have words (shocking, to use words), those words +% tend to become italic or slanted too, which I think looks silly. +% +% Two solutions: +% +% (1) Don't use a special style for theorems and lemmas. +% +% (2) Use \normalfont a lot. +% +\newcommand{\textt}[1]{\text{\normalfont\texttt{#1}}} + + +\newdimen{\zzzpbox} +\newcommand{\pbox}[1]{\settowidth\zzzpbox{#1}\parbox{\zzzpbox}{#1}} + +\newlength{\zzsplatboldwidth} +\newcommand{\xsplatbold}[2]{\settowidth{\zzsplatboldwidth}{{#2}}{#2}\addtolength{\zzsplatboldwidth}{-#1}\hspace{-\zzsplatboldwidth}\raisebox{#1}{{#2}}} +\newcommand{\splatbold}[1]{\xsplatbold{-0.04mm}{#1}} + +%\newcommand{\stupidfontsz}[1]{\zzfontsz=#1% +%{\fontsize{\zzfontsz}{1.2\zzfontsz}\selectfont}} + +\makeatletter +\def\url@MGstyle{% +\def\UrlFont{\tiny\ttfamily}% \tiny for 2-column dense papers (will not appear in final versions anyway) +\def\do@url@hyp{\do\-}%Break at hyphens +\Url@do +} +\makeatother +\newcommand{\marginPseudoURL}{\begingroup \urlstyle{MG}\Url} + +\newcommand{\marginnote}[1]{\marginpar{\raggedright\scriptsize{#1}}} + + + +% Yes, \Lam and \Fix are defined to take only one argument (the binder). +% This is on purpose: these macros don't render a closing delimiter, +% so it's misleading to have a closing delimiter (of a 2nd argument) in +% the LaTeX source. +% +% Since TeX doesn't care if you have extra {}, you won't get an error if you write +% +% \Lam{x}{blabla(x)} +% +% but you shouldn't, because it doesn't match the rendering +% +% \lambda x. blabla(x) +% +% Also, single-argument macros let you write (in an array) +% +% \Lam{x} \\ +% ~~blabla(x) +% +% without putting a silly {} after \Lam{x}. + +\newcommand{\xLam}{\lambda} +\newcommand{\Lam}[1]{\xLam#1.\,} +\newcommand{\lam}[1]{\Lam{#1}} + +\newcommand{\Pair}[2]{\textvtt{(}#1\textvtt{,}\;#2\textvtt{)}} +\newcommand{\xFix}{\ensuremath{\keyword{fix}}} +\newcommand{\Fix}[1]{\xFix~#1.\:} + + +% +% Characters that can't be easily quoted +% +\newcommand{\Backslash}{\char"5C} +\newcommand{\Lbrack}{\char"5B} +\newcommand{\Rbrack}{\char"5D} +\newcommand{\Lbrace}{\char"7B} +\newcommand{\Rbrace}{\char"7D} +\newcommand{\tildesym}{\textt{\char"7E}} +\newcommand{\caret}{\char94} + + +% +% Unfiled +% +\newcommand{\bnfas}{\ensuremath{\mathrel{::=}}} +\newcommand{\bnfalt}{\mathrel{\mid}} +\newcommand{\bnfaltbrk}{\hspace{-1.02ex}\bnfalt} +\newcommand{\bnfaltBRK}{\) \\ &&& \(\bnfaltbrk} + + +% +% References (``Lemma \ref{...}'', etc.) +% +% Lots of people would put nonbreaking spaces in these, such as +% +% Figure~\ref{fig:foo} +% +% These macros don't. Most of the time, the ~ makes no difference; +% when it does make a difference, the line-layout effects are +% unpredictable. Having ``Figure'' and ``5'' on different lines is bad, +% but it's less bad than having a line run into the gutter. The cure shouldn't +% be worse than the disease, and you always have the option of hand-writing +% ``Figure~\ref{fig:foo}'' when that's really the lesser evil. +% +% Notes: +% \Lemmaref{...} uses \nameref to add the lemma name (in parentheses). +% \Lemref doesn't. + +\newcommand{\Appendixref}[1]{Appendix \ref{#1}} +\newcommand{\Figureref}[1]{Figure \ref{#1}} +\newcommand{\Figref}[1]{Fig.\ \ref{#1}} +\newcommand{\Sectionref}[1]{Section \ref{#1}} +\newcommand{\Secref}[1]{Sec.\ \ref{#1}} + +\newcommand{\Lemmaref}[1]{Lemma \ref{#1} (\nameref{#1})} % name and number +\newcommand{\Lemref}[1]{\Lemma \ref{#1}} % number only + +\newcommand{\Chapterref}[1]{Chapter \ref{#1}} +\newcommand{\Listingref}[1]{Listing \ref{#1}} + +\newcommand{\Theoremref}[1]{Theorem \ref{#1}} +\newcommand{\Thmref}[1]{Thm.\ \ref{#1}} +\newcommand{\Corollaryref}[1]{Corollary \ref{#1}} +\newcommand{\Corref}[1]{Cor.\ \ref{#1}} + +\newcommand{\Conjectureref}[1]{Conjecture \ref{#1}} +\newcommand{\Propositionref}[1]{Proposition \ref{#1}} +\newcommand{\Propertyref}[1]{Property \ref{#1}} +\newcommand{\Remarkref}[1]{Remark \ref{#1}} +\newcommand{\Tableref}[1]{Table \ref{#1}} + +\newcommand{\Definitionref}[1]{Definition \ref{#1}} +\newcommand{\Defnref}[1]{Def.\ \ref{#1}} + +% Apparently I once used these macros, but I don't anymore: +\newcommand{\nestedref}[2]{\ref{#1}(\textit{\ref{#1:#2}})} +\newcommand{\nestedLemmaref}[2]{Lemma \nestedref{#1}{#2}} +\newcommand{\nestedPropertyref}[2]{Property \nestedref{#1}{#2}} + + + +% ``Semantic'' synonyms for symbols. +% +% Ideally, LaTeX source would be almost as readable as the rendered output. +% It never will be, but it should be made more readable wherever practical. +% It is *entirely* practical to write \union instead of \cup, etc., giving the reader +% (of the source) a fighting chance of reading the source out loud. Compare +% +% F(A) \union B +% +% to +% +% F(A) \cup B +% +% If you actually pronounce set union "cup", you're putting the LaTeX cart +% in front of the mathematical horse, and you're being rude to any members +% of your audience who don't know much LaTeX---which means you're being +% particularly and disproportionately rude to undergrads and early grad +% students. (Thus, if you're faculty, you're being rude to your students.) + +\newcommand{\union}{\cup} +\newcommand{\sect}{\cap} + +\newcommand{\Union}{\bigcup} + +\newcommand{\contradiction}{{\Longrightarrow}{\Longleftarrow}} + + +% +% Example of the \foosym / \foo convention for (among others) binary operators: +% +% \foosym is the ``raw'' symbol, usually with extra braces to defeat +% whatever spacing TeX might otherwise try to add. +% +% \foo is the symbol you usually want to use, with \mathrel{} so it gets +% +% Why do I defeat the spacing only to add it again? +% +% First, if I want to build some big symbol with \foosym as a component, +% I probably don't want any extra space around \foosym (though I +% probably want space around the big symbol). +% +% Second, TeX's spacing is not always correct. A particularly irritating +% example that I've seen in many published papers is !. TeX assumes +% that ! (in math mode) is factorial, which is a postfix operator. +% In programming languages, ! is usually a prefix operator. So if you write +% +% \Gamma \vdash !e : \tau +% +% TeX will slam the ! right up against the \vdash, making it look like ! is +% decorating the turnstile. If you write {!}, you avoid this; but why should +% you have to remember where to add random sets of braces? Give ! a name, +% and put the extra braces in the macro definition. +% +% (\usepackage{euler}, for some reason, solves this particular problem; +% but that seems fortuitous.) + +\newcommand{\subtypesym}{{\leq}} +\newcommand{\subtype}{\mathrel{\subtypesym}} + +\newcommand{\supertypesym}{{\geq}} +\newcommand{\supertype}{\mathrel{\supertypesym}} + +\newcommand{\set}[1]{\left\{ #1 \right\}} +\newcommand{\abs}[1]{\left| #1\right|} +\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} +\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil} +\newcommand{\twodots}{\mathinner{\ldotp\ldotp}} + + +\newcommand{\where}{~\text{where}~} +\newcommand{\forsome}{~\text{for some}~} +\newcommand{\AND}{\textrm{~and~}} +\newcommand{\OR}{\textrm{~or~}} + +%\renewcommand{\o}{\,{\circ}\,} % Clashes with slash-o +\newcommand{\composesubst}{\mathrel{\circ}} + +\newcommand{\arr}{\rightarrow} + +\newcommand{\entailssym}{{\vdash}} +\newcommand{\entails}{\mathrel{\entailssym}} % the |- symbol + +\newcommand{\notentailssym}{\printsavewidth{{\vdash}}\hspace{-0.9\zzskipwidthlen}\mathcolor{dRed}{\text{\tt /}}\hspace{0.1\zzskipwidthlen}} +\newcommand{\notentails}{\mathrel{\notentailssym}} % slashed |- symbol +\newcommand{\lnotentails}{{\notentailssym}\;} % slashed |- symbol +\newcommand{\Entails}{\models\,} % the |= symbol + +\newcommand{\aeq}{=_\alpha} +\newcommand{\beq}{=_\beta} + +\newcommand{\diff}{\mathrel{\setminus}} +\newcommand{\pfn}{\mathrel{\rightharpoonup}} % partial-function arrow + +\newcommand{\xFV}{\mathit{FV}} +\newcommand{\FV}[1]{\xFV(#1)} + +\newcommand{\sembr}[1]{\llbracket #1 \rrbracket} % ``semantics'' brackets + + +% +% +% Terser versions of array and tabular, with particular, tightly-spaced arguments +% +% + +\newcommand{\arrayenv}[1]{\renewcommand{\arraystretch}{1} \begin{array}[t]{@{}c@{}}#1\end{array}} +\newcommand{\arrayenvc}[1]{\renewcommand{\arraystretch}{1} \begin{array}[c]{@{}c@{}}#1\end{array}} +\newcommand{\arrayenvcl}[1]{\renewcommand{\arraystretch}{1} \begin{array}[c]{@{}l@{}}#1\end{array}} +\newcommand{\arrayenvr}[1]{\renewcommand{\arraystretch}{1} \begin{array}[t]{@{}r@{}}#1\end{array}} +\newcommand{\arrayenvbr}[1]{\renewcommand{\arraystretch}{1} \begin{array}[b]{@{}r@{}}#1\end{array}} +\newcommand{\arrayenvl}[1]{\renewcommand{\arraystretch}{1} \begin{array}[t]{@{}l@{}}#1\end{array}} +\newcommand{\arrayenvb}[1]{\renewcommand{\arraystretch}{1} \begin{array}[b]{@{}c@{}}#1\end{array}} +\newcommand{\arrayenvbl}[1]{\renewcommand{\arraystretch}{1} \begin{array}[b]{@{}l@{}}#1\end{array}} +\newcommand{\arrayenvblll}[1]{\renewcommand{\arraystretch}{1} \begin{array}[b]{@{}lll@{}}#1\end{array}} +\newcommand{\pfarr}[1]{\begin{array}[b]{@{}l@{}}#1\end{array}} + +\newcommand{\tabularenv}[1]{\renewcommand{\arraystretch}{1} \begin{tabular}[t]{@{}c@{}}#1\end{tabular}} +\newcommand{\tabularenvc}[1]{\renewcommand{\arraystretch}{1} \begin{tabular}[c]{@{}c@{}}#1\end{tabular}} +\newcommand{\tabularenvcl}[1]{\renewcommand{\arraystretch}{1} \begin{tabular}[c]{@{}l@{}}#1\end{tabular}} +\newcommand{\tabularenvr}[1]{\renewcommand{\arraystretch}{1} \begin{tabular}[t]{@{}r@{}}#1\end{tabular}} +\newcommand{\tabularenvbr}[1]{\renewcommand{\arraystretch}{1} \begin{tabular}[b]{@{}r@{}}#1\end{tabular}} +\newcommand{\tabularenvl}[1]{\renewcommand{\arraystretch}{1} \begin{tabular}[t]{@{}l@{}}#1\end{tabular}} +\newcommand{\tabularenvb}[1]{\renewcommand{\arraystretch}{1} \begin{tabular}[b]{@{}c@{}}#1\end{tabular}} +\newcommand{\tabularenvbl}[1]{\renewcommand{\arraystretch}{1} \begin{tabular}[b]{@{}l@{}}#1\end{tabular}} +\newcommand{\tabularenvblll}[1]{\renewcommand{\arraystretch}{1} \begin{tabular}[b]{@{}lll@{}}#1\end{tabular}} + + + +% +% Various formatting: +% +% `displ' for displayed text, +% `mathdispl' for displayed mathpar, +% `rclll' for equations (with extra left-aligned columns), +% `grammar' for BNF. +% + +\newenvironment{displ}{\vspace{1pt} \begin{center} ~\!\!}{\end{center}} +\newcommand{\vertcenter}[1]{\begin{tabular}{@{}c@{}}#1\end{tabular}} + +\newenvironment{rclll}{\vspace{1pt} \begin{center} ~\!\!\begin{array}[t]{@{}r@{~~}c@{~~}l@{}l@{}l@{}}}{\end{array}\end{center}} +\newenvironment{grammar}{\vspace{1pt} \begin{center} ~\!\!\begin{tabular}[t]{@{}lr@{~~}c@{~~}lll@{}}}{\end{tabular}\end{center}} + + +% \newenvironment{mathdispl}{\begin{displ}~$\begin{mathpar} ~\!\!}{\end{mathpar}$\end{displ}} +\newenvironment{mathdispl}{\begin{mathpar}}{\end{mathpar}\ignorespacesafterend} + +% \RequirePackage{environ} +% \NewEnviron{mathdispl}{% +% \begin{mathpar} +% \BODY +% \end{mathpar} +% } + + +\newcommand{\parasection}[1]{\paragraph*{#1.}} + + +% +% Letters +% + +\newcommand{\Cee}{\mathcal{C}} +\newcommand{\Dee}{\mathcal{D}} +\newcommand{\E}{\mathcal{E}} +\newcommand{\Judg}{\mathcal{J}} + + + +% +% Miscellaneous +% + +\newcommand{\hole}{\ensuremath{[\,]}} + +\newcommand{\Hand}{\text{\Pointinghand~~~~}} + +\newlength\zzskipwidthlen +\newcommand{\skipwidth}[1]{\settowidth{\zzskipwidthlen}{\ensuremath{#1}}\hspace{\zzskipwidthlen}} +\newcommand{\printsavewidth}[1]{\settowidth{\zzskipwidthlen}{\ensuremath{#1}}#1} +\newcommand{\skipsavedwidth}{\hspace{\zzskipwidthlen}} + +\newcommand{\raiseline}[1]{\raisebox{2.5ex}[0pt]{#1}} +\newcommand{\raisehalf}[1]{\raisebox{1.2ex}[0pt]{#1}} +\newcommand{\drophalf}[1]{\raisebox{-1.2ex}[0pt]{#1}} +\newcommand{\dropline}[1]{\raisebox{-2.5ex}[0pt]{#1}} +\newcommand{\ghostbox}[1]{\raisebox{0pt}[0pt][0pt]{#1}} + + +% +% Better (to me) inference rules. +% +% \Infer: +% +% -- takes three arguments, not two, which forces you to either give the rule +% a name (try to give your rules names, and I mean actual names, not +% random numbers), or at least to write {} for the name, which reminds you +% that you aren't giving the rule a name and you should probably fix that. +% (Every rule name should have its own macro, preferably with rulelinks.sty.) +% +% -- places the rule name to the right (which forces more space between +% rules, which I think is more readable, at the cost of discouraging long +% rule names); +% +% -- adds \strut/\mathstrut, making line heights consistent, so +% +% \Infer{} +% {} +% {e : \tau} % lowercase that nearly touches the horizontal line +% +% doesn't look strange next to +% +% \Infer{} +% {\text{UPPERCASE}} +% {\text{USING ENTIRE LINE HEIGHT}} +% + +\newcommand{\rulename}[1]{\texttt{#1}} +\def \TirNameStyle #1{\rulename{#1}} +\newcommand{\Infer}[3]{\inferrule*[right={\text{\strut#1}}]{{}#2\mathstrut}{{}#3\mathstrut}} + +% +% Inside a mathpar, I sometimes want to include some text, like +% +% ``no intro rule for $\botty$'' +% +% I'd like this to be centred, but mathpar bottom-aligns each row of rules. +% Thus, use \centerruleplaceholder. For example: +% +% \centerruleplaceholder{ +% \Infer{foo} +% {\text{Some rule}} +% {\text{that will appear next to your text}} +% }{ +% Some text, in which you may include +% \\ +% line breaks +% \\ +% to keep the width from getting out of control +% } + +\newcommand{\centerruleplaceholder}[2]{% + \text{% + \begin{tabular}[c]{c} + \ensuremath{#1} + \end{tabular} + ~~~~ + \begin{tabular}[c]{l} + #2 + \end{tabular} + }% +} + +% failed attempt: +\newcommand{\centerplaceholder}[1]{% + \text{% + \begin{tabular}[t]{c} + #1 + \end{tabular} + }% +} + +% +% ``Judgment boxes'' that should precede a mathpar, so the reader +% can instantly see which judgment is being defined by the rules. +% +% The first argument is the judgment form. +% The second argument tells the reader how to interpret the judgment. +% I don't mean *what* the judgment is, but literally how to *say* it: +% If you're defining \Gamma \entails e : \tau, don't write +% +% \judgbox{\Gamma \entails e : \tau}{Typing} +% +% Instead, write +% +% \judgbox{\Gamma \entails e : \tau} +% {Under context $\Gamma$, expression $e$ has type $\tau$} +% +% This tells the reader what the judgment is supposed to mean. It's also +% an opportunity to remind the reader what your metavariables (like $e$) +% stand for. The reader doesn't know your system as well as you do, +% and will lose track of all the kinds of animals you have in your zoo. +% +% For less familiar judgments than typing, this is even more important. +% + +\ifnum\OPTIONConf=1 + \newcommand{\judgboxfontsize}[1]{\mathsz{9pt}{#1}} +\else + \newcommand{\judgboxfontsize}[1]{\mathsz{12pt}{#1}} +\fi + +\newcommand{\judgbox}[2]{% + %% {\raggedright \textgraybox{\ensuremath{\judgboxfontsize{#1}}}\!\begin{tabular}[c]{l} #2 \end{tabular} % + %% \ifnum\OPTIONConf=1 \\[-1.5ex] \else \\[-1.8ex] \fi% + {\raggedright \textgraybox{\ensuremath{\judgboxfontsize{#1}}}\!\begin{tabular}[c]{l}\hspace{0.03in} #2 \end{tabular} % + \ifnum\OPTIONConf=1 \\[-1.0ex] \else \\[-1.8ex] \fi% +}} + +% Horribly named variants of \judgbox. The \judgbox spacing is supposed +% to be tuned for a following mathpar. But sometimes I define +% a judgment via equations, or something else, and the violent negative space +% that I like for mathpar leads to disaster. +% +% Use \judgboxa when you are about to use \[ ... \]. +% +% Use \judgboxx when you are about to write an ordinary paragraph. +% +\newcommand{\judgboxa}[2]{% + {\raggedright \textgraybox{\judgboxfontsize{\ensuremath{#1}}}\!\begin{tabular}[c]{l} #2 \end{tabular} % + \ifnum\OPTIONConf=1 \\[-1.3ex] \else \\[-1.0ex] \fi% +}} +\newcommand{\judgboxx}[2]{% + {\raggedright \textgraybox{\judgboxfontsize{\ensuremath{#1}}}\!\begin{tabular}[c]{l} #2 \end{tabular} % + \ifnum\OPTIONConf=1 \\[0.3ex] \else \\[0.6ex] \fi% +}} + +% +% Variant with another box, *after* the description, intended for a +% Twelf expression. +% +\newcommand{\judgboxtwelf}[3]{% + {\raggedright \textgraybox{\judgboxfontsize{\ensuremath{#1}}}\!\begin{tabular}[c]{l} #2 \end{tabular} % + \!\!\textgraybox{\fontsz{8pt}{#3}}% + \ifnum\OPTIONConf=1 \\[3pt] \else \\[6pt] \fi% +}} + + + + +% +% Showing derivations and other things in proofs. +% These macros assume you use an `itemize' or `enumerate' environment +% to list the cases of a proof. +% + +\newcommand{\ProofCaseRule}[1]{\item \textbf{Case }\textrm{{#1}}: ~ } +\newcommand{\ProofCaseThing}[1]{\ProofCaseRule{\ensuremath{#1}}} +\newcommand{\ProofCasesRules}[1]{\item \textbf{Cases }\textrm{{#1}}: ~ } +\newcommand{\ProofCaseRuleNoColon}[1]{\item \textbf{Case }\textrm{{#1}}} + +\gdef\xxDerivationProofCaseColor{N} +\newcommand{\Begincolorcases}[1]{\gdef\xxDerivationProofCaseColor{#1}} +\newcommand{\Endcolorcases}{\gdef\xxDerivationProofCaseColor{N}} + +%\newcommand{\DerivationProofCase}[3]{% +% \medskip +% \begin{samepage} +% \nopagebreak +% \item % +% \parbox[t]{100ex}{% +% \textbf{Case } \\[-0.95em] +% $~$\hspace{5ex} +% \if\xxDerivationProofCaseColor N% +% \ensuremath{% +% \Infer{#1}{#2}{#3}% +% } +% \else% +% \colorbox{\xxDerivationProofCaseColor}{% +% \ensuremath{% +% \Infer{#1}{#2}{#3}% +% }% +% }% +% \fi% +% }% +% \end{samepage} +% \nopagebreak \\[-2ex] +% } +\newcommand{\DerivationProofCase}[3]{% + \smallskip + \item % + \parbox[t]{100ex}{% + \textbf{Case } \\[-0.5em] + $~$\hspace{5ex} + \if\xxDerivationProofCaseColor N% + \ensuremath{% + \Infer{#1}{#2}{#3}% + } + \else% + \colorbox{\xxDerivationProofCaseColor}{% + \ensuremath{% + \Infer{#1}{#2}{#3}% + }% + }% + \fi% + }% + \nopagebreak \\[0.2ex] + } + +\newcommand{\DoubleDerivationProofCase}[6]{% + \smallskip + \item % + \parbox[t]{100ex}{% + \textbf{Case } \\[-0.5em] + $~$\hspace{5ex} + \if\xxDerivationProofCaseColor N% + \ensuremath{% + \Infer{#1}{#2}{#3}% + ~~~~~ + \Infer{#4}{#5}{#6}% + } + \else% + \colorbox{\xxDerivationProofCaseColor}{% + \ensuremath{% + \Infer{#1}{#2}{#3}% + ~~~~~ + \Infer{#4}{#5}{#6}% + }% + }% + \fi% + }% + \nopagebreak \\[-0.8ex] + } diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f77f38e1173c44d6312a69f39b380f1f86de7314 GIT binary patch literal 140515 zcma&MQ;;r9(5>0_ZriqP+qQPwwr$(oZQHhO+x@nU`DSANiHSJp;?&iuioDItwW1y} zMNx4&CVDm)viaqabr=>xMnZcdD;Qp07zSBWJ98HcLMARQPQw4^z%Yng+PIiH5i*F| z{B(;`;)1|%9c&+LPz2EHZ)qDH%!6eh9fHjkcVDWTzw6H(GLd859 zp%P_Y9FL_E#rY*kBUe5pUbb;RaMToS(1+hk(BpF_Ti;pl5CRRW*0IhrX+boiR9{mZL^gstESVpn?^=ybFU3ZBj<1&k@2x7N<*rVE=K zRZ`N0tXGXE5BBWRg{3sfH$FkYjS;iIl5jc@7?y=@#m5|KrHC|Z%}M?`d6uvmQ9T6} zE3KBqZThlMnC91L3*73PsX65oK(72*@!+Kh(S*tw4r$ettwR0hK-AI^$L*d~R(rft zFZ;;Mt5vhNnA)<(P_(+IE%o^zGD}h&MOjJ_i%d0CA78QP4)k~Lz0Br2o@37UzMv-E zOYh4`%Sk&`@aaqtb|HM@#@2P|hDp-F?0)T27$G)H z_*Kexjpljeb=m%(Ux0g}t$X@fEdI{`ivHZwS`l#B11zM36+&K;+jSYn3vp7YYFp)h&aX*gn@%Wy>T;a`q)k)&hqK}h(# zDhZOYONmx7>=#w+{2!`DWpOTYv^C6d>|o%QYH?6ju`UbVXYP$DXu58uN*ZuwkfPG= z+l7^P&qg>RLRWr~H{o7rJDq+y9+-j|tUmIVUMUym=_5W}+`P zbcC|Tj`7w;t2wVGHBiC~vdhc@e}t3g-SN!vQ8RR&NmPLJN5L3g#F$H=29?ZE`uQJS zmR;{ckdu}VQc4S1%-{5!`=U`AO1|>nYO5}O(&oVa>NxECdR!3n^z-c?V>s#>n^uWH zV%;UomF}FyKw=%FgJOJXb{!UZ{j3h`>68`>B}=Ao%r!NSYI^MPR%~jvMi7@Jz2AM$ z&1!$#z{{BK#JEf9Y}XJCDkJ^SXrdE7(%AfplNk&e&R(bC!%E7ue2VBMe~K#Z5BDi- z#59;1_FCY-BizUZRBK$BalhQShvgeS-L&w74NYl_Hll~vN(-iw&WPs;- zvr4@O)}t64L5hFM2qSBZ^<<9|hLzpCN>oR2|Ei9}ATlW>2oof+f~~i2Fq*=oVn!4_ z#4`Ew>8Qb~qfA)P7w5!JLoeH~n_uTzR}I^FO)V5w{SbFSWHX0%qsQ2C$$Iu-MC2GVm&LV*4 z*!Z((s$Aku;*QNwG#7S~H!9=A1Bp3;&>uBRv1>hdxmf*3I7XN#Dkbq~9%5?7P;p`>Na41w~ie2y=cv$bsx? 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+\usepackage{joshuadunfield} + + +\newcommand{\elabAna}[6]{#1 \vdash #2 \Leftarrow #3 \leadsto #4 : #5 \dashv #6} +\newcommand{\elabSyn}[5]{#1 \vdash #2 \Rightarrow #3 \leadsto #4 \dashv #5} +\newcommand{\kconsistent}[2]{#1 \sim #2} + +\newcommand{\hPhi}{\Phi} +\newcommand{\EmptyhPhi}{\emptyset} % From hand-written notes, Canonical forms lemma; page 14 +\newcommand{\EmptyDelta}{\cdot} % From hand-written notes, ES-Const rule, page 1 + +\newcommand{\htau}{\hat{\tau}} +\newcommand{\hkappa}{\kappa} +\newcommand{\dtau}{\tau} + +\newcommand{\Ty}{\textt{Ty}} +\newcommand{\KHole}{\textt{KHole}} +\newcommand{\hlist}[1]{\textt{list(}#1\textt{)}} + +\newcommand{\llparenthesiscolor}{\textcolor{violet}{\llparenthesis}} +\newcommand{\rrparenthesiscolor}{\textcolor{violet}{\rrparenthesis}} +\newcommand{\hthole}[1]{\llparenthesiscolor\rrparenthesiscolor^{#1}} +\newcommand{\hhole}[2]{\llparenthesiscolor#1\rrparenthesiscolor^{#2}} +\newcommand{\kehole}{\llparenthesiscolor\rrparenthesiscolor} + +\newcommand{\idof}[1]{\mathsf{id}(#1)} +\newcommand{\Dbinding}[3]{#1 :: #3[#2]} + + +\begin{document} + +\begin{figure}[t] + $\arraycolsep=4pt\begin{array}{rllllll} + \mathsf{BinOp} & \oplus & ::= & + \textt{Product} ~\vert~ \textt{Sum} ~\vert~ \textt{Arrow} \\ + \mathsf{Kind} & \kappa & ::= & + \Ty ~\vert~ \KHole \\ + \mathsf{Constant Types} & c & ::= & + \textt{Int} ~\vert~ \textt{Float} ~\vert~ \textt{Bool} \\ + \mathsf{User HTyp} & \htau & ::= & + c ~\vert~ \htau_1 \oplus \htau_2 ~\vert~ \hlist{\htau} ~\vert~ \hthole{u} ~\vert~ \hhole{\htau}{u} \\ + \mathsf{Internal HTyp} & \dtau & ::= & + c ~\vert~ \dtau_1 \oplus \dtau_2 ~\vert~ \hlist{\dtau} ~\vert~ \hthole{u} ~\vert~ \hhole{\dtau}{u} \\ + \end{array}$ +\end{figure} + + +\begin{minipage}{\linewidth} + \judgbox + {\kconsistent{\hkappa_1}{\hkappa_2}} + {$\hkappa_1$ is consistent with $\hkappa_2$} + \begin{mathpar} + \inferrule[KCHole]{ }{ + \kconsistent{\KHole}{\Ty} + } + \and + \inferrule[KCSymm]{ + \kconsistent{\hkappa_1}{\hkappa_2} + }{ + \kconsistent{\hkappa_2}{\hkappa_1} + } + \and + \inferrule[KCRefl]{ }{ + \kconsistent{\hkappa}{\hkappa} + } + \end{mathpar} +\end{minipage} +\\ +\\ + +\begin{minipage}{\linewidth} + \judgbox + {\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}} + {$\htau$ synthesizes kind $\hkappa$ and elaborates to $\dtau$} + \begin{mathpar} + \inferrule[TElabSConst]{ }{ + \elabSyn{\hPhi}{c}{\Ty}{c}{\EmptyDelta} + } + \and + \inferrule[TElabSBinOp]{ + \elabAna{\hPhi}{\htau_1}{\Ty}{\dtau_1}{\Ty}{\Delta_1} \\ + \elabAna{\hPhi}{\htau_2}{\Ty}{\dtau_2}{\Ty}{\Delta_2} + }{ + \elabSyn{\hPhi}{\htau_1 \oplus \htau_2}{\Ty}{\dtau_1 \oplus \dtau_2}{\Delta_1 \cup \Delta_2} + } + \and + \inferrule[TElabSList]{ + \elabAna{\hPhi}{\htau}{\Ty}{\dtau}{\Ty}{\Delta} + } { + \elabSyn{\hlist{\htau}}{\Ty}{\hlist{\dtau}}{\Delta} + } + \and + \inferrule[TElabSVar]{ + t : \hkappa \in \hPhi + } { + \elabSyn{\hPhi}{t}{\hkappa}{t}{\EmptyDelta} + } + \and + \inferrule[TElabSUVar]{ + t \not\in \hPhi + } { + \elabSyn{\hPhi}{t}{\KHole}{\hhole{t}{u}_{\idof{\hPhi}}}{\Dbinding{u}{\hPhi}{\kehole} } + } + \and + \inferrule[TElabSHole]{ } { + \elabSyn{\hPhi}{\hthole{u}}{\KHole}{\hthole{u}_{\idof{\hPhi}}}{\Dbinding{u}{\hPhi}{\kehole}} + } + \and + \inferrule[TElabSNEHole]{ + \elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta} + } { + \elabSyn{\hPhi}{\hhole{\htau}{u}}{\KHole}{\hhole{\dtau}{u}_{\idof{\hPhi}}}{\Delta,\Dbinding{u}{\hPhi}{\kehole}} + } + \end{mathpar} +\end{minipage} +\\ +\\ +\begin{minipage}{\linewidth} + \judgbox + {\elabAna{\hPhi}{\htau}{\hkappa_1}{\dtau}{\hkappa_2}{\Delta}} + {$\htau$ analyzes against type $\hkappa_1$ and + elaborates to $\dtau$ of consistent type $\hkappa_2$} + \begin{mathpar} + \inferrule[TElabASubsume]{ + \htau \neq \hthole{u} \\ + \htau \neq \hhole{\htau'}{u} \\ + \elabSyn{\hPhi}{\htau}{\hkappa'}{\dtau}{\Delta} \\ + \kconsistent{\hkappa}{\hkappa'} + }{ + \elabAna{\hPhi}{\htau}{\hkappa}{\dtau}{\hkappa'}{\Delta} + } + \and + \inferrule[TElabAEHole]{ } { + \elabAna{\hPhi}{\hthole{u}}{\hkappa}{\hthole{u}_{\idof{\hPhi}}}{\hkappa}{\Dbinding{u}{\hPhi}{\kappa}} + } + \and + \inferrule[TElabANEHole]{ + \elabSyn{\hPhi}{\htau}{\hkappa'}{\dtau}{\Delta} + } { + \elabAna{\hPhi}{\hhole{\htau}{u}}{\hkappa}{\hhole{\dtau}{u}_{\idof{\hPhi}}}{\hkappa}{\Delta,\Dbinding{u}{\hPhi}{\kappa}} + } + \end{mathpar} +\end{minipage} + + +\end{document} From 8b3f0dd7471bd867e0c2689a86244ebdfdcc55bc Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Sat, 20 Mar 2021 20:53:11 -0500 Subject: [PATCH 002/181] Adds note about type holes --- 9-type-aliases-redux/9-type-aliases-redux.md | 3 +++ 1 file changed, 3 insertions(+) diff --git a/9-type-aliases-redux/9-type-aliases-redux.md b/9-type-aliases-redux/9-type-aliases-redux.md index 28903f7..0f1f32a 100644 --- a/9-type-aliases-redux/9-type-aliases-redux.md +++ b/9-type-aliases-redux/9-type-aliases-redux.md @@ -86,6 +86,9 @@ and `TyVarCtx.t |- UHtyp.t <= Kind.t_1 ~> HTyp.t : Kind.t_2` for consistent `Kind.t_1` and `Kind.t_2`. These judgements are shown in the attached [latex document](./latex/kind-judgements.pdf). +`Delta.t`'s `hole_sort` will be expanded to include `TypeHole` so we can track +type holes there too. + # Action Semantics Adding support for type variables requires adding support for character-level From aaa112a4ae019379f02238293cb1fccd5fdaf7d0 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Sun, 21 Mar 2021 20:11:06 -0500 Subject: [PATCH 003/181] Delta supports htyp/varctx or kind/tyvarctx --- 9-type-aliases-redux/9-type-aliases-redux.md | 14 +++++++++++++- 1 file changed, 13 insertions(+), 1 deletion(-) diff --git a/9-type-aliases-redux/9-type-aliases-redux.md b/9-type-aliases-redux/9-type-aliases-redux.md index 0f1f32a..ab30c21 100644 --- a/9-type-aliases-redux/9-type-aliases-redux.md +++ b/9-type-aliases-redux/9-type-aliases-redux.md @@ -87,7 +87,19 @@ and `TyVarCtx.t |- UHtyp.t <= Kind.t_1 ~> HTyp.t : Kind.t_2` for consistent [latex document](./latex/kind-judgements.pdf). `Delta.t`'s `hole_sort` will be expanded to include `TypeHole` so we can track -type holes there too. +type holes there too. Since Delta for TypeHoles needs `Kind.t` and `TyVarCtx.t` instead of `HTyp.t` and `VarCtx.t` but we still want to have the same keyspace (the `u`s are consistent and incrementing across all the holes), wrap the value for the `MetaVarMap.t` in `Delta.t` to pack this dependency with the associated `hole_sort`: + +```reasonml +type hole_sort(_, _) = + | ExpressionHole: hole_sort(HTyp.t, VarCtx.t) + | TypeHole: hole_sort(Kind.t, TyVarCtx.t) + | PatternHole: hole_sort(HTyp.t, VarCtx.t); + +type value = + | V(hole_sort('s, 'ctx), 's, 'ctx): value; + +type t = MetaVarMap.t(value); +``` # Action Semantics From 696d9d98e4e2ebe743ca9c1731e516be31cdf4c9 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Tue, 23 Mar 2021 20:39:01 -0500 Subject: [PATCH 004/181] Adds kind assignment, fixes a few bugs --- .../latex/kind-judgements.pdf | Bin 140515 -> 141703 bytes .../latex/kind-judgements.tex | 57 +++++++++++++++++- 2 files changed, 54 insertions(+), 3 deletions(-) diff --git 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a/9-type-aliases-redux/9-type-aliases-redux.md +++ b/9-type-aliases-redux/9-type-aliases-redux.md @@ -78,6 +78,12 @@ Kind consistency is defined by a judgement of the form `Kind.t ~ Kind.t` with the rules presented in the attached [latex document](./latex/kind-judgements.pdf). +## Kind Assignment + +Kind assignment is defined by a judgement +`Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t` with rules presented in the attached +[latex document](./latex/kind-judgements.pdf). + ## Type Elaboration Expansion from unexpanded types to expanded types is defined by @@ -86,19 +92,19 @@ and `TyVarCtx.t |- UHtyp.t <= Kind.t_1 ~> HTyp.t : Kind.t_2` for consistent `Kind.t_1` and `Kind.t_2`. These judgements are shown in the attached [latex document](./latex/kind-judgements.pdf). -`Delta.t`'s `hole_sort` will be expanded to include `TypeHole` so we can track -type holes there too. Since Delta for TypeHoles needs `Kind.t` and `TyVarCtx.t` instead of `HTyp.t` and `VarCtx.t` but we still want to have the same keyspace (the `u`s are consistent and incrementing across all the holes), wrap the value for the `MetaVarMap.t` in `Delta.t` to pack this dependency with the associated `hole_sort`: +`Delta.t`'s `hole_sort` will be changed to a `Hole.t` so we can track type +holes there too. `Hole.t` is an ADT that captures type holes' different `Kind.t` +and `TyVarCtx.t` instead of `HTyp.t` and `VarCtx.t` and we'll still have the +same keyspace (the `u`s are consistent and incrementing across all the holes): ```reasonml -type hole_sort(_, _) = - | ExpressionHole: hole_sort(HTyp.t, VarCtx.t) - | TypeHole: hole_sort(Kind.t, TyVarCtx.t) - | PatternHole: hole_sort(HTyp.t, VarCtx.t); - -type value = - | V(hole_sort('s, 'ctx), 's, 'ctx): value; +module Hole = { +type t = + | Expression(HTyp.t, VarCtx.t) + | Type(Kind.t, TyVarCtx.t) + | Pattern(HTyp.t, VarCtx.t); -type t = MetaVarMap.t(value); +type t = MetaVarMap.t(Hole.t); ``` # Action Semantics From f57dd56cfa90c64ceea82416743d82e24b59790a Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Tue, 23 Mar 2021 20:51:02 -0500 Subject: [PATCH 006/181] Uses cdot for both empty sets --- 9-type-aliases-redux/latex/kind-judgements.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/9-type-aliases-redux/latex/kind-judgements.tex b/9-type-aliases-redux/latex/kind-judgements.tex index b3c44c9..9e2e451 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -15,8 +15,8 @@ \newcommand{\kindAssign}[3]{#1 \vdash #2 : #3} \newcommand{\hPhi}{\Phi} -\newcommand{\EmptyhPhi}{\emptyset} % From hand-written notes, Canonical forms lemma; page 14 -\newcommand{\EmptyDelta}{\cdot} % From hand-written notes, ES-Const rule, page 1 +\newcommand{\EmptyhPhi}{\cdot} +\newcommand{\EmptyDelta}{\cdot} \newcommand{\htau}{\hat{\tau}} \newcommand{\hkappa}{\kappa} From b14fc0cb291dba570f4d4b3f4c3d558be8ba7223 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Tue, 23 Mar 2021 21:04:50 -0500 Subject: [PATCH 007/181] Attempts to add Ana uvar rule --- .../latex/kind-judgements.pdf | Bin 141703 -> 141888 bytes .../latex/kind-judgements.tex | 8 ++++++++ 2 files changed, 8 insertions(+) diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index ea45442d367bcebc4eaf084e5672de1e27cb3721..f77b08c6fc42c3630c3a3edbf774891654fade37 100644 GIT binary patch delta 4889 zcmV+!6XxuP(+I%S2(UQ_0y#93kPj(;?OI818#ffb>sQRtK8PhBcLzNrg_9ylTeNX| zNnAvU6<4Suw_XsWzrG(w8%c3SGg_P!=*7#3dVK5mw&B%f#j9rnKL7pXZ1BxfqI!Mwh6{dte)jyEr&>fuJtLt(ratp_b|9hUm~m!*ap3nk z1s)w3a;E%}#2YSE;Tp?~QmZqX7YGT#tc?iKoDkUxVpwa!As}DZ6Ul^-NtL#2d8I)zX#zluLPvx%xP5_ zqnYBmdjE0o`kYr6`0zQ%OYf?GPxaA{mEewX9k{=$UJZUuhzNWFB3dxVO&bv$mrTRS zavCHYv)t}PV=2sL8o4laP>>E*_8QNjJtEHx4^d$l_lZN3ix0w9hP!poNXdaWNlR;t zrlYIR|BP>shmy?E=yEif-rj|O?9rc->9IJvDBUC=I5}5vAOqL)mW#Q6Z%9`S;jdmL zPMlxxE`t zj|2JHGYDUl+$$!y>>#*L1fPeF5rGBIeSK`GphjhF1rZsMoE<$oh18E<-M_oN|2QrS zA)qk=ePT+>EgV9ML?mW^mr!9!7xKlY-xhrVT+kAa?#dPxOO<{4V|+6%(UFolVKX`@ zr*;Qcz6a{DPg^7~59n^rW(uoIiKiAX!crW*}xq0+b^S!(( zl`UrJq(sIrZE4$R5be>>%ZsxbMW7%Lh8vP*T-mrr`SrKaO;ID9gVC%&*>6J6+#0^> zm8`S`99r4x)?}D8rp>LF)KHaiw-u_oW<_t-!^&e(r!+c$+Kcb6uf`>pm5Qw*m52C$Y9pcr{GUsV_VscTe$)%UEii1U?+?2Wiq7iiv zg$GH@rWOe?D@z6ujRqNQ_K^EBet5UM>g1%Sx;=BG_EA!1JcXKaPW5%r=Xn`IJg>{O z03pVR7(z~e7fMS5noy++7(hA5ZG2lmhr*uM6-jfRT2QlSM4app&?X7!KJgPQX%1Js zo5V#@&KSWtgFHXi8}1OWZ#`v)<}|rGXl+kY9aZxEN(QwQ~rzPQ4S2xhNfIq@ol%V#x&9K^VD5&~?M+&c^s-j_EmQoAe$T6)&A&n4|CFs@uBk=YlUk{6E~W<9vj$l|0YWh|L+ z>Jdj+(hbzpn%pjO%xFG@01yDT$yVeluD=}&(Z6$~sk^&-uG zS$jBnBlL)Kw`SbQE=y$&e~eCb4~9%yVv?&kA3va=sO(Q~T*uGmth%jX( z*A^4fBG=;2L&rpC)rs=dRXP`|ZpByrxs|zvtwpCjvT<(&j<5v5wBHEI>AjoMma_>o>N42o;TXqSZw?%5`cT3NEQx30s)m6p{_<6NOyC zAWfYdQf#T|&lRKM4QKIhd>Vjw?Gd95)OZ*wWSFks=zNg#x21A;%waV0->^n6rt@FBGRZWBXU17#g@Y? 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(Explicit Polymorphism). This PHI adds support for type variables and type aliases to Hazel. -TODO: Merge with Type-alias content - # Type Language + ## Type Identifiers + We define a type `TyId.t` as follows: ``` @@ -19,6 +19,19 @@ variable identifiers, but this may change in the future. We restrict type variable identifiers from using keywords in the same way (to support extraction). +## Type Pattern + +Type patterns are the part of the language between `type` and `=` in `type a = in e` + +It is represented as greek letter `rho` in the attached +[latex document](./latex/kind-judgements.pdf). + +``` +type TPat.t = + | EmptyHole(MetaVar.t) + | TyVar(VarErrStatus.t, TyId.t) +``` + ## Unexpanded Types We extend the syntax of `UHTyp.t` as follows: @@ -52,15 +65,57 @@ as a type variable hole. Like other holes, each type variable hole has a metavariable. ## Kinds + We will need kinds to distinguish types from type-level holes, such as `TyVarHole`. We define a new type, `Kind.t`, as follows: ``` type Kind.t = KHole | Type + | Singleton(HTyp.t) ``` +## Unexpanded Expressions + +To support type aliases, we need to add a new line item to the language and the proposed syntax is + +`TyAliasLine(TPat.t, UHTyp.t)` for abstract syntax + +`type t = in e` for GUI syntax + +### `ZExp.re` : + +By considering the cursor positions of `TyAlias`, we need to add two more zline items which are + +`TyAliasLineT(TPat.t, ZTyp.t)` (when cursor is on the type) + +`TyAliasLineP(ZTPat.t, UHTyp.t)` (when cursor is on the type pattern) + + +## Expanded Expressions + +So that we can still assign a type properly to the resulting `DExp.t` we add the following new syntax: + +`TyBindings(TyVarCtx.t, DHExp.t)` for abstract syntax + +(TODO: is it ever necessary to display this to the user? If so -- maybe something +like `type bindings [type t = \tau, type x] in e`) + # Static Semantics + +This PHI introduces a gradual kind system supporting singletons into Hazel. + +The addition of type variables demands we have a type variable context to store them. + +We can define bidirectional type elaboration judgements which will replace the existing `UHTyp.expand` function. + +Type aliases are given meaning with singleton kinds. A singleton kind is kind dependent on a type. + +Singleton kinds to the language demand a subkinding relation and in the context of gradual typing, subkinding is replaced with consistent subkinding relation derived from PFPL chapter 43 and the [Consistent Subtyping For All](https://i.cs.hku.hk/~bruno/papers/gradual-esop18.pdf) paper. In addition to the subkinding relation, we also add the following mutually recursive auxilliary judgements in order to properly assign singleton kinds in the type system: Kind Formation, Kind Equivalence, Kind Assignment, Kind Constructor Equivalence, (along with Consistent Subkinding). + +We extend bidirectional expression elaboration to support the new type alias form using these auxilliary judgements, and add a new type pattern analysis judgement that yields new bindings. + ## Type Variable Contexts + Type variable contexts are values of type `TyVarCtx.t`, which is defined as follows: @@ -72,10 +127,26 @@ The position within a type variable context determines the de Bruijn index (most recent bindings at the head). For this reason, type identifiers *can* be duplicated. -## Kind Consistency +## Consistent Subkinding + +Consistent Subkinding is defined by a judgement of the form `Kind.t <~ Kind.t` +with the rules presented in the attached +[latex document](./latex/kind-judgements.pdf). + +This judgement handles both consistency and subkinding at the same time. + +## Kind Formation + +Kind Formation is defined by a judgement of the form +`Delta.t; TyVarCtx.t |- Kind.t kind` there is a single rule that says if +`HTyp.t : kind Ty` then we can form the kind `Singleton(Kind.t)`. It is +presented in the attached [latex document](./latex/kind-judgements.pdf). + +## Kind Equivalence -Kind consistency is defined by a judgement of the form `Kind.t ~ Kind.t` with -the rules presented in the attached +Kind Equivalence is a reflexive, symmetric, and transitive relation that caries +kind constructor equivalence through to singleton kinds. +`Delta.t; TyVarCtx.t |- Kind.t_1 === Kind.t_2` It is presented in the attached [latex document](./latex/kind-judgements.pdf). ## Kind Assignment @@ -84,6 +155,15 @@ Kind assignment is defined by a judgement `Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t` with rules presented in the attached [latex document](./latex/kind-judgements.pdf). +## Kind Constructor Equivalence + +Currently, Kind Constructor Equivalence is exactly type equivalence because +Hazel (even after applying this proposal) doesn't support higher kinds. Kind +Constructor Equivalence is a reflexive, symmetric, and transitive relation +where the "type aliasing" using the singleton kind occurs. The judgement +`Delta.t; TyVarCtx.t |- HTyp.t_1 === HTyp.t_2` is defined in the attached +[latex document](./latex/kind-judgements.pdf). + ## Type Elaboration Expansion from unexpanded types to expanded types is defined by @@ -107,8 +187,22 @@ type t = type t = MetaVarMap.t(Hole.t); ``` +## Type Pattern Analysis + +Type patterns analyze with a type to create new tyvar bindings and hole bindings +`Delta.t_1 ; TyVarCtx.t_1 |- TPat.t <- HTyp.t -| TyVarCtx.t_2 ; Delta.t_2`. This +judgement is in the attached [latex document](./latex/kind-judgements.pdf). + +## Expression Elaboration for Type Aliases + +The typical expansion from unexpanded expressions to expanded expressions +handles the new type alias case relying on the Type Pattern Analysis judgement +to generate the bindings for the remaining subexpression. See the attached [latex document](./latex/kind-judgements.pdf). + # Action Semantics +## Type Variables + Adding support for type variables requires adding support for character-level editing at the type-level. Currently, types are bound to individual uppercase characters, e.g. `I` inserts the `Int` type immediately. 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zEj`UH<68SixOlg+T@k&W&TQ#xMK*hcxmRn+71`^)=6Zb1JVNEGe}!CEcvy2bEBl0} z&4sU)PAkIKe9d)oT3QV*Qmv{>Ta+s5T9oQ8 zBqmlM28Kq47GPj#17ZVKU}|PyAQ9|A49R;sdGh8Y=Nw7;OH$b&O9fFo(1s1@z#42p z7uKN#o3IZDunk9W2)j}Pooy?v#+SS zebtLvvhVtTm+j||sD}N18?|Ep=A)WUrHiOl=s8s% Date: Mon, 12 Apr 2021 18:25:53 -0400 Subject: [PATCH 009/181] Requested changes --- 9-type-aliases-redux/9-type-aliases-redux.md | 7 ++++++- .../latex/kind-judgements.pdf | Bin 162213 -> 162291 bytes .../latex/kind-judgements.tex | 4 ++++ 3 files changed, 10 insertions(+), 1 deletion(-) diff --git a/9-type-aliases-redux/9-type-aliases-redux.md b/9-type-aliases-redux/9-type-aliases-redux.md index f407452..d255488 100644 --- a/9-type-aliases-redux/9-type-aliases-redux.md +++ b/9-type-aliases-redux/9-type-aliases-redux.md @@ -27,9 +27,14 @@ It is represented as greek letter `rho` in the attached [latex document](./latex/kind-judgements.pdf). ``` +module VarPatErrStatus = { + type t = + | Keyword(ExpandingKeyword.t) +} + type TPat.t = | EmptyHole(MetaVar.t) - | TyVar(VarErrStatus.t, TyId.t) + | TyVar(option(VarPatErrStatus.t), TyId.t) ``` ## Unexpanded Types diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index 27262afa91874e2f1850189a00b05320f4356e30..01cd3f8e2257a5594dd706cf54ee59d55d71d6db 100644 GIT binary patch delta 2285 zcmVexyU zaqM8r(ER&;tBb9bSGE%zN@w~KCCbs+bG|*7@2o_V^Moc(k7(=q=;Y|$u@nho#3{v- zle2^wMkLjVR+>&7q!HjPhDv5-GLi$BE9am7;05D37lWV}ZJP=?f*D)uVCLDXYx05fP%_eDdqb1Rr z-2n7=RM8j!46xG%07AxeX#nchLW+Myf#*~a#$|~7ROf?475XAIQ= zC7RmJ4c-gbP*n)*D29e5>umZMTgVVA*zF9~c-R@r3cm@p0#LzwQ3|q{Ifj1->yn04 zXG!ElX?YiR@i{l^Hvd~gsO0tk0|jS!2g_| zC+k06KlS~Lb|%Bi^5MD*b}E7BFM%lBa}C%X;m#(K2vK#&Wz2AgQisiHh#k=Qjm#um z-DXQ}V2$6c)$?2NL5)%oc-nvUTRUQr-~(ofv1(O6)A@8bX;q=X6HpXI`M+6XZ09cB zT>%OEsQ*=eFV$g#z`IDc?Hrd!8ef%8ux#`WL#7qgY z&h9+Ah(+blW|O-F>%1z~$$dSE)dxq36KZ?OWQ6Kw$>awrG#hhX!K$FuzV*omFwv` zJHHIGUi&2cUC0Acm9B0|$Q=dJwV45Ee8m 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5a04f77..a442cdb 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -123,6 +123,10 @@ {\kformation{\Delta;\hPhi}{\hkappa}} {$\hkappa$ forms a kind} \;\\ \begin{mathpar} + \inferrule[KFTy] { }{ \kformation{\Delta;\hPhi}{\Ty} } + \and + \inferrule[KFHole] { }{ \kformation{\Delta;\hPhi}{\KHole} } + \and \inferrule[KFSing]{ \kindAssign{\Delta;\hPhi}{\dtau}{\Ty} } { From 1506086b08a8cce0c11904075f9bcee005de1c41 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Sat, 17 Apr 2021 00:17:01 -0500 Subject: [PATCH 010/181] Fixes issues in PHI, adds theorems for reasoning --- 9-type-aliases-redux/9-type-aliases-redux.md | 12 +- .../latex/kind-judgements.pdf | Bin 162291 -> 205166 bytes .../latex/kind-judgements.tex | 166 +++++++++++++----- 3 files changed, 136 insertions(+), 42 deletions(-) diff --git a/9-type-aliases-redux/9-type-aliases-redux.md b/9-type-aliases-redux/9-type-aliases-redux.md index d255488..096a33f 100644 --- a/9-type-aliases-redux/9-type-aliases-redux.md +++ b/9-type-aliases-redux/9-type-aliases-redux.md @@ -119,6 +119,9 @@ Singleton kinds to the language demand a subkinding relation and in the context We extend bidirectional expression elaboration to support the new type alias form using these auxilliary judgements, and add a new type pattern analysis judgement that yields new bindings. +There are several accompanying theorems presented in the attached +[latex document](./latex/kind-judgements.pdf). + ## Type Variable Contexts Type variable contexts are values of type `TyVarCtx.t`, which is defined @@ -158,7 +161,12 @@ kind constructor equivalence through to singleton kinds. Kind assignment is defined by a judgement `Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t` with rules presented in the attached -[latex document](./latex/kind-judgements.pdf). +[latex document](./latex/kind-judgements.pdf). Kind assignment has an affinity for Ty over singletons. + +## Tau Self-recognition and unrecognizing + +Tau self-recognition is defined by a judgement +`Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t =>> Kind.t` and unrecognizing is defined by `Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t <<= Kind.t`. These rules explicitly recognize and unrecognize the singleton within a Ty. ## Kind Constructor Equivalence @@ -175,7 +183,7 @@ Expansion from unexpanded types to expanded types is defined by bidirectional judgements `TyVarCtx.t |- UHTyp.t => Kind.t ~> HTyp.t -| Delta` and `TyVarCtx.t |- UHtyp.t <= Kind.t_1 ~> HTyp.t : Kind.t_2` for consistent `Kind.t_1` and `Kind.t_2`. These judgements are shown in the attached -[latex document](./latex/kind-judgements.pdf). +[latex document](./latex/kind-judgements.pdf). Type elaboration synthesis has an affinity for singletons over Ty. `Delta.t`'s `hole_sort` will be changed to a `Hole.t` so we can track type holes there too. `Hole.t` is an ADT that captures type holes' different `Kind.t` diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index 01cd3f8e2257a5594dd706cf54ee59d55d71d6db..511009671aa2f0d87b4dd898fef4c4e3e7a32e45 100644 GIT binary patch delta 125800 zcmZs>V|3tc5ak)$NyoNr+jd7Cb!;b{7O}ruu@LW4vD4@plr)PFApBVzqMOJ1VD^G1iGa!ttGeJHB z*lDC2n;*AcbHs;Z^9eV8J-y7i+3ap)jbS9i@50`lemq;-orC2BJHn7;`Kw>jXV3Wauvj zg5pQe3S!7#BnOsPf{I2C!0-=&p^p0;<}%FxO)DB&lU~@na$+y>!am8bwuqwswZSfs zK<29$X%e(C=FLJAvlm_(ImD0OqDp4Rd-|C(vNnO+0Ya>dv4|ThBvux~lAvwR>%nB` zqaj$gfP#z5vG%~z6epSyKfp=9XH>}S=tfqBBMiCRs2Ug4Qi!qu_&Jd^HG`2qF#4QO2RK#Sl z3We+a>igwtwN)>)3S=m<0dZiA9A?68tVoKMU4k1KmSYF7Xu2Sno1U;(3ANAJ7Fdtu zV;nHTX|^vD8??pUKAM#kMqm+Iw#66xHb2|vo4z-Vw@VjHGgxVPkzSB;?2Kw<=G;o_ z7Fb>~t9qXRb)@j>Dy7nGt0sAXxtvWfZKk}DL6HQ>;qqbLM$3sl>%{Zm4?PzN9iz_) zzQ6+y?qCCMR;qrt?nDdIvV#D%HA03RJ)OVIje2mWke@Dwe_~&%YS&O1F*^G~d9a|Q zv&e7ax9Fw^jm46U{+Ty66K&2ZoYhz+{?vdSO#xB&6a%LchC9trsmmj?L%_?V7?zG@ z7;m9W8-nd!g~{)Af`T1(XOjB6l=TfgGXn)6SLFhEVF3%hWI+-E!XVbT3boKTLi-j_ zzq$him`~BU6d!mK(Aag+Pv*{cTVs$cT@9nyQ+#-ZvjQ>O%fZpYEIfq>ni)D>U2Gho zOR|S!AO}M=C{gQMd4*eHc{~tC)ePZo)t1qmwkZ!~1aaAD{n`5@5KAN3p}`1CQ+6?q zy1W5b-`~gt*_->MX=+h@aid^;vppjfxWY&vBwVX;=MRV>KVTkN5@!=CKRiAsbBMcP zgJO55simz}!OujeHNbgpS*PTooqBke$GMA2Fnq0Kmt?1=N$^)o#JPAZ%@5-gnr*T} zMUP)g9TdJrb!GEyl(;#T1tYRZ7^12VlNtfepc_t-qKvhH)a=+)%%j) zHM1XyMpq(;>@}LjJ9#sa;U+5=<5M^@BwEI!#H!BP3Vi*i;S;zcxS5Mh6nL;2csrn$ zA_F%0H)efd$+Nq|029&vAut7WL;V+-)j-r@p5)ZQZFr3GB0#WI{!5kwNjAR=jLL`) zE-kO?v2wY%Ptfnkz~RBwFVr*sXe7_^!$^z4=3(*XjX&q)h4yxjWgrHtH5R9r$@VKe zQ!SP$(UWyU@hPooY(p+CO)_eNumKneDtGkF^m7_dyUAUOE7VTJ(GC6VD zZbS@(QMvMQu&MI0<6Pl``}CunBaC+w>g^H$f~AC+3Am;1adkT+sV9Lz)6Z?;6T zn1#?WlVlJ56x26mp&#Og^ug4Isw}I!Utrk9CL*xx((Ia8V14A9%P{RQELj=vR=gRB za_=xOV9aK`L6M#cr9B>Y`c#V1tns9+sc+z4r)^wzxh%F^F3;6daBs3`9hdLVR-VP= z+6tdo_$oWpkJfw+_UL?57Y)E%3YPi&GE^$E7Ii>7&lmTD#3Y7WQUsn<*`o<=&It-> ze{+g2{u`lQv(h#Z8Po*9cp#5^D#=UC_~$MiL@?@ab_EJDi?EnSvsW zxE>Gy+K?A6G?A4>dG?!mnOjxcd&VWsMCbU8Fvs+V`I?~Z;yB@YveZeSt8tsz=5xT# z=k%ll+_&(ryr7nnj7L^6zuTVA;wvq<-*%?XE>6FVZ2w1PZ)^p}%0|pg{67kQemExi z-*)CM7Q`&vTpX#6KS8O1C7lhs4M`NA6|LJj-&r?zv)Z#nwKkglCJ7TVCK*5878{u6 z;$bJ9qF+0WMT~k@GT{w2R4{R)(-v+glj(Fckr$$vwNOr9HxrG0$U=B(9;j7ILu(rO z+OSL-M$xDw80M)%?}5mw6-3CP<22stcee*?*4`6K-4d*D6S5z`OuJt-tVcc=1B9UT zd@@2i*a)|IVH9drgwdiY9ZxdP7+h&Y27*iZq^Ah?ozF}mH9NN;o;Tqx!iP#QDO1ya zzYbXw`_RzU?x)mGy?voy2e|m&|LkeJbhW9HnqNHBvXH(kPU}=cRYWspi^2DG>RK*% z!R#)GN{o%h%?J}Zn}#rwn{Sx?b=9$wO8w&zZA8qHgzu3k{fO)@Z= zc~}RfQ`=tmLSwld(K*rw42~B z_Q3;AeA7-j5c$1=(j%Km3@$|_ro$qJObBUK%B$MdsN?MYkunU12J)UKjcYGntQ}E8 z?SW%1tXU*lJ5krAiL$%3{|;FvjOFvvrnwMF%3!d?3>V^9>Zc=*#^#h3#x4a9%&?vj zN@k(JiJedALldT3oKtkE7yMK@M=I&DUJrpON_LqZ8b_|V#yT95fVARZP%5dL>6-?3 zQSx<9Eq*1a=k`b*qb7Sj`lTejsvEcbYMb*I(a>0o#dGnO-(i?L_6H1sWy=*>y1B8j z#F~os!NGs*xsU6#H()0^FByA@Md)v`qx;+B7CP}XDdyxo2f7M@PrvNzq1XA=D*aBGWYKe*k%-_ z$|PFUM&&GDeb{n---n9^+Ai8zmlFU44duR%hg6+yAKMW*3F3LP>6<6xPkl3!S#3cF zP}dfjO}pHwV$e-{WU-IuS*um0G+#9 z;F>KdMs=lng@&ql#w?leUA66DRw6K-R1E9Ez4s^Xtep|!FX_hG=k^UXS;# znsB>1UFL>{?X`49{-Z5j=JDR^kJ&z9`k3vwJeogT^dSq0NCz1kDil}C)(n3aTaq*qf_! zcNr^8&v_)FyH35U{xIQ(oH1G)TP55RjWYp*X`RoFSbAZTut53FNrBIVf@JEAjZ0OC zz#X)!P5VK&|Cs%%^1*S~SP7EqE%^oXnRZ6QPbn+e>*CU~NcD8Pzm7i3n+uiK3~sOL zrYq22woIyy9mcuQb~QkVdtv0+qv^+v@q({V_1C>IZ@N){%ywOd&+DSwul`+$kmshP z{kFu|8G|v{g7(v~vyG1wC3}J~ zxrJQ>(SR>_C@3a&0;A`<(lbrK;mf)~ADgU$^GJKPSJl z6%Nq3ys`Tne)|5^Vbr`|0tReYXmhZbstCVvqhDv%K5LqqS@BDywgt!jM9V-srUt+T zuVH7$Ti|r9%T(91`+V$eF-{1fppkps~XwB^y& z*e54PSM(b-@;uutz>$R|%#PPWcWp|1KVEg}v?`HDpwVQ)Re<}^mHKUq)Ec_Yl?-~q z==wMjctX2!Ioa^~@(ewhoF|HtB?fr3LLgV^@?f~{q0g>6YXB$g5?T}g8wq-RWn!=^ zTfu~^=6S@bR_%!f!*lopeIoB-UgSvNEQxjaYU=YJaT;q}z$_)DMq|)q&r^`odhMpL zJ|(Lum>Uc7XA*?1rS6LJV70NVKxmf>bqTM_(~;xXbJ*nV9cO$d@&rDjN8Iv>@`gNY zV$wrRQd!RYk+IJ3*MXD9`L4d~%H5iwoc8+w^SlU+#BTH=oSp z4*t+sgW;aKzbp*ltI_oz|QZ;u_H-oI!?Q>0~

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%% Joshua Dunfield macros \def\OPTIONConf{1}% @@ -11,26 +13,26 @@ \newcommand{\elabAna}[6]{#1 \vdash #2 \Leftarrow #3 \leadsto #4 : #5 \dashv #6} \newcommand{\elabSyn}[5]{#1 \vdash #2 \Rightarrow #3 \leadsto #4 \dashv #5} -\newcommand{\patAna}[4]{#1 \vdash #2 \leftarrow #3 \dashv #4} -\newcommand{\kconsistent}[2]{#1 \sim #2} -\newcommand{\kconsubkind}[3]{#1 \vdash #2 <\sim #3} +\newcommand{\patMatch}[4]{#1 \vdash #2 \vartriangleright #3 \dashv #4} +\newcommand{\kconsubkind}[3]{#1 \vdash #2 \lesssim #3} \newcommand{\kindAssign}[3]{#1 \vdash #2 : #3} \newcommand{\kformation}[2]{#1 \vdash #2 \textt{ kind} } \newcommand{\kequiv}[3]{#1 \vdash #2 \equiv #3} \newcommand{\kcequiv}[4]{#1 \vdash #2 \equiv #3 : #4} \newcommand{\ksubkind}[3]{#1 \vdash #2 <:: #3} \newcommand{\tyvarValid}[1]{#1 \textt{ valid} } -\newcommand{\tyvarNotValid}[1]{#1 \textt{ }\lnot\textt{valid} } +\newcommand{\tyvarNotValid}[1]{\lnot(#1 \textt{ }\textt{valid}) } \newcommand{\dexpTypeAssign}[3]{#1 \vdash #2 : #3} - +\newcommand{\tauSelfRecognized}[4]{#1 \vdash #2 : #3 \Rrightarrow \floor{#4}} +\newcommand{\tauUnrecognize}[4]{#1 \vdash #2 : #3 \Rrightarrow \ceil{#4}} \newcommand{\hPhi}{\Phi} \newcommand{\hGamma}{\Gamma} \newcommand{\EmptyhPhi}{\cdot} \newcommand{\EmptyDelta}{\cdot} -\newcommand{\hexp}{\hat{e}} -\newcommand{\dexp}{e} +\newcommand{\hexp}{e} +\newcommand{\dexp}{d} \newcommand{\htau}{\hat{\tau}} \newcommand{\hkappa}{\kappa} \newcommand{\dtau}{\tau} @@ -51,7 +53,7 @@ \newcommand{\Dbinding}[3]{#1 :: #3[#2]} \newcommand{\htdefine}[3]{\textt{type }#1 \textt{ = } #2\textt{ in }#3} -\newcommand{\dtdefine}[2]{\textt{type bindings } #1 \textt{ in } #2} +\newcommand{\dtdefine}[3]{\textt{type }#1 \textt{ = } #2\textt{ in }#3} \begin{document} @@ -73,7 +75,7 @@ \mathsf{User Expression} & \hexp & ::= & \htdefine{\hrho}{\htau}{\hexp} ~\vert~ elided \\ \mathsf{Internal Expression} & \dtau & ::= & - \dtdefine{\hPhi}{\dexp} ~\vert~ elided \\ + \dtdefine{\hrho}{\dtau}{\dexp} ~\vert~ elided \\ \end{array}$ \end{figure} @@ -172,20 +174,21 @@ \begin{minipage}{\linewidth} \judgbox {\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa}} - {$\dtau$ is assigned kind $\hkappa$} + {$\dtau$ is assigned non-singleton kind $\hkappa$} \begin{mathpar} \inferrule[KAConst]{ }{ \kindAssign{\Delta;\hPhi}{c}{\Ty} } \and \inferrule[KAVar]{ - t : \hkappa \in \hPhi + t : \hkappa_1 \in \hPhi \\ + \tauUnrecognize{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2} } { - \kindAssign{\Delta;\hPhi}{t}{\hkappa} + \kindAssign{\Delta;\hPhi}{t}{\hkappa_2} } \and \inferrule[KABinOp]{ - \kindAssign{\Delta;\hPhi}{\dtau_1}{\Ty} + \kindAssign{\Delta;\hPhi}{\dtau_1}{\Ty} \\ \kindAssign{\Delta;\hPhi}{\dtau_2}{\Ty} } { \kindAssign{\Delta;\hPhi}{\dtau_1 \oplus \dtau_2}{\Ty} @@ -211,25 +214,54 @@ } { \kindAssign{\Delta;\hPhi}{\hhole{\dtau}{u}_{\sigma}}{\hkappa} } - \and - \inferrule[KASelfRecognition]{ + + \end{mathpar} +\end{minipage} +\\ +\\ + +\begin{minipage}{\linewidth} + \judgbox + {\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2}} + {$\dtau$ of kind $\hkappa_1$ is self-recognized to consistent subkind $\hkappa_2$} + \begin{mathpar} + \inferrule[KSRTy]{ \kindAssign{\Delta;\hPhi}{\dtau}{\Ty} }{ - \kindAssign{\Delta;\hPhi}{\dtau}{\KSing{\dtau}} + \tauSelfRecognized{\Delta;\hPhi}{\dtau}{\Ty}{\KSing{\dtau}} } \and - \inferrule[KASubkind]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_1} \\ - \kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} - } { - \kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_2} + \inferrule[KSRHole]{ + \kindAssign{\Delta;\hPhi}{\dtau}{\KHole} + }{ + \tauSelfRecognized{\Delta;\hPhi}{\dtau}{\KHole}{\KHole} } - \end{mathpar} \end{minipage} \\ \\ +\begin{minipage}{\linewidth} + \judgbox + {\tauUnrecognize{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2}} + {$\dtau$ of kind $\hkappa_1$ is unrecognized to consistent superkind $\hkappa_2$} + \begin{mathpar} + \inferrule[KUSing]{ + \kindAssign{\Delta;\hPhi}{\dtau}{\Ty} \\ + \tauSelfRecognized{\Delta;\hPhi}{\dtau}{\Ty}{\KSing{\dtau}} + }{ + \tauUnrecognize{\Delta;\hPhi}{\dtau}{\KSing{\dtau}}{\Ty} + } + \and + \inferrule[KUHole]{ + \kindAssign{\Delta;\hPhi}{\dtau}{\KHole} + }{ + \tauUnrecognize{\Delta;\hPhi}{\dtau}{\KHole}{\KHole} + } + \end{mathpar} +\end{minipage} +\\ +\\ @@ -238,7 +270,9 @@ {\kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\hkappa}} {$\dtau_1$ is equivalent to $\dtau_2$ and has kind $\hkappa_2$} \;\\ \begin{mathpar} - \inferrule[KCERefl]{ } { + \inferrule[KCERefl]{ + \kindAssign{\Delta;\hPhi}{\dtau}{\hkappa} + } { \kcequiv{\Delta;\hPhi}{\dtau}{\dtau}{\hkappa} } \and @@ -271,20 +305,23 @@ {$\htau$ synthesizes kind $\hkappa$ and elaborates to $\dtau$} \begin{mathpar} \inferrule[TElabSConst]{ }{ - \elabSyn{\hPhi}{c}{\Ty}{c}{\EmptyDelta} + \elabSyn{\hPhi}{c}{\KSing{c}}{c}{\EmptyDelta} } \and \inferrule[TElabSBinOp]{ \elabAna{\hPhi}{\htau_1}{\Ty}{\dtau_1}{\Ty}{\Delta_1} \\ - \elabAna{\hPhi}{\htau_2}{\Ty}{\dtau_2}{\Ty}{\Delta_2} + \tauSelfRecognized{\Delta_1;\hPhi}{\dtau_1}{\Ty}{\KSing{\dtau_1}} \\ + \elabAna{\hPhi}{\htau_2}{\Ty}{\dtau_2}{\Ty}{\Delta_2} \\ + \tauSelfRecognized{\Delta_2;\hPhi}{\dtau_2}{\Ty}{\KSing{\dtau_2}} }{ - \elabSyn{\hPhi}{\htau_1 \oplus \htau_2}{\Ty}{\dtau_1 \oplus \dtau_2}{\Delta_1 \cup \Delta_2} + \elabSyn{\hPhi}{\htau_1 \oplus \htau_2}{\KSing{\dtau_1 \oplus \dtau_2}}{\dtau_1 \oplus \dtau_2}{\Delta_1 \cup \Delta_2} } \and \inferrule[TElabSList]{ - \elabAna{\hPhi}{\htau}{\Ty}{\dtau}{\Ty}{\Delta} + \elabAna{\hPhi}{\htau}{\Ty}{\dtau}{\Ty}{\Delta} \\ + \tauSelfRecognized{\Delta;\hPhi}{\dtau}{\Ty}{\KSing{\dtau}} } { - \elabSyn{\hPhi}{\hlist{\htau}}{\Ty}{\hlist{\dtau}}{\Delta} + \elabSyn{\hPhi}{\hlist{\htau}}{\KSing{\hlist{\dtau}}}{\hlist{\dtau}}{\Delta} } \and \inferrule[TElabSVar]{ @@ -316,14 +353,14 @@ \judgbox {\elabAna{\hPhi}{\htau}{\hkappa_1}{\dtau}{\hkappa_2}{\Delta}} {$\htau$ analyzes against kind $\hkappa_1$ and - elaborates to $\dtau$ of consistent kind $\hkappa_2$} + elaborates to $\dtau$ of consistent subkind $\hkappa_2$} \begin{mathpar} \inferrule[TElabASubsume]{ \htau \neq t \text{ where } t \not\in \hPhi \\ \htau \neq \hthole{u} \\ \htau \neq \hhole{\htau'}{u} \\ \elabSyn{\hPhi}{\htau}{\hkappa'}{\dtau}{\Delta} \\ - \kconsistent{\hkappa}{\hkappa'} + \kconsubkind{\Delta;\hPhi}{\hkappa'}{\hkappa} }{ \elabAna{\hPhi}{\htau}{\hkappa}{\dtau}{\hkappa'}{\Delta} } @@ -351,24 +388,25 @@ \begin{minipage}{\linewidth} \judgbox - {\patAna{\Delta_1;\hPhi_1}{\hrho}{\dtau}{\hPhi_2;\Delta_2}} + {\patMatch{\Delta_1;\hPhi_1}{\dtau}{\hrho}{\hPhi_2;\Delta_2}} {$\hrho$ analyzes against $\dtau$ yielding new tyvar and hole bindings} \begin{mathpar} \inferrule[RESVar]{ \tyvarValid{t} \\ - \kindAssign{\Delta;\hPhi}{\dtau}{\hkappa} + \kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_1} \\ + \tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2} }{ - \patAna{\Delta;\hPhi}{t}{\dtau}{t :: \hkappa;\EmptyDelta} + \patMatch{\Delta;\hPhi}{\dtau}{t}{t :: \hkappa_2;\EmptyDelta} } \and \inferrule[RESEHole]{ }{ - \patAna{\Delta;\hPhi}{\hthole{u}}{\dtau}{\EmptyhPhi;\Dbinding{u}{\hPhi}{\kehole}} + \patMatch{\Delta;\hPhi}{\dtau}{\hthole{u}}{\EmptyhPhi;\Dbinding{u}{\hPhi}{\kehole}} } \and \inferrule[RESVarHole]{ \tyvarNotValid{t} }{ - \patAna{\Delta;\hPhi}{\hhole{t}{u}}{\dtau}{\EmptyhPhi;\Dbinding{u}{\hPhi}{\kehole}} + \patMatch{\Delta;\hPhi}{\dtau}{\hhole{t}{u}}{\EmptyhPhi;\Dbinding{u}{\hPhi}{\kehole}} } \end{mathpar} \end{minipage} @@ -382,10 +420,10 @@ \begin{mathpar} \inferrule[ESDefine]{ \elabSyn{\hPhi_1}{\htau}{\hkappa}{\dtau}{\Delta_1} \\ - \patAna{\Delta_1;\hPhi_1}{\hrho}{\dtau}{\hPhi_2;\Delta_2} \\ + \patMatch{\Delta_1;\hPhi_1}{\dtau}{\hrho}{\hPhi_2;\Delta_2} \\ \elabSyn{\hGamma;\hPhi_1 \cup \hPhi_2}{\hexp}{\dtau_1}{\dexp}{\Delta_3} }{ - \elabSyn{\hGamma;\hPhi_1}{\htdefine{\hrho}{\htau}{\hexp}}{\dtau_1}{\dtdefine{\hPhi_2}{\dexp}}{\Delta_1 \cup \Delta_2 \cup \Delta_3} + \elabSyn{\hGamma;\hPhi_1}{\htdefine{\hrho}{\htau}{\hexp}}{\dtau_1}{\dtdefine{\hrho}{\dtau}{\dexp}}{\Delta_1 \cup \Delta_2 \cup \Delta_3} } @@ -394,24 +432,72 @@ \begin{minipage}{\linewidth} \judgbox - {\dexpTypeAssign{\hGamma;\hPhi}{\dexp}{\dtau}} + {\dexpTypeAssign{\Delta;\hGamma;\hPhi}{\dexp}{\dtau}} {$\dexp$ is assigned type $\dtau$} \begin{mathpar} \inferrule[DEDefine]{ - \dexpTypeAssign{\hGamma;\hPhi_1 \cup \hPhi_2}{\dexp}{\dtau} + \patMatch{\Delta;\hPhi_1}{\dtau}{\hrho}{\hPhi_2;\Delta} \\ + \dexpTypeAssign{\Delta;\hGamma;\hPhi_1 \cup \hPhi_2}{\dexp}{\dtau} }{ - \dexpTypeAssign{\hGamma;\hPhi_1}{\dtdefine{\hPhi_2}{\dexp}}{\dtau} + \dexpTypeAssign{\Delta;\hGamma;\hPhi_1}{\dtdefine{\hrho}{\dtau}{\dexp}}{\dtau} } \end{mathpar} \end{minipage} +\\ +\\ + +\newtheorem{thm}{Theorem} +\begin{thm}[Well-Kinded Elaboration]\ \\ + (1) If $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ and $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa'}$ then $\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa'}{\hkappa}$ \\ + (2) If $\elabAna{\hPhi}{\htau}{\hkappa_1}{\dtau}{\hkappa_2}{\Delta}$ then $\kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}$ and $\exists \hkappa_3$ such that $\kconsubkind{\Delta;\hPhi}{\hkappa_3}{\hkappa_2}$ and $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_3}$ +\end{thm} +\noindent +This is like the Typed Elaboration theorem in the POPL19 paper except that (1) synthesis produces the most refined kind while assignment prefers Ty (see \textit{Kind Assignment Ty Affinity} theorem) and (2) analysis doesn't necessarily produce the most consistent subkind for $\tau$. + +\begin{thm}[Type Elaboration Synthesis Singleton Affinity]\ \\ + If $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ then $\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa}{\hkappa}$ +\end{thm} +\noindent +Type Elaboration synthesis always synthesizes the most self-recognized kind. + +\begin{thm}[Kind Assignment Unicity]\ \\ + If $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa}$ and $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa'}$ then $\hkappa = \hkappa'$ +\end{thm} +\noindent +This is like the Type Unicity theorem in the POPL19 paper. + +\begin{thm}[Kind Assignment Ty Affinity]\ \\ + If $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa}$ then $\tauUnrecognize{\Delta;\hPhi}{\dtau}{\hkappa}{\hkappa}$ +\end{thm} +\noindent +Kind assignment assigns $\Ty$ rather than $\KSing{\dtau}$. The kind is explicitly refined where needed in other places using the recognition judgment: $\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2}$. + + +\begin{thm}[Type Elaboration Unicity]\ \\ + (1) If $\elabSyn{\hPhi}{\htau}{\hkappa_1}{\dtau_1}{\Delta_1}$ and $\elabSyn{\hPhi}{\htau}{\hkappa_2}{\dtau_2}{\Delta_2}$ then $\hkappa_1 = \hkappa_2$, $\dtau_1 = \dtau_2$, $\Delta_1 = \Delta_2$ \\ + (2) If $\elabAna{\hPhi}{\htau}{\hkappa}{\dtau_1}{\hkappa_1}{\Delta_1}$ and $\elabAna{\hPhi}{\htau}{\hkappa}{\dtau_2}{\hkappa_2}{\Delta_2}$ then $\dtau_1 = \dtau_2$, $\hkappa_1 = \hkappa_2$, $\Delta_1 = \Delta_2$ +\end{thm} +\noindent +This is like the Elaboration Unicity theorem in the POPL19 paper. + +\begin{thm}[Kind Precision]\ \\ + If $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ and $\elabAna{\hPhi}{\htau}{\hkappa'}{\dtau}{\hkappa''}{\Delta}$ then $\kconsubkind{\Delta;\hPhi}{\hkappa}{\hkappa'}$ +\end{thm} +\noindent +Kind Precision says that any kind $\hkappa$ that successfully analyzes against $\htau$ is a consistent superkind of the one that is synthesized from $\htau$. In other words, the kind that is synthesized is the most precise. +\begin{thm}[Recognize Unrecognize Self-inverse]\ \\ + $\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa_2}{\hkappa_1}$ iff $\tauUnrecognize{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2}$ +\end{thm} +\noindent +Self recognition and unrecognizing are self-inverse with eachother From 2759c7e4685bb7f9f7b400a84ce71ebd38ffdf25 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Sat, 17 Apr 2021 12:14:21 -0500 Subject: [PATCH 011/181] Unrecognition tweak --- .../latex/kind-judgements.pdf | Bin 205166 -> 204396 bytes .../latex/kind-judgements.tex | 22 ++++++++---------- 2 files changed, 10 insertions(+), 12 deletions(-) diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index 511009671aa2f0d87b4dd898fef4c4e3e7a32e45..75d11c32fb8d847c8b76708e08dd58cc313e2a27 100644 GIT binary patch delta 54550 zcmV(rK<>Zp!VK)Z46q{ze@$=OMi9OGSMX>bwB4EgPEStZrijxPjg4HA0CjBH5Nb<} 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zK>kNH0G#xxlEBTKY7i)#s#3tqfhrArKd6S_LcgkExLjCJm4Q2dRE@x0y{Sgw3Z<$s zxTj0iINaxlY64!MxW^>fBm3||hXya6vwv4kErAMUZe(+Ga%Ev{3T19&Z(?c+b97;H Wba--QW(qPeG&V3c3MC~)Peuxm;51$U diff --git a/9-type-aliases-redux/latex/kind-judgements.tex b/9-type-aliases-redux/latex/kind-judgements.tex index 4d44b79..cfab4bc 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -24,7 +24,7 @@ \newcommand{\tyvarNotValid}[1]{\lnot(#1 \textt{ }\textt{valid}) } \newcommand{\dexpTypeAssign}[3]{#1 \vdash #2 : #3} \newcommand{\tauSelfRecognized}[4]{#1 \vdash #2 : #3 \Rrightarrow \floor{#4}} -\newcommand{\tauUnrecognize}[4]{#1 \vdash #2 : #3 \Rrightarrow \ceil{#4}} +\newcommand{\tauUnrecognize}[3]{#1 \vdash #2 \Rrightarrow \ceil{#3}} \newcommand{\hPhi}{\Phi} \newcommand{\hGamma}{\Gamma} @@ -182,7 +182,7 @@ \and \inferrule[KAVar]{ t : \hkappa_1 \in \hPhi \\ - \tauUnrecognize{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2} + \tauUnrecognize{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} } { \kindAssign{\Delta;\hPhi}{t}{\hkappa_2} } @@ -243,20 +243,18 @@ \begin{minipage}{\linewidth} \judgbox - {\tauUnrecognize{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2}} - {$\dtau$ of kind $\hkappa_1$ is unrecognized to consistent superkind $\hkappa_2$} + {\tauUnrecognize{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}} + {$\hkappa_1$ is unrecognized to consistent superkind $\hkappa_2$} \begin{mathpar} \inferrule[KUSing]{ \kindAssign{\Delta;\hPhi}{\dtau}{\Ty} \\ \tauSelfRecognized{\Delta;\hPhi}{\dtau}{\Ty}{\KSing{\dtau}} }{ - \tauUnrecognize{\Delta;\hPhi}{\dtau}{\KSing{\dtau}}{\Ty} + \tauUnrecognize{\Delta;\hPhi}{\KSing{\dtau}}{\Ty} } \and - \inferrule[KUHole]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\KHole} - }{ - \tauUnrecognize{\Delta;\hPhi}{\dtau}{\KHole}{\KHole} + \inferrule[KUHole]{ }{ + \tauUnrecognize{\Delta;\hPhi}{\KHole}{\KHole} } \end{mathpar} \end{minipage} @@ -492,12 +490,12 @@ Kind Precision says that any kind $\hkappa$ that successfully analyzes against $\htau$ is a consistent superkind of the one that is synthesized from $\htau$. In other words, the kind that is synthesized is the most precise. -\begin{thm}[Recognize Unrecognize Self-inverse]\ \\ - $\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa_2}{\hkappa_1}$ iff $\tauUnrecognize{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2}$ +\begin{thm}[Reognition Inversion]\ \\ + If $\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa_2}{\hkappa_1}$ then $\tauUnrecognize{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}$ \end{thm} \noindent -Self recognition and unrecognizing are self-inverse with eachother +Unrecognizing inverts self-recognition. From c43241ee9508229fd55f1e41b564a34d470cb2c2 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Mon, 19 Apr 2021 22:09:06 -0500 Subject: [PATCH 012/181] Fixes for broken handling of tyvars --- 9-type-aliases-redux/9-type-aliases-redux.md | 11 +- .../latex/kind-judgements.pdf | Bin 204396 -> 209908 bytes .../latex/kind-judgements.tex | 120 +++++++++++------- 3 files changed, 81 insertions(+), 50 deletions(-) diff --git a/9-type-aliases-redux/9-type-aliases-redux.md b/9-type-aliases-redux/9-type-aliases-redux.md index 096a33f..e5333be 100644 --- a/9-type-aliases-redux/9-type-aliases-redux.md +++ b/9-type-aliases-redux/9-type-aliases-redux.md @@ -163,10 +163,15 @@ Kind assignment is defined by a judgement `Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t` with rules presented in the attached [latex document](./latex/kind-judgements.pdf). Kind assignment has an affinity for Ty over singletons. -## Tau Self-recognition and unrecognizing +## Kind AssignmentS -Tau self-recognition is defined by a judgement -`Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t =>> Kind.t` and unrecognizing is defined by `Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t <<= Kind.t`. These rules explicitly recognize and unrecognize the singleton within a Ty. +Kind assignmentS is defined by a judgement +`Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t` with rules presented in the attached +[latex document](./latex/kind-judgements.pdf). Kind assignmentS has an affinity for singletons over Ty. + +## Tau Unrecognizing + +Tau unrecognizing is defined by a judgement `Delta.t ; TyVarCtx.t |- Kind.t <<= Kind.t`. This judgment explicitly unrecognizes the singleton in exchange for a Ty. ## Kind Constructor Equivalence diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index 75d11c32fb8d847c8b76708e08dd58cc313e2a27..dc2dbcc89f540bbf6ac4c93cb72c3dadd36e3b55 100644 GIT binary patch delta 90872 zcmZs?Q*bU^v@9Ciwr$(?if!B07u&XN+csB{72CFx`|ne=t8UdfZ#`efL(d-FJ+6A- zo=)Ks$bmVMnaOAp6-fY1a#?`g=IYBSn{_8V7*@YXG?%lj;5B0G{vyfs;(5+4Z zeq25tWHu5oY~Z!#=m@GHuCzon6eHB!*`n9sa_Nst=coItm15Mf&7m2hA+!o-V?w6E zP_&gk8L`sJ5&}VgWhm%Hj3PMbQrg1yb>p_YlS1}RF^MRdI~RZyW{sp4xl>mkLS-Qu zbK$GTqe>dRy$m&C12Zy+6Np9}S9<8vC0TB2H3}kAOv*mIVG1F1L-S@rvKR@BPiK2_ zy#d3ClVdS=QqmUi6QbrG5)hc-I6V*+;&~EKg;fX`G{~Pup%7=ncZ+D()IUpKwk79x z?%X&k{4pBDDr^9Wlr;hQ_!eZr@^|K;Ytz1rqw#-{<J=6qsU3==}L?6+VFdT~*R}f0Yq;XiZyhl8y9f8V_ z*~=)rMfbRwkriB~vNwjY+qEC>`o0AckwARgAN24`3410*iAzOw`A%sHC;sRU{J>Ep zOy7t$a^rxU7Y2S!;PWS?&-yQks$q)p#9xaaQ1ia{cG)%us%mgoNYd$=6LNmL=0-t@ 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\newcommand{\patMatch}[4]{#1 \vdash #2 \vartriangleright #3 \dashv #4} \newcommand{\kconsubkind}[3]{#1 \vdash #2 \lesssim #3} \newcommand{\kindAssign}[3]{#1 \vdash #2 : #3} +\newcommand{\kindAssignS}[3]{#1 \vdash #2 :_s #3} \newcommand{\kformation}[2]{#1 \vdash #2 \textt{ kind} } \newcommand{\kequiv}[3]{#1 \vdash #2 \equiv #3} \newcommand{\kcequiv}[4]{#1 \vdash #2 \equiv #3 : #4} @@ -23,7 +24,6 @@ \newcommand{\tyvarValid}[1]{#1 \textt{ valid} } \newcommand{\tyvarNotValid}[1]{\lnot(#1 \textt{ }\textt{valid}) } \newcommand{\dexpTypeAssign}[3]{#1 \vdash #2 : #3} -\newcommand{\tauSelfRecognized}[4]{#1 \vdash #2 : #3 \Rrightarrow \floor{#4}} \newcommand{\tauUnrecognize}[3]{#1 \vdash #2 \Rrightarrow \ceil{#3}} \newcommand{\hPhi}{\Phi} @@ -222,20 +222,47 @@ \begin{minipage}{\linewidth} \judgbox - {\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2}} - {$\dtau$ of kind $\hkappa_1$ is self-recognized to consistent subkind $\hkappa_2$} + {\kindAssignS{\Delta;\hPhi}{\dtau}{\hkappa}} + {$\dtau$ is assigned kind (with singleton affinity) $\hkappa$} \begin{mathpar} - \inferrule[KSRTy]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\Ty} - }{ - \tauSelfRecognized{\Delta;\hPhi}{\dtau}{\Ty}{\KSing{\dtau}} + \inferrule[KAConst]{ }{ + \kindAssignS{\Delta;\hPhi}{c}{\KSing{c}} } \and - \inferrule[KSRHole]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\KHole} - }{ - \tauSelfRecognized{\Delta;\hPhi}{\dtau}{\KHole}{\KHole} + \inferrule[KAVar]{ + t : \hkappa \in \hPhi + } { + \kindAssignS{\Delta;\hPhi}{t}{\hkappa} + } + \and + \inferrule[KABinOp]{ + \kindAssignS{\Delta;\hPhi}{\dtau_1}{\KSing{\dtau_1'}} \\ + \kindAssignS{\Delta;\hPhi}{\dtau_2}{\KSing{\dtau_2'}} + } { + \kindAssignS{\Delta;\hPhi}{\dtau_1 \oplus \dtau_2}{\KSing{\dtau_1' \oplus \dtau_2'}} + } + \and + \inferrule[KAList]{ + \kindAssignS{\Delta;\hPhi}{\dtau}{\KSing{\dtau'}} + } { + \kindAssignS{\Delta;\hPhi}{\hlist{\dtau}}{\KSing{\hlist{\dtau'}}} + } + \and + \inferrule[KAEHole]{ + \Dbinding{u}{\hPhi'}{\hkappa} \in \Delta \\ + \kindAssignS{\Delta;\hPhi}{\sigma}{\hPhi'} + } { + \kindAssignS{\Delta;\hPhi}{\hthole{u}_{\sigma}}{\hkappa} + } + \and + \inferrule[KANEHole]{ + \kindAssignS{\Delta;\hPhi}{\dtau}{\hkappa'} \\ + \Dbinding{u}{\hPhi'}{\hkappa} \in \Delta \\ + \kindAssignS{\Delta;\hPhi}{\sigma}{\hPhi'} + } { + \kindAssignS{\Delta;\hPhi}{\hhole{\dtau}{u}_{\sigma}}{\hkappa} } + \end{mathpar} \end{minipage} \\ @@ -247,10 +274,14 @@ {$\hkappa_1$ is unrecognized to consistent superkind $\hkappa_2$} \begin{mathpar} \inferrule[KUSing]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\Ty} \\ - \tauSelfRecognized{\Delta;\hPhi}{\dtau}{\Ty}{\KSing{\dtau}} + \kindAssignS{\Delta;\hPhi}{\dtau}{\KSing{\dtau'}} + }{ + \tauUnrecognize{\Delta;\hPhi}{\KSing{\dtau'}}{\Ty} + } + \and + \inferrule[KUTy]{ }{ - \tauUnrecognize{\Delta;\hPhi}{\KSing{\dtau}}{\Ty} + \tauUnrecognize{\Delta;\hPhi}{\Ty}{\Ty} } \and \inferrule[KUHole]{ }{ @@ -288,7 +319,7 @@ } \and \inferrule[KCESingEquiv]{ - \kindAssign{\Delta;\hPhi}{\dtau_1}{\KSing{\dtau_2}} + \kindAssignS{\Delta;\hPhi}{\dtau_1}{\KSing{\dtau_2}} } { \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\Ty} } @@ -307,19 +338,16 @@ } \and \inferrule[TElabSBinOp]{ - \elabAna{\hPhi}{\htau_1}{\Ty}{\dtau_1}{\Ty}{\Delta_1} \\ - \tauSelfRecognized{\Delta_1;\hPhi}{\dtau_1}{\Ty}{\KSing{\dtau_1}} \\ - \elabAna{\hPhi}{\htau_2}{\Ty}{\dtau_2}{\Ty}{\Delta_2} \\ - \tauSelfRecognized{\Delta_2;\hPhi}{\dtau_2}{\Ty}{\KSing{\dtau_2}} + \elabAna{\hPhi}{\htau_1}{\Ty}{\dtau_1}{\KSing{\dtau_1'}}{\Delta_1} \\ + \elabAna{\hPhi}{\htau_2}{\Ty}{\dtau_2}{\KSing{\dtau_2'}}{\Delta_2} }{ - \elabSyn{\hPhi}{\htau_1 \oplus \htau_2}{\KSing{\dtau_1 \oplus \dtau_2}}{\dtau_1 \oplus \dtau_2}{\Delta_1 \cup \Delta_2} + \elabSyn{\hPhi}{\htau_1 \oplus \htau_2}{\KSing{\dtau_1' \oplus \dtau_2'}}{\dtau_1 \oplus \dtau_2}{\Delta_1 \cup \Delta_2} } \and \inferrule[TElabSList]{ - \elabAna{\hPhi}{\htau}{\Ty}{\dtau}{\Ty}{\Delta} \\ - \tauSelfRecognized{\Delta;\hPhi}{\dtau}{\Ty}{\KSing{\dtau}} + \elabAna{\hPhi}{\htau}{\Ty}{\dtau}{\KSing{\dtau'}}{\Delta} } { - \elabSyn{\hPhi}{\hlist{\htau}}{\KSing{\hlist{\dtau}}}{\hlist{\dtau}}{\Delta} + \elabSyn{\hPhi}{\hlist{\htau}}{\KSing{\hlist{\dtau'}}}{\hlist{\dtau}}{\Delta} } \and \inferrule[TElabSVar]{ @@ -391,10 +419,9 @@ \begin{mathpar} \inferrule[RESVar]{ \tyvarValid{t} \\ - \kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_1} \\ - \tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2} + \kindAssignS{\Delta;\hPhi}{\dtau}{\hkappa} }{ - \patMatch{\Delta;\hPhi}{\dtau}{t}{t :: \hkappa_2;\EmptyDelta} + \patMatch{\Delta;\hPhi}{\dtau}{t}{t :: \hkappa;\EmptyDelta} } \and \inferrule[RESEHole]{ }{ @@ -448,29 +475,37 @@ \newtheorem{thm}{Theorem} \begin{thm}[Well-Kinded Elaboration]\ \\ - (1) If $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ and $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa'}$ then $\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa'}{\hkappa}$ \\ - (2) If $\elabAna{\hPhi}{\htau}{\hkappa_1}{\dtau}{\hkappa_2}{\Delta}$ then $\kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}$ and $\exists \hkappa_3$ such that $\kconsubkind{\Delta;\hPhi}{\hkappa_3}{\hkappa_2}$ and $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_3}$ + (1) If $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ and $\kindAssignS{\Delta;\hPhi}{\dtau}{\hkappa}$ \\ + (2) If $\elabAna{\hPhi}{\htau}{\hkappa_1}{\dtau}{\hkappa_2}{\Delta}$ then $\kconsubkind{\Delta;\hPhi}{\hkappa_2}{\hkappa_1}$ and $\kindAssignS{\Delta;\hPhi}{\dtau}{\hkappa_2}$ \end{thm} \noindent -This is like the Typed Elaboration theorem in the POPL19 paper except that (1) synthesis produces the most refined kind while assignment prefers Ty (see \textit{Kind Assignment Ty Affinity} theorem) and (2) analysis doesn't necessarily produce the most consistent subkind for $\tau$. +This is like the Typed Elaboration theorem in the POPL19 paper. Note that analysis produces the most precise kind (preferring singletons) even when analyzing against unprecise $\Ty$. This is because the relevant rules are handled by synthesis. -\begin{thm}[Type Elaboration Synthesis Singleton Affinity]\ \\ - If $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ then $\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa}{\hkappa}$ +\begin{thm}[Type Elaboration Singleton Affinity]\ \\ + (1) If $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ then $\hkappa = \KSing{\dtau'}$ or $\hkappa = \KHole$. \\ + (2) If $\elabAna{\hPhi}{\htau}{\hkappa_1}{\dtau}{\hkappa_2}{\Delta}$ then $\hkappa_2 = \KSing{\dtau'}$ or $\hkappa_2 = \KHole$. \end{thm} \noindent -Type Elaboration synthesis always synthesizes the most self-recognized kind. +Type Elaboration always elaborates to the most precise kind -- it never elaborates to $\Ty$ directly even when analyzed against $\Ty$. \begin{thm}[Kind Assignment Unicity]\ \\ - If $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa}$ and $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa'}$ then $\hkappa = \hkappa'$ + (1) If $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa}$ and $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa'}$ then $\hkappa = \hkappa'$ \\ + (2) If $\kindAssignS{\Delta;\hPhi}{\dtau}{\hkappa}$ and $\kindAssignS{\Delta;\hPhi}{\dtau}{\hkappa'}$ then $\hkappa = \hkappa'$ \end{thm} \noindent This is like the Type Unicity theorem in the POPL19 paper. \begin{thm}[Kind Assignment Ty Affinity]\ \\ - If $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa}$ then $\tauUnrecognize{\Delta;\hPhi}{\dtau}{\hkappa}{\hkappa}$ + If $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa}$ then $\tauUnrecognize{\Delta;\hPhi}{\hkappa}{\hkappa}$ \end{thm} \noindent -Kind assignment assigns $\Ty$ rather than $\KSing{\dtau}$. The kind is explicitly refined where needed in other places using the recognition judgment: $\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa_1}{\hkappa_2}$. +Kind assignment assigns $\Ty$ rather than a singleton. When the kind needs to be a singleton rather than $\Ty$, instead use: $\kindAssignS{\Delta;\hPhi}{\dtau}{\hkappa}$. + +\begin{thm}[Kind AssignmentS Singleton Affinity]\ \\ + If $\kindAssignS{\Delta;\hPhi}{\dtau}{\hkappa}$ then $\hkappa = \KSing{\dtau'}$ or $\hkappa = \KHole$. +\end{thm} +\noindent +Kind assignment assigns a singleton rather than $\Ty$. When the kind needs to be a $\Ty$, instead use: $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa}$. \begin{thm}[Type Elaboration Unicity]\ \\ @@ -482,21 +517,12 @@ -\begin{thm}[Kind Precision]\ \\ - If $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ and $\elabAna{\hPhi}{\htau}{\hkappa'}{\dtau}{\hkappa''}{\Delta}$ then $\kconsubkind{\Delta;\hPhi}{\hkappa}{\hkappa'}$ +\begin{thm}[Kind Assignment Consistency]\ \\ + If $\kindAssignS{\Delta;\hPhi}{\dtau}{\hkappa_1}$ and $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_2}$ then $\kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}$ and $\hkappa_1,\hkappa_2$ are either both $\KHole$ or neither is. \end{thm} \noindent -Kind Precision says that any kind $\hkappa$ that successfully analyzes against $\htau$ is a consistent superkind of the one that is synthesized from $\htau$. In other words, the kind that is synthesized is the most precise. - - -\begin{thm}[Reognition Inversion]\ \\ - If $\tauSelfRecognized{\Delta;\hPhi}{\dtau}{\hkappa_2}{\hkappa_1}$ then $\tauUnrecognize{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}$ -\end{thm} - -\noindent -Unrecognizing inverts self-recognition. - +Kind Assignment Consistency says that both the singleton form and non-singleton form of kind assignment differ only in that the singleton form is more precise. But both must agree on holes. \end{document} From b2ea0f12dceeb79e3032d296b5c25ab0c3c8eaf4 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Tue, 20 Apr 2021 16:31:15 -0500 Subject: [PATCH 013/181] Made suggested changes --- 9-type-aliases-redux/9-type-aliases-redux.md | 23 +- .../latex/kind-judgements.pdf | Bin 209908 -> 195626 bytes .../latex/kind-judgements.tex | 229 +++++++----------- 3 files changed, 94 insertions(+), 158 deletions(-) diff --git a/9-type-aliases-redux/9-type-aliases-redux.md b/9-type-aliases-redux/9-type-aliases-redux.md index e5333be..3edd1da 100644 --- a/9-type-aliases-redux/9-type-aliases-redux.md +++ b/9-type-aliases-redux/9-type-aliases-redux.md @@ -159,19 +159,9 @@ kind constructor equivalence through to singleton kinds. ## Kind Assignment -Kind assignment is defined by a judgement +Kind assignment is defined by a declarative judgement `Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t` with rules presented in the attached -[latex document](./latex/kind-judgements.pdf). Kind assignment has an affinity for Ty over singletons. - -## Kind AssignmentS - -Kind assignmentS is defined by a judgement -`Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t` with rules presented in the attached -[latex document](./latex/kind-judgements.pdf). Kind assignmentS has an affinity for singletons over Ty. - -## Tau Unrecognizing - -Tau unrecognizing is defined by a judgement `Delta.t ; TyVarCtx.t |- Kind.t <<= Kind.t`. This judgment explicitly unrecognizes the singleton in exchange for a Ty. +[latex document](./latex/kind-judgements.pdf). Kind assignment can derive both singleton and Ty kinds for valid types. ## Kind Constructor Equivalence @@ -186,8 +176,7 @@ where the "type aliasing" using the singleton kind occurs. The judgement Expansion from unexpanded types to expanded types is defined by bidirectional judgements `TyVarCtx.t |- UHTyp.t => Kind.t ~> HTyp.t -| Delta` -and `TyVarCtx.t |- UHtyp.t <= Kind.t_1 ~> HTyp.t : Kind.t_2` for consistent -`Kind.t_1` and `Kind.t_2`. These judgements are shown in the attached +and `TyVarCtx.t |- UHtyp.t <= Kind.t ~> HTyp.t -| Delta`. These judgements are shown in the attached [latex document](./latex/kind-judgements.pdf). Type elaboration synthesis has an affinity for singletons over Ty. `Delta.t`'s `hole_sort` will be changed to a `Hole.t` so we can track type @@ -208,14 +197,12 @@ type t = MetaVarMap.t(Hole.t); ## Type Pattern Analysis Type patterns analyze with a type to create new tyvar bindings and hole bindings -`Delta.t_1 ; TyVarCtx.t_1 |- TPat.t <- HTyp.t -| TyVarCtx.t_2 ; Delta.t_2`. This +`Delta.t_1 ; TyVarCtx.t_1 |- TPat.t <- HTyp.t : Kind.t -| TyVarCtx.t_2 ; Delta.t_2`. This judgement is in the attached [latex document](./latex/kind-judgements.pdf). ## Expression Elaboration for Type Aliases -The typical expansion from unexpanded expressions to expanded expressions -handles the new type alias case relying on the Type Pattern Analysis judgement -to generate the bindings for the remaining subexpression. See the attached [latex document](./latex/kind-judgements.pdf). +The expanded type alias expression also contains an explicit ascription for the kind associated with the internal type. 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zQb%F5L%g>ZJ+q3#dD2^e+NsI$M(O`5R`|3j?f{gcvW9h17pZ6{sjc&D3aQe(<)kUa zk?I(E^YJ?9RN&irJG zIPw)}t#{rFH0uhIoG?$y&rPV8gK9BJjIAx%y&vM|B>QDdv;X(qJG&Sz5oCK diff --git a/9-type-aliases-redux/latex/kind-judgements.tex b/9-type-aliases-redux/latex/kind-judgements.tex index 9feb215..3360c1b 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -44,18 +44,26 @@ \newcommand{\llparenthesiscolor}{\textcolor{violet}{\llparenthesis}} \newcommand{\rrparenthesiscolor}{\textcolor{violet}{\rrparenthesis}} +\newcommand{\llparenthesiscolorc}{\textcolor{cyan}{\llparenthesis}} +\newcommand{\rrparenthesiscolorc}{\textcolor{cyan}{\rrparenthesis}} \newcommand{\hthole}[1]{\llparenthesiscolor\rrparenthesiscolor^{#1}} \newcommand{\hhole}[2]{\llparenthesiscolor#1\rrparenthesiscolor^{#2}} +\newcommand{\patehole}{\llparenthesiscolorc\rrparenthesiscolorc} +\newcommand{\pathole}[1]{\llparenthesiscolorc#1\rrparenthesiscolorc} \newcommand{\kehole}{\llparenthesiscolor\rrparenthesiscolor} \newcommand{\idof}[1]{\mathsf{id}(#1)} \newcommand{\Dbinding}[3]{#1 :: #3[#2]} +\newcommand{\domOf}[1]{\mathsf{dom}(#1)} \newcommand{\htdefine}[3]{\textt{type }#1 \textt{ = } #2\textt{ in }#3} \newcommand{\dtdefine}[4]{\textt{type }#1 \textt{ = } #2 : #3\textt{ in }#4} \begin{document} +TODO: +1. think about decidability of analysis, and equivalence + \begin{figure}[t] $\arraycolsep=4pt\begin{array}{rllllll} \mathsf{BinOp} & \oplus & ::= & @@ -70,7 +78,7 @@ c ~\vert~ \dtau_1 \oplus \dtau_2 ~\vert~ \hlist{\dtau} ~\vert~ \hthole{u} ~\vert~ \hhole{\dtau}{u} \\ \mathsf{Type Vars} & t & & \\ \mathsf{Type Pattern} & \hrho & ::= & - t ~\vert~ \hthole{u} ~\vert~ \hhole{t}{u} \\ + t ~\vert~ \patehole ~\vert~ \pathole{t} \\ \mathsf{User Expression} & \hexp & ::= & \htdefine{\hrho}{\htau}{\hexp} ~\vert~ elided \\ \mathsf{Internal Expression} & \dtau & ::= & @@ -78,43 +86,6 @@ \end{array}$ \end{figure} -\begin{minipage}{\linewidth} - \judgbox - {\kindLeqLattice{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}} - {$\hkappa_1$ is more precise than $\hkappa_2$} - \begin{mathpar} - \inferrule[KLTrans]{ - \kindLeqLattice{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} \\ - \kindLeqLattice{\Delta;\hPhi}{\hkappa_2}{\hkappa_3} - } { - \kindLeqLattice{\Delta;\hPhi}{\hkappa_1}{\hkappa_3} - } - \and - \inferrule[KLTyHole]{ } - { - \kindLeqLattice{\Delta;\hPhi}{\Ty}{\KHole} - } - \and - \inferrule[KLSingletonTy]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\Ty} - } { - \kindLeqLattice{\Delta;\hPhi}{\KSing{\dtau}}{\Ty} - } - \and - \inferrule[KLRespectEquiv]{ - \kequiv{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} \\ - } { - \kindLeqLattice{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} - } - - - \end{mathpar} -\end{minipage} -\\ -\\ - - - \begin{minipage}{\linewidth} \judgbox {\kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}} @@ -216,9 +187,9 @@ } \and \inferrule[KAVar]{ - t : \hkappa_1 \in \hPhi + t : \hkappa \in \hPhi } { - \kindAssign{\Delta;\hPhi}{t}{\hkappa_2} + \kindAssign{\Delta;\hPhi}{t}{\hkappa} } \and \inferrule[KABinOp]{ @@ -255,7 +226,7 @@ \kindAssign{\Delta;\hPhi}{\dtau}{\KSing{\dtau}} } \and - \inferrule[KASubkind]{ + \inferrule[KASubsumption]{ \kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_1} \\ \kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} } { @@ -274,12 +245,6 @@ {\kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\hkappa}} {$\dtau_1$ is equivalent to $\dtau_2$ and has kind $\hkappa_2$} \;\\ \begin{mathpar} - \inferrule[KCERefl]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\hkappa} - } { - \kcequiv{\Delta;\hPhi}{\dtau}{\dtau}{\hkappa} - } - \and \inferrule[KCESymm]{ \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\hkappa} } { @@ -298,6 +263,51 @@ } { \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\Ty} } + \and + \inferrule[KCEConst]{ }{ + \kcequiv{\Delta;\hPhi}{c}{c}{\Ty} + } + \and + \inferrule[KCEVar]{ + t : \hkappa \in \hPhi + } { + \kcequiv{\Delta;\hPhi}{t}{t}{\hkappa} + } + \and + \inferrule[KCEBinOp]{ + \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\Ty} \\ + \kcequiv{\Delta;\hPhi}{\dtau_3}{\dtau_4}{\Ty} + } { + \kcequiv{\Delta;\hPhi}{\dtau_1 \oplus \dtau_3}{\dtau_2 \oplus \dtau_4}{\Ty} + } + \and + \inferrule[KAList]{ + \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\Ty} + } { + \kcequiv{\Delta;\hPhi}{\hlist{\dtau_1}}{\hlist{\dtau_2}}{\Ty} + } + \and + \inferrule[KAEHole]{ + \Dbinding{u_1}{\hPhi'_1}{\hkappa_1} \in \Delta \\ + \kindAssign{\Delta;\hPhi}{\sigma_2}{\hPhi'_1} \\ + \Dbinding{u_2}{\hPhi'_2}{\hkappa_2} \in \Delta \\ + \kindAssign{\Delta;\hPhi}{\sigma_2}{\hPhi'_2} \\ + \kequiv{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} + } { + \kcequiv{\Delta;\hPhi}{\hthole{u_1}_{\sigma_1}}{\hthole{u_2}_{\sigma_2}}{\hkappa} + } + \and + \inferrule[KANEHole]{ + \kindAssign{\Delta;\hPhi}{\dtau_1}{\hkappa'_1} \\ + \kindAssign{\Delta;\hPhi}{\dtau_2}{\hkappa'_2} \\ + \Dbinding{u_1}{\hPhi'_1}{\hkappa_1} \in \Delta \\ + \kindAssign{\Delta;\hPhi}{\sigma_2}{\hPhi'_1} \\ + \Dbinding{u_2}{\hPhi'_2}{\hkappa_2} \in \Delta \\ + \kindAssign{\Delta;\hPhi}{\sigma_2}{\hPhi'_2} \\ + \kequiv{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} + } { + \kcequiv{\Delta;\hPhi}{\hhole{\dtau_1}{u_1}_{\sigma_1}}{\hhole{\dtau_2}{u_2}_{\sigma_2}}{\hkappa} + } \end{mathpar} \end{minipage} \\ @@ -332,7 +342,7 @@ } \and \inferrule[TElabSUVar]{ - t \not\in \hPhi + t \not\in \domOf{\hPhi} } { \elabSyn{\hPhi}{t}{\KHole}{\hhole{t}{u}_{\idof{\hPhi}}}{\Dbinding{u}{\hPhi}{\kehole} } } @@ -350,13 +360,14 @@ \end{minipage} \\ \\ + \begin{minipage}{\linewidth} \judgbox {\elabAna{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}} {$\htau$ analyzes against kind $\hkappa_1$ and elaborates to $\dtau$} \begin{mathpar} \inferrule[TElabASubsume]{ - \htau \neq t \text{ where } t \not\in \hPhi \\ + \htau \neq t \text{ where } t \not\in \domOf{\hPhi} \\ \htau \neq \hthole{u} \\ \htau \neq \hhole{\htau'}{u} \\ \elabSyn{\hPhi}{\htau}{\hkappa'}{\dtau}{\Delta} \\ @@ -366,7 +377,7 @@ } \and \inferrule[TElabAUVar]{ - t \not\in \hPhi + t \not\in \domOf{\hPhi} } { \elabAna{\hPhi}{t}{\KHole}{\hhole{t}{u}_{\idof{\hPhi}}}{\Dbinding{u}{\hPhi}{\kehole} } } @@ -388,23 +399,23 @@ \begin{minipage}{\linewidth} \judgbox - {\patMatch{\Delta_1;\hPhi_1}{\dtau}{\hkappa}{\hrho}{\hPhi_2;\Delta_2}} - {$\hrho$ matches against $\dtau : \hkappa$ extending the relevant contexts} + {\patMatch{\hPhi_1}{\dtau}{\hkappa}{\hrho}{\hPhi_2}} + {$\hrho$ matches against $\dtau : \hkappa$ extending $\hPhi$ if necessary} \begin{mathpar} \inferrule[RESVar]{ \tyvarValid{t} }{ - \patMatch{\Delta;\hPhi}{\dtau}{\hkappa}{t}{\hPhi,t :: \hkappa;\Delta} + \patMatch{\hPhi}{\dtau}{\hkappa}{t}{\hPhi,t :: \hkappa} } \and \inferrule[RESEHole]{ }{ - \patMatch{\Delta;\hPhi}{\dtau}{\hkappa}{\hthole{u}}{\hPhi;\Delta,\Dbinding{u}{\hPhi}{\kehole}} + \patMatch{\hPhi}{\dtau}{\hkappa}{\patehole}{\hPhi} } \and \inferrule[RESVarHole]{ \tyvarNotValid{t} }{ - \patMatch{\Delta;\hPhi}{\dtau}{\hkappa}{\hhole{t}{u}}{\hPhi;\Delta,\Dbinding{u}{\hPhi}{\kehole}} + \patMatch{\hPhi}{\dtau}{\hkappa}{\pathole{t}}{\hPhi} } \end{mathpar} \end{minipage} @@ -418,10 +429,10 @@ \begin{mathpar} \inferrule[ESDefine]{ \elabSyn{\hPhi_1}{\htau}{\hkappa}{\dtau}{\Delta_1} \\ - \patMatch{\Delta_1;\hPhi_1}{\dtau}{\hkappa}{\hrho}{\hPhi_2;\Delta_2} \\ - \elabSyn{\hGamma;\hPhi_2}{\hexp}{\dtau_1}{\dexp}{\Delta_3} + \patMatch{\hPhi_1}{\dtau}{\hkappa}{\hrho}{\hPhi_2} \\ + \elabSyn{\hGamma;\hPhi_2}{\hexp}{\dtau_1}{\dexp}{\Delta_2} }{ - \elabSyn{\hGamma;\hPhi_1}{\htdefine{\hrho}{\htau}{\hexp}}{\dtau_1}{\dtdefine{\hrho}{\dtau}{\hkappa}{\dexp}}{\Delta_2 \cup \Delta_3} + \elabSyn{\hGamma;\hPhi_1}{\htdefine{\hrho}{\htau}{\hexp}}{\dtau_1}{\dtdefine{\hrho}{\dtau}{\hkappa}{\dexp}}{\Delta_1 \cup \Delta_2} } @@ -436,8 +447,8 @@ {$\dexp$ is assigned type $\dtau$} \begin{mathpar} \inferrule[DEDefine]{ - \patMatch{\Delta_1;\hPhi_1}{\dtau_1}{\hkappa}{\hrho}{\hPhi_2;\Delta_2} \\ - \dexpTypeAssign{\Delta_2;\hGamma;\hPhi_2}{\dexp}{\dtau_2} + \patMatch{\hPhi_1}{\dtau_1}{\hkappa}{\hrho}{\hPhi_2} \\ + \dexpTypeAssign{\Delta;\hGamma;\hPhi_2}{\dexp}{\dtau_2} }{ \dexpTypeAssign{\Delta;\hGamma;\hPhi_1}{\dtdefine{\hrho}{\dtau_1}{\hkappa}{\dexp}}{\dtau_2} } @@ -465,11 +476,11 @@ \begin{thm}[Kind Synthesis Precision]\ \\ - If $\elabSyn{\hPhi}{\htau}{\hkappa_1}{\dtau}{\Delta_1}$ and $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_2}$ then $\kindLeqLattice{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}$ + If $\elabSyn{\hPhi}{\htau}{\hkappa_1}{\dtau}{\Delta_1}$ and $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_2}$ for $t : \Ty \not\in \hPhi$ then $\kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}$ \end{thm} \noindent -Kind Synthesis Precision says that elaboration synthesizes the most precise kappa possible for a given input type. The proof goes by induction on the elaboration rules and then for each tau, induction on all valid kind assignments for that tau ensuring that each one assignment is greater in the lattice than the kappa synthesized by elaboration. +Kind Synthesis Precision says that elaboration synthesizes the most precise kappa possible for a given input type. The proof goes by induction on the elaboration rules and then for each tau, induction on all valid kind assignments for that tau ensuring that each one assignment is a consistent supertype of the kappa synthesized by elaboration. The interesting rules in the kind assignments are: \texttt{KASubsumption} and \texttt{KASelfRecognition}. \texttt{KASubsumption} holds because this rule only makes the kind a consistent supertype of what it was. \texttt{KASelfRecognition} holds because elaboration always synthesizes the singleton-version of the kind for every type when possible. We need the side-condition to protect agaisnt $\Ty$ deriving from elaboration and $\KSing{t}$ from \texttt{KASelfRecognition}. From ce7d4feff7a700a3e02321a1c535d2aecbce371a Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Tue, 27 Apr 2021 19:56:17 -0500 Subject: [PATCH 015/181] Moves to bidirectional kind assignment --- 9-type-aliases-redux/9-type-aliases-redux.md | 7 +- .../latex/kind-judgements.pdf | Bin 198076 -> 189457 bytes .../latex/kind-judgements.tex | 154 +++++++++--------- 3 files changed, 80 insertions(+), 81 deletions(-) diff --git a/9-type-aliases-redux/9-type-aliases-redux.md b/9-type-aliases-redux/9-type-aliases-redux.md index 3edd1da..ba576d3 100644 --- a/9-type-aliases-redux/9-type-aliases-redux.md +++ b/9-type-aliases-redux/9-type-aliases-redux.md @@ -157,11 +157,10 @@ kind constructor equivalence through to singleton kinds. `Delta.t; TyVarCtx.t |- Kind.t_1 === Kind.t_2` It is presented in the attached [latex document](./latex/kind-judgements.pdf). -## Kind Assignment +## Kind Synthesis/Analysis -Kind assignment is defined by a declarative judgement -`Delta.t ; TyVarCtx.t |- HTyp.t : Kind.t` with rules presented in the attached -[latex document](./latex/kind-judgements.pdf). Kind assignment can derive both singleton and Ty kinds for valid types. +Kind synthesis and analysis is defined by bidirectional judgements +`Delta.t ; TyVarCtx.t |- HTyp.t => Kind.t` and `Delta.t ; TyVarCtx.t |- HTyp.t <= Kind.t` with rules presented in the attached [latex document](./latex/kind-judgements.pdf). Kind synthesis derives the more precise singletons over Ty wherever possible. ## Kind Constructor Equivalence diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index bd3485760774220edbe38dd1017c73b5fefd5cc4..c9586363299176c41db994c1bd1315e499662413 100644 GIT binary patch delta 94682 zcmZU(Q*bT}uq+zewr$(CZQD*(k}tMx+g`D4+qRPx=luKbdtdIEr>wFhqcD<#zl{`aYViCp{$|CrcvW9FU;W{iJG?#+5*191FGYBp{+ zWb$ep$twUo#y)o@EBkBY7=iMYt) zSVMG}agvA5+tmrTN@4HTr`!ba#}*rvWM-5 zU!+O!V6T(sv6z$mhU;s;u3 zicw{kX7k>B@pV5*JpH-*8eKrus-$mCVuDynEFg-ZWZHLu zTAn1A^T$(+Ai*agb3WsaO0+Kcsdm|O11osV0uqA*IGLHSOUGwV)RAQP z0uO6IfMc*qG5N2J=)vLl@bP|h(WC>>gnkpzp@4ztEm?W^Js+UPo!-xJ8i9hsxEzVq zlTr&eyE`H05`3>N5!WKbB#ydBK}NXFlW@8V1tHTDmKDF6|BcIIymf0@vug}EZ8rqDqi6p|w{m(8+4{dmJsZL=|=a)yX&9Koy`& z*9T*b68?)$=d{deXR5AD^bq)sV<+Xl44w=O_7Ng|-l*=8)7>!U?g)B7+U^RYf@4SI zH{M}G%-KLa{g&{r%Ca^_)I{chvP^9UdPML6WH05syzKk|@Mps#dG>&mEDvzJ^2*s# 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C_Ym4rwKEaTTAOd=#oVx*IFk!}(vJtRSTNgwGa17wgS$q*SPBV?3} z!LpuSOTo_nzdH`Q{ZLK79&)NQ?6soGz*- zMo^UnnwP2^@LN>P!=-Af1-P>x)gs)rPa?Ghcb`|~;htYq%W&^uRRQkvMzumt;lADs do>#X&&kZZ3m+0dH9Sk`)H8U^@B_%~qMhc3W35);$ diff --git a/9-type-aliases-redux/latex/kind-judgements.tex b/9-type-aliases-redux/latex/kind-judgements.tex index 3360c1b..ac09ef2 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -16,6 +16,8 @@ \newcommand{\patMatch}[5]{#1 \vdash #2 : #3 \vartriangleright #4 \dashv #5} \newcommand{\kconsubkind}[3]{#1 \vdash #2 \lesssim #3} \newcommand{\kindAssign}[3]{#1 \vdash #2 : #3} +\newcommand{\kindSyn}[3]{#1 \vdash #2 \Rightarrow #3} +\newcommand{\kindAna}[3]{#1 \vdash #2 \Leftarrow #3} \newcommand{\kformation}[2]{#1 \vdash #2 \textt{ kind} } \newcommand{\kequiv}[3]{#1 \vdash #2 \equiv #3} \newcommand{\kcequiv}[4]{#1 \vdash #2 \equiv #3 : #4} @@ -50,10 +52,8 @@ \newcommand{\hhole}[2]{\llparenthesiscolor#1\rrparenthesiscolor^{#2}} \newcommand{\patehole}{\llparenthesiscolorc\rrparenthesiscolorc} \newcommand{\pathole}[1]{\llparenthesiscolorc#1\rrparenthesiscolorc} -\newcommand{\kehole}{\llparenthesiscolor\rrparenthesiscolor} -\newcommand{\idof}[1]{\mathsf{id}(#1)} -\newcommand{\Dbinding}[3]{#1 :: #3[#2]} +\newcommand{\Dbinding}[2]{#1 :: #2} \newcommand{\domOf}[1]{\mathsf{dom}(#1)} \newcommand{\htdefine}[3]{\textt{type }#1 \textt{ = } #2\textt{ in }#3} @@ -61,9 +61,6 @@ \begin{document} -TODO: -1. think about decidability of analysis, and equivalence - \begin{figure}[t] $\arraycolsep=4pt\begin{array}{rllllll} \mathsf{BinOp} & \oplus & ::= & @@ -106,7 +103,7 @@ } \and \inferrule[KCSubsumption]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\Ty} + \kindAna{\Delta;\hPhi}{\dtau}{\Ty} } { \kconsubkind{\Delta;\hPhi}{\KSing{\dtau}}{\Ty} } @@ -136,7 +133,7 @@ \inferrule[KFHole] { }{ \kformation{\Delta;\hPhi}{\KHole} } \and \inferrule[KFSing]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\Ty} + \kindAna{\Delta;\hPhi}{\dtau}{\Ty} } { \kformation{\Delta;\hPhi}{\KSing{\dtau}} } @@ -179,67 +176,79 @@ \begin{minipage}{\linewidth} \judgbox - {\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa}} - {$\dtau$ is assigned kind $\hkappa$} + {\kindSyn{\Delta;\hPhi}{\dtau}{\hkappa}} + {$\dtau$ synthesizes kind $\hkappa$} \begin{mathpar} - \inferrule[KAConst]{ }{ - \kindAssign{\Delta;\hPhi}{c}{\Ty} + \inferrule[KSConst]{ }{ + \kindSyn{\Delta;\hPhi}{c}{\KSing{c}} } \and - \inferrule[KAVar]{ + \inferrule[KSVar]{ t : \hkappa \in \hPhi } { - \kindAssign{\Delta;\hPhi}{t}{\hkappa} + \kindSyn{\Delta;\hPhi}{t}{\hkappa} } \and - \inferrule[KABinOp]{ - \kindAssign{\Delta;\hPhi}{\dtau_1}{\Ty} \\ - \kindAssign{\Delta;\hPhi}{\dtau_2}{\Ty} + \inferrule[KSUVar]{ + t \not \in \domOf{\hPhi} } { - \kindAssign{\Delta;\hPhi}{\dtau_1 \oplus \dtau_2}{\Ty} + \kindSyn{\Delta;\hPhi}{t}{\KHole} } \and - \inferrule[KAList]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\Ty} + \inferrule[KSBinOp]{ + \kindAna{\Delta;\hPhi}{\dtau_1}{\KSing{\dtau_1}} \\ + \kindAna{\Delta;\hPhi}{\dtau_2}{\KSing{\dtau_2}} } { - \kindAssign{\Delta;\hPhi}{\hlist{\dtau}}{\Ty} + \kindSyn{\Delta;\hPhi}{\dtau_1 \oplus \dtau_2}{\KSing{\dtau_1 \oplus \dtau_2}} } \and - \inferrule[KAEHole]{ - \Dbinding{u}{\hPhi'}{\hkappa} \in \Delta \\ - \kindAssign{\Delta;\hPhi}{\sigma}{\hPhi'} + \inferrule[KSList]{ + \kindAna{\Delta;\hPhi}{\dtau}{\KSing{\dtau}} } { - \kindAssign{\Delta;\hPhi}{\hthole{u}_{\sigma}}{\hkappa} + \kindSyn{\Delta;\hPhi}{\hlist{\dtau}}{\KSing{\hlist{\dtau}}} } \and - \inferrule[KANEHole]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\hkappa'} \\ - \Dbinding{u}{\hPhi'}{\hkappa} \in \Delta \\ - \kindAssign{\Delta;\hPhi}{\sigma}{\hPhi'} + \inferrule[KSEHole]{ + \Dbinding{u}{\hkappa} \in \Delta } { - \kindAssign{\Delta;\hPhi}{\hhole{\dtau}{u}_{\sigma}}{\hkappa} - } - \and - \inferrule[KASelfRecognition]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\Ty} - }{ - \kindAssign{\Delta;\hPhi}{\dtau}{\KSing{\dtau}} + \kindSyn{\Delta;\hPhi}{\hthole{u}}{\hkappa} } \and - \inferrule[KASubsumption]{ - \kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_1} \\ - \kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} + \inferrule[KSNEHole]{ + \Dbinding{u}{\hkappa} \in \Delta \\ + \kindSyn{\Delta;\hPhi}{\dtau}{\hkappa'} } { - \kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_2} + \kindSyn{\Delta;\hPhi}{\hhole{\dtau}{u}}{\hkappa} } + \end{mathpar} +\end{minipage} +\\ +\\ + + + +\begin{minipage}{\linewidth} + \judgbox + {\kindAna{\Delta;\hPhi}{\dtau}{\hkappa}} + {$\dtau$ analyzes against kind $\hkappa$} + \begin{mathpar} + \inferrule[KAASubsume]{ + \kindSyn{\hPhi}{\dtau}{\hkappa'} \\ + \kconsubkind{\Delta;\hPhi}{\hkappa'}{\hkappa} + }{ + \kindAna{\Delta;\hPhi}{\dtau}{\hkappa} + } + \end{mathpar} \end{minipage} \\ \\ + + \begin{minipage}{\linewidth} \judgbox {\kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\hkappa}} @@ -259,7 +268,7 @@ } \and \inferrule[KCESingEquiv]{ - \kindAssign{\Delta;\hPhi}{\dtau_1}{\KSing{\dtau_2}} + \kindAna{\Delta;\hPhi}{\dtau_1}{\KSing{\dtau_2}} } { \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\Ty} } @@ -281,32 +290,23 @@ \kcequiv{\Delta;\hPhi}{\dtau_1 \oplus \dtau_3}{\dtau_2 \oplus \dtau_4}{\Ty} } \and - \inferrule[KAList]{ + \inferrule[KCEList]{ \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\Ty} } { \kcequiv{\Delta;\hPhi}{\hlist{\dtau_1}}{\hlist{\dtau_2}}{\Ty} } \and - \inferrule[KAEHole]{ - \Dbinding{u_1}{\hPhi'_1}{\hkappa_1} \in \Delta \\ - \kindAssign{\Delta;\hPhi}{\sigma_2}{\hPhi'_1} \\ - \Dbinding{u_2}{\hPhi'_2}{\hkappa_2} \in \Delta \\ - \kindAssign{\Delta;\hPhi}{\sigma_2}{\hPhi'_2} \\ - \kequiv{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} + \inferrule[KCEEHole]{ + \Dbinding{u}{\hkappa} \in \Delta } { - \kcequiv{\Delta;\hPhi}{\hthole{u_1}_{\sigma_1}}{\hthole{u_2}_{\sigma_2}}{\hkappa} + \kcequiv{\Delta;\hPhi}{\hthole{u}}{\hthole{u}}{\hkappa} } \and - \inferrule[KANEHole]{ - \kindAssign{\Delta;\hPhi}{\dtau_1}{\hkappa'_1} \\ - \kindAssign{\Delta;\hPhi}{\dtau_2}{\hkappa'_2} \\ - \Dbinding{u_1}{\hPhi'_1}{\hkappa_1} \in \Delta \\ - \kindAssign{\Delta;\hPhi}{\sigma_2}{\hPhi'_1} \\ - \Dbinding{u_2}{\hPhi'_2}{\hkappa_2} \in \Delta \\ - \kindAssign{\Delta;\hPhi}{\sigma_2}{\hPhi'_2} \\ - \kequiv{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} + \inferrule[KCENEHole]{ + \Dbinding{u}{\hkappa} \in \Delta \\ + \kindAna{\Delta;\hPhi}{\dtau}{\hkappa'} } { - \kcequiv{\Delta;\hPhi}{\hhole{\dtau_1}{u_1}_{\sigma_1}}{\hhole{\dtau_2}{u_2}_{\sigma_2}}{\hkappa} + \kcequiv{\Delta;\hPhi}{\hhole{\dtau}{u}}{\hhole{\dtau}{u}}{\hkappa} } \end{mathpar} \end{minipage} @@ -342,19 +342,19 @@ } \and \inferrule[TElabSUVar]{ - t \not\in \domOf{\hPhi} + t \not\in \domOf{\hPhi} } { - \elabSyn{\hPhi}{t}{\KHole}{\hhole{t}{u}_{\idof{\hPhi}}}{\Dbinding{u}{\hPhi}{\kehole} } + \elabSyn{\hPhi}{t}{\KHole}{\hhole{t}{u}}{\Dbinding{u}{\KHole} } } \and \inferrule[TElabSHole]{ } { - \elabSyn{\hPhi}{\hthole{u}}{\KHole}{\hthole{u}_{\idof{\hPhi}}}{\Dbinding{u}{\hPhi}{\kehole}} + \elabSyn{\hPhi}{\hthole{u}}{\KHole}{\hthole{u}}{\Dbinding{u}{\KHole}} } \and \inferrule[TElabSNEHole]{ - \elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta} + \elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta} } { - \elabSyn{\hPhi}{\hhole{\htau}{u}}{\KHole}{\hhole{\dtau}{u}_{\idof{\hPhi}}}{\Delta,\Dbinding{u}{\hPhi}{\kehole}} + \elabSyn{\hPhi}{\hhole{\htau}{u}}{\KHole}{\hhole{\dtau}{u}}{\Delta,\Dbinding{u}{\KHole}} } \end{mathpar} \end{minipage} @@ -367,7 +367,6 @@ {$\htau$ analyzes against kind $\hkappa_1$ and elaborates to $\dtau$} \begin{mathpar} \inferrule[TElabASubsume]{ - \htau \neq t \text{ where } t \not\in \domOf{\hPhi} \\ \htau \neq \hthole{u} \\ \htau \neq \hhole{\htau'}{u} \\ \elabSyn{\hPhi}{\htau}{\hkappa'}{\dtau}{\Delta} \\ @@ -376,20 +375,14 @@ \elabAna{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta} } \and - \inferrule[TElabAUVar]{ - t \not\in \domOf{\hPhi} - } { - \elabAna{\hPhi}{t}{\KHole}{\hhole{t}{u}_{\idof{\hPhi}}}{\Dbinding{u}{\hPhi}{\kehole} } - } - \and \inferrule[TElabAEHole]{ } { - \elabAna{\hPhi}{\hthole{u}}{\hkappa}{\hthole{u}_{\idof{\hPhi}}}{\Dbinding{u}{\hPhi}{\kappa}} + \elabAna{\hPhi}{\hthole{u}}{\hkappa}{\hthole{u}}{\Dbinding{u}{\kappa}} } \and \inferrule[TElabANEHole]{ - \elabSyn{\hPhi}{\htau}{\hkappa'}{\dtau}{\Delta} + \elabSyn{\hPhi}{\htau}{\hkappa'}{\dtau}{\Delta} } { - \elabAna{\hPhi}{\hhole{\htau}{u}}{\hkappa}{\hhole{\dtau}{u}_{\idof{\hPhi}}}{\Delta,\Dbinding{u}{\hPhi}{\kappa}} + \elabAna{\hPhi}{\hhole{\htau}{u}}{\hkappa}{\hhole{\dtau}{u}}{\Delta,\Dbinding{u}{\kappa}} } \end{mathpar} @@ -461,12 +454,19 @@ \newtheorem{thm}{Theorem} \begin{thm}[Well-Kinded Elaboration]\ \\ - (1) If $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ then $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa}$ \\ - (2) If $\elabAna{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ then $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa}$ + (1) If $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ then $\kindSyn{\Delta;\hPhi}{\dtau}{\hkappa}$ \\ + (2) If $\elabAna{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ then $\kindAna{\Delta;\hPhi}{\dtau}{\hkappa}$ \end{thm} \noindent This is like the Typed Elaboration theorem in the POPL19 paper. +\begin{thm}[Elaborability]\ \\ + (1) $\exists \Delta$ s.t. if $\kindSyn{\Delta;\hPhi}{\dtau}{\hkappa}$ then $\exists \htau$ such that $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ \\ + (2) $\exists \Delta$ s.t. if $\kindAna{\Delta;\hPhi}{\dtau}{\hkappa}$ then $\exists \htau$ such that $\elabAna{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ +\end{thm} +\noindent +This is similar but a little different from Elaborability theorem in the POPL19 paper. Choose the $\Delta$ that is emitted from elaboration and then there's an $\htau$ that elaborates to any of the $\dtau$ forms. Elaborability and Well-Kinded Elaboration implies we can just rely on the elaboration forms for the premises of any rules that demand kind synthesis/analysis. + \begin{thm}[Type Elaboration Unicity]\ \\ (1) If $\elabSyn{\hPhi}{\htau}{\hkappa_1}{\dtau_1}{\Delta_1}$ and $\elabSyn{\hPhi}{\htau}{\hkappa_2}{\dtau_2}{\Delta_2}$ then $\hkappa_1 = \hkappa_2$, $\dtau_1 = \dtau_2$, $\Delta_1 = \Delta_2$ \\ (2) If $\elabAna{\hPhi}{\htau}{\hkappa}{\dtau_1}{\Delta_1}$ and $\elabAna{\hPhi}{\htau}{\hkappa}{\dtau_2}{\Delta_2}$ then $\dtau_1 = \dtau_2$, $\Delta_1 = \Delta_2$ @@ -476,11 +476,11 @@ \begin{thm}[Kind Synthesis Precision]\ \\ - If $\elabSyn{\hPhi}{\htau}{\hkappa_1}{\dtau}{\Delta_1}$ and $\kindAssign{\Delta;\hPhi}{\dtau}{\hkappa_2}$ for $t : \Ty \not\in \hPhi$ then $\kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}$ + If $\kindSyn{\Delta;\hPhi}{\dtau}{\hkappa_1}$ and $\kindAna{\Delta;\hPhi}{\dtau}{\hkappa_2}$ then $\kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}$ \end{thm} \noindent -Kind Synthesis Precision says that elaboration synthesizes the most precise kappa possible for a given input type. The proof goes by induction on the elaboration rules and then for each tau, induction on all valid kind assignments for that tau ensuring that each one assignment is a consistent supertype of the kappa synthesized by elaboration. The interesting rules in the kind assignments are: \texttt{KASubsumption} and \texttt{KASelfRecognition}. \texttt{KASubsumption} holds because this rule only makes the kind a consistent supertype of what it was. \texttt{KASelfRecognition} holds because elaboration always synthesizes the singleton-version of the kind for every type when possible. We need the side-condition to protect agaisnt $\Ty$ deriving from elaboration and $\KSing{t}$ from \texttt{KASelfRecognition}. +Kind Synthesis Precision says that synthesis finds the most precise kappa possible for a given input type. This is somewhat trivial, but interesting to note because it means we can expect singletons wherever possible. From 3f6986b9785c6923fc0c84aaaa932df6a5903691 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Fri, 30 Apr 2021 16:57:54 -0500 Subject: [PATCH 016/181] Tweaks to design docs --- 9-type-aliases-redux/9-type-aliases-redux.md | 15 +++++++-------- 1 file changed, 7 insertions(+), 8 deletions(-) diff --git a/9-type-aliases-redux/9-type-aliases-redux.md b/9-type-aliases-redux/9-type-aliases-redux.md index ba576d3..288aae7 100644 --- a/9-type-aliases-redux/9-type-aliases-redux.md +++ b/9-type-aliases-redux/9-type-aliases-redux.md @@ -33,7 +33,7 @@ module VarPatErrStatus = { } type TPat.t = - | EmptyHole(MetaVar.t) + | EmptyHole | TyVar(option(VarPatErrStatus.t), TyId.t) ``` @@ -83,7 +83,7 @@ type Kind.t = KHole To support type aliases, we need to add a new line item to the language and the proposed syntax is -`TyAliasLine(TPat.t, UHTyp.t)` for abstract syntax +`TyAliasLine(TPat.t, UHTyp.t, UHExp.t)` for abstract syntax `type t = in e` for GUI syntax @@ -91,19 +91,18 @@ To support type aliases, we need to add a new line item to the language and the By considering the cursor positions of `TyAlias`, we need to add two more zline items which are -`TyAliasLineT(TPat.t, ZTyp.t)` (when cursor is on the type) +`TyAliasLineP(ZTPat.t, UHTyp.t, UHExp.t)` (when cursor is on the type pattern) -`TyAliasLineP(ZTPat.t, UHTyp.t)` (when cursor is on the type pattern) +`TyAliasLineT(TPat.t, ZTyp.t, UHExp.t)` (when cursor is on the type) + +`TyAliasLineE(TPat.t, UHTyp.t, ZExp.t)` (when cursor is on the body) ## Expanded Expressions So that we can still assign a type properly to the resulting `DExp.t` we add the following new syntax: -`TyBindings(TyVarCtx.t, DHExp.t)` for abstract syntax - -(TODO: is it ever necessary to display this to the user? If so -- maybe something -like `type bindings [type t = \tau, type x] in e`) +`TyAlias(TPat.t, HTyp.t, Kind.t, DHExp.t)` for abstract syntax # Static Semantics From f1f1ed1954ffc5d34c106c95a776d392b25fc3f8 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Fri, 30 Apr 2021 20:29:34 -0500 Subject: [PATCH 017/181] Removes the extra recursive param in lines --- 9-type-aliases-redux/9-type-aliases-redux.md | 9 +++------ 1 file changed, 3 insertions(+), 6 deletions(-) diff --git a/9-type-aliases-redux/9-type-aliases-redux.md b/9-type-aliases-redux/9-type-aliases-redux.md index 288aae7..f1721f6 100644 --- a/9-type-aliases-redux/9-type-aliases-redux.md +++ b/9-type-aliases-redux/9-type-aliases-redux.md @@ -83,7 +83,7 @@ type Kind.t = KHole To support type aliases, we need to add a new line item to the language and the proposed syntax is -`TyAliasLine(TPat.t, UHTyp.t, UHExp.t)` for abstract syntax +`TyAliasLine(TPat.t, UHTyp.t)` for abstract syntax `type t = in e` for GUI syntax @@ -91,12 +91,9 @@ To support type aliases, we need to add a new line item to the language and the By considering the cursor positions of `TyAlias`, we need to add two more zline items which are -`TyAliasLineP(ZTPat.t, UHTyp.t, UHExp.t)` (when cursor is on the type pattern) - -`TyAliasLineT(TPat.t, ZTyp.t, UHExp.t)` (when cursor is on the type) - -`TyAliasLineE(TPat.t, UHTyp.t, ZExp.t)` (when cursor is on the body) +`TyAliasLineP(ZTPat.t, UHTyp.t)` (when cursor is on the type pattern) +`TyAliasLineT(TPat.t, ZTyp.t)` (when cursor is on the type) ## Expanded Expressions From 99db5129a2ecbdb92cfa9af3b9efff8ae27adbd1 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Mon, 17 May 2021 16:00:41 +0200 Subject: [PATCH 018/181] Remove kind ascription on type equivalence --- .../latex/kind-judgements.pdf | Bin 189457 -> 189343 bytes .../latex/kind-judgements.tex | 38 +++++++++--------- 2 files changed, 19 insertions(+), 19 deletions(-) diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index c9586363299176c41db994c1bd1315e499662413..37a32ecdaab9a64701fc220e5333ac7afa8d6327 100644 GIT binary patch delta 3278 zcmV;<3^DVO$P1sx3$OtQ0yi|1fgLD+&05=T+eQ$5*H`e-e$aMj_MSc^g_9yqTQqX} zk_4z4E3Ob*ZeqJd`t>~&MVU)3DN+~PFkn;iaCUd*%$W-*vx`h-uSRn9_r=-hyO+ji zUIc4(cJ?VFMZ!zUtT7^l?Cd;ycl2IL`u=!Kj&Mox_`})j?_OG2xr=2kEDP~}KMDa8 zDgs_m;8NhdWL?N8nPC~{Y%XJjqHLUNvV|NaR{(qtd6M(;kz>>U4mjcuslvwxSOSDAH!ti|PJHW~d|(_wQRhJ?f} zJ9I!?3#DySN5F3VZYs*7Y}QeKJIWFzuoaOA#&F?qBdm2$W`zuOGjqrs%nTS69#bJ3 zlW;*teuaZ78et=)vx?OG=6I~w9?ibo&Hp(z`e=GNTO6}Hx-GuHy<^XO}eZb|c zV6VEsRVdpMT>21J=0~u9G9|H^Tg2yBmAZuL+1KgfrVFbqw9yX{qHVmoH>{?DdBdWd zvJ6rUX&tZnpzsQNhSwH`QOzh;VD)_eJ)$^IJ;8P62<0_yF;6IrAA>AaI&=H5H0PL= zV6zV8$$W8fvaBak;T!~NqwR9fFmtNZ(TtE@A26PKE}BX{s0W>Y=FDhdvKzHB2=7g2 z@+p-e7AWjUo5r-$=H$IJvc)h;>TSIdFNrEmaqJC~bXyKb)vTv#2Z&s@?*s6_14Mlx z0tw*(oR1a4%@Fk=5+9a-IlGG z$G&&dusTT(9kr8xB$_!2Xm7mB4APRK2C955#uQU&=9_OtNA}n3X8zx7)v$e@FV2&# znH9?FP2Iwn6e8Usm*THnWwq1E57&#E+k{7PI>uqOL8qov7dIhWn$EI#txIcEf9`Mz z;A5KCl?e%q#{^;E?V$|!F{KIgpnP+=*caLgf*wRXm-aY+o%}xi+OnpjvKn4-$!!Wj zP8hvj`B_Sp6~^(P1~plP;*ta1>nXffOKmm;MppfbS(P$!jr7S14-r&JOD5?c1%`IBVVY(@S_mbT z*EbxVYH&EAe28sBX@@K?4+j={c(6RCL3-0bouTo6!8@um^qcujvZ?}~lk{;GJ}e09 z{cS$~k_V-PSeZow)F_y1-GM_@4Ld2#!lP^Hnq1hjs%Y;IN^OczJe%EYoS0m!ZF~tI zx7h|YS+#x}V4BoKg12Zmx^|}(it^!}YYPJSgk8N3J-IJxYOv9OzKcWPE)^>Qh2DCc z@GRMXf&}12@4p3+sS?!(VWsHJ@io?;xM`>2TR*87ZEnc%MFeIFng>{9cfr&Fy7L2uiQ0=V^mCo8#sSFdTvbVZjpDD2TCfk?ZE9?dPg z?h&z;WQS^U|U(X|G(9p|M$h*VthCzq?O~UTU zfDyzfeZ3JTm`7>Kbd$>ZfTfZS;xmmEA+JD?<6xGpY~`78c?&H1KaICb@RQ39Cj&Aw zHIsoID1Y5qO>Y}F5WVYH@Ms;x@MAb6(^~>3MVhu~E%Xuxh~c<~0Dm;L(xU&qZ+9i_ za>ZS(WGQkA1GX%WW;iqN&Ag?RxE4yh9jV37SJTmpH!g@EBQuHVpF&$LgVKVXlQD|v zXYt|WTq*s-c%r?=ORdHqr|(|8Vb%DQQH8Op%zr;s1}-)Xg4EG04bGLN%El^zFy3=v zLsB^wlfo5OhB_09rPo?CjLwzOnTSz(;|E0aNkmgvjZUYdFCzrOmloOtN!ki#CrLSR zc{}>>QHjs^<2#(g#vuMFO>YI0WLMPTMx2d)Yx&^9htmid`+UHhq$OVW5seJ~i9VLO zY=8FA%B4h#g$FI8bEay#m73v;rHbu3n17m8L(vK0W*Q_BNM;2G9Xw)Ia!;(tn42gFZ!~jZy;D2%Hiq%?gL&9H6_hvPq%S zZ~@J?qG45Ys1?)zwsGsASr$dffC&RM44Ac$N4`d$H*jrJ

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zm@nBMkoCMzlNT`s+1AW^UEEZqqWDcf7IB$4I3tCp{ttWuK88tyQWS?$6aj})6au$W z6ayyi1~oDMmR$>HbOB$H!?XeI72u=HAOQ-L_sk%IX)m=lkqnm ze=<2ZAU-}IARr(hARr(Lb98cLVQmU{+8xbFFN0we#qoRI_chhfsGe}(4;^s-AFm@lbukyH>57CO?did4d4Um91) zGN~ceu*8!NYh;7e5+5x4qeF(&k$TcV{3JlKq>%(k6A6(piI8RzB`u_t#7La9k#^ES zI!PBS_n^ Date: Tue, 13 Jul 2021 02:54:54 +0000 Subject: [PATCH 022/181] let's backup before any possible hard drive failures! --- 9-type-aliases-redux/latex/hejohns-colors.sty | 40 ++ 9-type-aliases-redux/latex/hejohns-hazel.sty | 216 +++++++ .../latex/kind-judgements.pdf | Bin 266781 -> 148553 bytes .../latex/kind-judgements.tex | 610 ++++-------------- 4 files changed, 380 insertions(+), 486 deletions(-) create mode 100644 9-type-aliases-redux/latex/hejohns-colors.sty create mode 100644 9-type-aliases-redux/latex/hejohns-hazel.sty diff --git a/9-type-aliases-redux/latex/hejohns-colors.sty b/9-type-aliases-redux/latex/hejohns-colors.sty new file mode 100644 index 0000000..ac27e73 --- /dev/null +++ b/9-type-aliases-redux/latex/hejohns-colors.sty @@ -0,0 +1,40 @@ +\ProvidesPackage{hejohns-colors.sty}[2021/07/02 - my color defs] + \RequirePackage{xcolor} + \RequirePackage{amsmath} + \definecolor{red}{HTML}{800000} + \definecolor{green}{HTML}{008000} + \definecolor{blue}{HTML}{000080} + \definecolor{orange}{HTML}{D08000} + \definecolor{purple}{HTML}{800080} + \definecolor{teal}{HTML}{008080} + \definecolor{black}{HTML}{000000} + \definecolor{grey}{HTML}{808080} + \newcommand*{\red}[1]{\textcolor{red}{#1}} + \newcommand*{\green}[1]{\textcolor{green}{#1}} + \newcommand*{\blue}[1]{\textcolor{blue}{#1}} + \newcommand*{\orange}[1]{\textcolor{orange}{#1}} + \newcommand*{\purple}[1]{\textcolor{purple}{#1}} + \newcommand*{\teal}[1]{\textcolor{teal}{#1}} + \newcommand*{\black}[1]{\textcolor{black}{#1}} + \newcommand*{\grey}[1]{\textcolor{grey}{#1}} + \newcommand*{\blackrm}[1]{\black{\mathtt{#1}}} + \newcommand*{\redtt}[1]{\red{\mathtt{#1}}} + \newcommand*{\greentt}[1]{\green{\mathtt{#1}}} + \newcommand*{\bluett}[1]{\blue{\mathtt{#1}}} + \newcommand*{\orangett}[1]{\orange{\mathtt{#1}}} + \newcommand*{\purplett}[1]{\purple{\mathtt{#1}}} + \newcommand*{\tealtt}[1]{\teal{\mathtt{#1}}} + \newcommand*{\blacktt}[1]{\black{\mathtt{#1}}} + \newcommand*{\greytt}[1]{\grey{\mathtt{#1}}} + \newcommand*{\redit}[1]{\red{\mathit{#1}}} + \newcommand*{\greenit}[1]{\green{\mathit{#1}}} + \newcommand*{\blueit}[1]{\blue{\mathit{#1}}} + \newcommand*{\orangeit}[1]{\orange{\mathit{#1}}} + \newcommand*{\purpleit}[1]{\purple{\mathit{#1}}} + \newcommand*{\tealit}[1]{\teal{\mathit{#1}}} + \newcommand*{\blackit}[1]{\black{\mathit{#1}}} + \newcommand*{\greyit}[1]{\grey{\mathit{#1}}} + % traditional hazel hole color + \newcommand*{\violettt}[1]{\textcolor{violet}{\mathtt{#1}}} + \newcommand*{\violetit}[1]{\textcolor{violet}{\mathit{#1}}} +\endinput diff --git a/9-type-aliases-redux/latex/hejohns-hazel.sty b/9-type-aliases-redux/latex/hejohns-hazel.sty new file mode 100644 index 0000000..7db2283 --- /dev/null +++ b/9-type-aliases-redux/latex/hejohns-hazel.sty @@ -0,0 +1,216 @@ +\ProvidesPackage{hejohns-hazel}[2021/07/02 - for phis] +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% packages + \RequirePackage{hejohns-colors} + \RequirePackage{mathpartir} + \RequirePackage{latexsym} + \RequirePackage{stmaryrd} + \RequirePackage{amssymb} + \RequirePackage{thmtools} + \RequirePackage{tikz} + % https://tex.stackexchange.com/questions/12678/squiggly-arrows-in-tikz + \usetikzlibrary{decorations.pathmorphing} + \usetikzlibrary{arrows.meta} + % https://tex.stackexchange.com/questions/317121/automatically-adding-page-breaks-into-long-array-environments + \RequirePackage{array} + \RequirePackage{longtable} + \newcolumntype{L}{>{\(}l<{\)}} + \newcolumntype{C}{>{\(}c<{\)}} + \newcolumntype{R}{>{\(}r<{\)}} + \setlength\tabcolsep{5pt} + % https://tex.stackexchange.com/questions/29973/more-than-one-optional-argument-for-newcommand + \RequirePackage{xparse} + + %% Joshua Dunfield macros + \def\OPTIONConf{1}% + \RequirePackage{joshuadunfield} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% syntax + \newcommand*{\knd}[1][]{\redit{\kappa_{#1}}} + \newcommand*{\utyp}[1][]{\greenit{\hat{\tau_{#1}}}} + \newcommand*{\typ}[1][]{\greenit{\tau_{#1}}} + \newcommand*{\utypvar}[1][]{\greenit{t_{#1}}} + \newcommand*{\typvar}[1][]{\greenit{t_{#1}}} + \newcommand*{\uexp}[1][]{\blueit{\hat{\delta_{#1}}}} + \renewcommand*{\exp}[1][]{\blueit{\delta_{#1}}} + \newcommand*{\uexpvar}[1][]{\blueit{x_{#1}}} + \newcommand*{\expvar}[1][]{\blueit{x_{#1}}} + \newcommand*{\sigknd}[1][]{\orangeit{\xi_{#1}}} + \newcommand*{\usig}[1][]{\purpleit{\hat{\sigma_{#1}}}} + \newcommand*{\sig}[1][]{\purpleit{\sigma_{#1}}} + \newcommand*{\usigvar}[1][]{\purpleit{s_{#1}}} + \newcommand*{\sigvar}[1][]{\purpleit{s_{#1}}} + \newcommand*{\umod}[1][]{\tealit{\hat{\mu_{#1}}}} + \renewcommand*{\mod}[1][]{\tealit{\mu_{#1}}} + \newcommand*{\umodvar}[1][]{\tealit{m_{#1}}} + \newcommand*{\modvar}[1][]{\tealit{m_{#1}}} + \newcommand*{\lab}[1][]{\mathit{lab_{#1}}} + + \newcommand*{\Type}{\redtt{Type}} + \NewDocumentCommand{\SKind}{O{\knd}O{\typ}}{\redtt{S_{#1}(#2)}} + \newcommand*{\KHole}{\redtt{KHole}} + \NewDocumentCommand{\DepFunKind}{O{}O{\knd[1]}O{\knd[2]}}{\redtt{\Pi_{\typvar[#1]\KndAssump#2}.{#3}}} + \NewDocumentCommand{\DepProdKind}{O{}O{\knd[1]}O{\knd[2]}}{\redtt{\Sigma_{\typvar[#1]\KndAssump#2}.{#3}}} + + \NewDocumentCommand{\TypeFun}{O{}O{\knd}O{\typ}}{\greentt{\lambda\typvar[#1]\KndAssump#2.#3}} + \NewDocumentCommand{\TypeAp}{O{\typ[1]}O{\typ[2]}}{\greentt{#1~#2}} + \newcommand*{\ListType}[1]{\greentt{list~#1}} + % use violet instead of green? + \NewDocumentCommand{\TypeHole}{O{}O{u}}{\greentt{\llparenthesis#1\rrparenthesis^{#2}}} + \newcommand*{\ETypeHole}{\TypeHole} + \NewDocumentCommand{\NETypeHole}{O{\typ}}{\TypeHole[#1]} + \NewDocumentCommand{\UnboundTypeVar}{O{\typvar}}{\TypeHole[#1]} + \newcommand{\ModTypePrj}[2]{\greentt{#1.#2}} + + \newcommand*{\bse}[1][]{\greentt{bse_{#1}}} + \newcommand*{\Int}{\greentt{Int}} + \newcommand*{\Float}{\greentt{Float}} + \newcommand*{\Bool}{\greentt{Bool}} + + \NewDocumentCommand{\ubinop}{O{\utyp[1]}O{\utyp[2]}}{\greentt{#1\oplus#2}} + \NewDocumentCommand{\binop}{O{\typ[1]}O{\typ[2]}}{\greentt{#1\oplus#2}} + \newcommand*{\ProdType}{\greentt{\times}} + \newcommand*{\SumType}{\greentt{+}} + \newcommand*{\FunType}{\greentt{\rightarrow}} + + \newcommand{\uLetSig}[2]{\purplett{signature~#1=#2}\bluett{~in~\uexp}} + \newcommand{\LetSig}[2]{\purplett{signature~#1=#2}\bluett{~in~\exp}} + \newcommand{\uLetMod}[2]{\tealtt{module~#1=#2}\bluett{~in~\uexp}} + \newcommand{\LetMod}[2]{\tealtt{module~#1=#2}\bluett{~in~\exp}} + \newcommand{\uLetFunctor}[2]{\tealtt{functor~#1=#2}\bluett{~in~\uexp}} + \newcommand{\LetFunctor}[2]{\tealtt{functor~#1=#2}\bluett{~in~\exp}} + \newcommand{\uModTermPrj}[2]{\bluett{#1.#2}} + \newcommand{\ModTermPrj}[2]{\bluett{#1.#2}} + + \newcommand{\SSigKind}[1]{\orangett{SSigKind(#1)}} + \newcommand{\SigKHole}{\orangett{SigKHole}} + + \newcommand{\uSig}{\Sig} + \newcommand{\Sig}[1]{\purplett{\{#1\}}} + \newcommand{\uFunctorSig}{\FunctorSig} + \newcommand{\FunctorSig}[2]{\purplett{\Pi_{#1}.{#2}}} + \newcommand{\uSigHole}{\SigHole} + \NewDocumentCommand{\SigHole}{O{}O{^{\hole}}}{\purplett{\llparenthesis#1\rrparenthesis#2}} + \newcommand{\UnboundSigVar}[1][]{\uwave{\hphantom{.}\sigvar\hphantom{.}}} + + \newcommand{\Struct}[1]{\tealtt{\{#1\}}} + \newcommand{\Functor}[2]{\tealtt{\lambda #1.#2}} + \newcommand{\FunctorAp}[2]{\tealtt{#1~#2}} + \newcommand{\SubModulePrj}[2]{\tealtt{#1.#2}} + \newcommand{\ModHole}[1][]{\tealtt{\llparenthesis#1\rrparenthesis}} + + + \newcommand{\usbnd}{\greyit{\hat{sbnd}}} + \newcommand{\sbnd}{\greyit{sbnd}} + \newcommand{\usbnds}{\greyit{\hat{sbnds}}} + \newcommand{\sbnds}{\greyit{sbnds}} + \newcommand{\usdec}{\greyit{\hat{sdec}}} + \newcommand{\sdec}[1][]{\greyit{sdec#1}} + \newcommand{\usdecs}{\greyit{\hat{sdecs}}} + \newcommand{\sdecs}[1][]{\greyit{sdecs#1}} + + \newcommand{\uOpaqueTypeSdec}[1]{\greentt{type~#1}} + \newcommand{\uTransparentTypeSdec}{\TypeSdec} + \newcommand{\uValSdec}{\ValSdec} + \newcommand{\TypeSdec}[2]{\greentt{type~#1\KndAnn #2}} + \newcommand{\ValSdec}[2]{\bluett{val~#1\TypAnn #2}} + \newcommand{\uModSdec}{\ModSdec} + \newcommand{\ModSdec}[2]{\tealtt{module~#1\SigAnn #2}} + \newcommand{\uFunctorSdec}{\FunctorSdec} + \newcommand{\FunctorSdec}[2]{\tealtt{functor~#1\SigAnn #2}} + + \newcommand{\TypeSbnd}[2]{\greentt{type~#1 = #2}} + \newcommand{\ValSbnd}[2]{\bluett{let~#1 = #2}} + \newcommand{\ModSbnd}[2]{\tealtt{module~#1 = #2}} + \newcommand{\FunctorSbnd}[2]{\tealtt{functor~#1 = #2}} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% misc + \newcommand*{\ExpVarCtx}[1][;]{\Gamma#1} + \newcommand*{\TypVarCtx}[1][;]{\Phi#1} + \newcommand*{\ModVarCtx}[1][;]{\Xi#1} + \newcommand*{\SigVarCtx}[1][;]{\Psi#1} + \newcommand*{\HoleCtx}[1][;]{\Delta#1} + \newcommand*{\dom}[1]{\mathsf{dom(}#1\mathsf{)}} + \newcommand*{\val}[1]{\mathsf{val(}#1\mathsf{)}} + \newcommand*{\type}[2]{\mathsf{type(}#1, #2\mathsf{)}} + \newcommand*{\submodule}[1]{\mathsf{submodule(}#1\mathsf{)}} + \renewcommand*{\hole}[1][]{\blackrm{u#1}} + + % https://tex.stackexchange.com/questions/128991/frac-or-inference-rule-with-dotted-line + % for mathpartir's inferrule + \makeatletter + \newcommand*{\dotfrac}[2]{ + \mathchoice + {\ooalign{$\genfrac{}{}{0pt}{0}{#1}{#2}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\displaystyle\cdot$\hss}\hfill\kern\z@\cr}} + {\ooalign{$\genfrac{}{}{0pt}{1}{#1}{#2}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\textstyle\cdot$\hss}\hfill\kern\z@\cr}} + {\ooalign{$\genfrac{}{}{0pt}{2}{#1}{#2}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptstyle\cdot$\hss}\hfill\kern\z@\cr}} + {\ooalign{$\genfrac{}{}{0pt}{3}{#1}{#2}$\cr\leavevmode\cleaders\hb@xt@ .22em{\hss $\scriptscriptstyle\cdot$\hss}\hfill\kern\z@\cr}} + } + \makeatother + \newcommand*{\inferruledotfrac}[2]{ + \hbox{$\dotfrac{#1}{#2}$} + } + \newcommand{\hintpagebreak}{\pagebreak[3]} % 0-4; 2 too weak, 4 too strong +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% judgements + % declarative + \newcommand*{\TypAsc}{\black{:}} + \newcommand*{\TypAssump}{\TypAsc} + \newcommand*{\TypAnn}{\TypAsc} + \newcommand*{\KndAsc}{\black{::}} + \newcommand*{\KndAssump}{\KndAsc} + \newcommand*{\KndAnn}{\KndAsc} % why do we have this? + \newcommand*{\SigAsc}{\black{:_{\mod}}} + \newcommand*{\SigAssump}{\SigAsc} + \newcommand*{\SigAnn}{\SigAsc} + \newcommand*{\SigKndAsc}{\black{::_{\sig}}} + \newcommand*{\SigKndAssump}{\SigKndAsc} + + % kinds + \newcommand*{\ConsistentSubKindOp}{\lesssim} + \NewDocumentCommand{\ConsistentSubKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mm}{\mathrm{#1 \vdash #2 \ConsistentSubKindOp #3}} + + \newcommand*{\KindEquivOp}{\equiv} + \NewDocumentCommand{\KindEquiv}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mm}{\mathrm{#1 \vdash #2 \KindEquivOp #3}} + + \newcommand*{\MatchedPiKindOp}{\stackrel{\blacktriangleright}{\Pi}} + \NewDocumentCommand{\MatchedPiKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\DepFunKind}}{\mathrm{#1 \vdash #2 \MatchedPiKindOp #3}} + % types + \newcommand*{\TypeEquivOp}{\equiv} + \newcommand*{\TypeEquivAtKindOp}[1][\knd]{\stackrel{#1}{\TypeEquivOp}} + \NewDocumentCommand{\TypeEquivAtKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mmO{\knd}}{\mathrm{#1 \vdash #2 \TypeEquivAtKindOp[#4] #3}} + \newcommand*{\TypeEquiv}[2]{\TypeEquivAtKind{#1}{#2}[]} + + \newcommand*{\PrincipalKindOp}{::>} + \NewDocumentCommand{\PrincipalKind}{O{}mO{}}{\mathrm{#1 \vdash #2 \PrincipalKindOp #3}} + + \NewDocumentCommand{\WellFormedAtKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\knd}}{\mathrm{#1 \vdash #2 \KndAsc #3}} + % signature kinds + \newcommand{\ConsistentSubSigKindOp}{\lesssim_{\sigknd}} + \newcommand{\ConsistentSubSigKind}[3]{\mathrm{#1 \vdash #2 \ConsistentSubSigKindOp #3}} + + \newcommand*{\SigKindEquivOp}{\equiv} + \NewDocumentCommand{\SigKindEquiv}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mm}{\mathrm{#1 \vdash #2 \equiv #3}} + % signatures + \newcommand*{\SigEquivOp}{\equiv} + \NewDocumentCommand{\SigEquiv}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mmO{\sigknd}}{\mathrm{#1 \vdash #2 \stackrel{#4}{\SigEquivOp} #3 #4}} + + \NewDocumentCommand{\WellFormedAtSigKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\sigknd}}{\mathrm{#1 \vdash #2 \SigKndAsc #3}} + % modules + \NewDocumentCommand{\WellFormedAtSig}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\sig}}{\mathrm{#1 \vdash #2 \SigAsc #3}} + % sdecs + \newcommand{\SubSdecOp}{\le_{\sdec}} + \newcommand{\SubSdec}[3]{\mathrm{#1 \vdash #2 \SubSdecOp #3}} + % ??? + \newcommand*{\WellFormedAtType}[3]{\mathrm{#1 \vdash #2 \TypAsc #3}} + % algorithmic + \newcommand{\Syn}[3]{#1 \vdash #2~\Rightarrow~#3} + \NewDocumentCommand{\SynKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\knd}}{\Syn{#1}{#2}{#3}} + \newcommand{\Ana}[3]{#1 \vdash #2~\Leftarrow~#3} + \NewDocumentCommand{\AnaKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\knd}}{\Ana{#1}{#2}{#3}} + \newcommand{\SynElab}[5]{#1 \vdash #2~\Rightarrow~#3 \rightsquigarrow #4 \dashv #5} + \NewDocumentCommand{\SynKindElab}{O{\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\knd}mO{\HoleCtx[]}}{\SynElab{#1}{#2}{#3}{#4}{#5}} + \newcommand{\AnaElab}[5]{#1 \vdash #2~\Leftarrow~#3 \rightsquigarrow #4 \dashv #5} + \NewDocumentCommand{\AnaKindElab}{O{\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\knd}mO{\HoleCtx[]}}{\AnaElab{#1}{#2}{#3}{#4}{#5}} + \newcommand*{\ElabDegenerate}[4]{#1 \vdash #2 \rightsquigarrow #3 \dashv #4} +\endinput diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index 604c5659f269ebc8eaf228e4d630b692f061605e..8466d26329c01a4f11a187e1cb6a6b0c59cb67cf 100644 GIT binary patch delta 147004 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z;AbOp#u~eO`IyxEX^-y0#FNZ-(C4{2jQ}sVMIm0IW@|28q(RRmU9!lQl8T@e-AX{* z_I)xIXLr^AMDXEc+CD#P-JC7vS0P;VAl5%ef;7l}F|*YMTeFfBA7Ao98a>lfumdavoL@#*uJi}Or z2D&@fsuaCD*Z8yhe6F3@{Y;>`R{2Drx%T&pyVUt}j~i*S5u&EC{Fe6VQ{;>03VutA WnIprx4}uJymji*CT1rJ4;r{?JdxW6? diff --git a/9-type-aliases-redux/latex/kind-judgements.tex b/9-type-aliases-redux/latex/kind-judgements.tex index f4ac428..b196c59 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -82,22 +82,47 @@ \section{syntax} \section{Declaratives} \subsection*{} \begin{minipage}{\textwidth} - \judgbox{\ConsistentSubKind{\knd[1]}{\knd[2]}}{$\knd[1]$ is a consistent subkind of $\knd[2]$} + \judgbox{\PrincipalKind{\typ}}{$\typ$ has principal kind $\knd$} \begin{mathpar} - \inferrule{\KindWellFormed}{\ConsistentSubKind{\KHole}{\knd}} + \inferrule{ }{\PrincipalKind{\bse}[\SKind[\Type][\bse]]} \and - \inferrule{\KindWellFormed}{\ConsistentSubKind{\knd}{\KHole}} - \\ - \inferrule{\KindEquiv{\knd[1]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} + \inferrule{\typvar\KndAssump\knd\in\TypVarCtx[]}{\PrincipalKind{\typvar}[\SKind[\knd][\typvar]]} \and - \inferrule{\KindEquiv{\knd[1]}{\knd[3]} \\ \ConsistentSubKind{\knd[3]}{\knd[4]} \\ \KindEquiv{\knd[4]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} + \inferrule{\WellFormedAtKind{\typ[1]}[\Type] \\ \WellFormedAtKind{\typ[2]}[\Type]}{\PrincipalKind{\binop}[\SKind[\Type][\binop]]} \and - \inferrule{\WellFormedAtKind{\typ}}{\ConsistentSubKind{\SKind}{\knd}} + \inferrule{\hole\KndAssump\knd\in\HoleCtx[]}{\PrincipalKind{\ETypeHole}} \and - \inferrule{\ConsistentSubKind{\knd[3]}{\knd[1]} \\ \ConsistentSubKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[3]]{\knd[2]}{\knd[4]}}{\ConsistentSubKind{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}} - \mprset{myfraction=\inferruledotfrac} + \inferrule{\hole\KndAssump\knd\in\HoleCtx[] \\ \WellFormedAtKind{\typ}[\knd[1]]}{\PrincipalKind{\NETypeHole}} + \and + \inferrule{\hole\KndAssump\knd\in\HoleCtx[] \\ \typvar\notin\dom{\TypVarCtx[]}}{\PrincipalKind{\UnboundTypeVar}} + \and + \inferrule{\PrincipalKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} + \and + \inferrule{\PrincipalKind{\typ[1]} \\ \MatchedPiKind{\knd} \\ \WellFormedAtKind{\typ[2]}[\knd[1]]}{\PrincipalKind{\TypeAp}[\subst{\typ[2]}{\typvar}{\knd[2]}]} + \end{mathpar} + \end{minipage} +\subsection*{} + \begin{minipage}{\textwidth} + \judgbox{\WellFormedAtKind{\typ}}{$\typ$ is well formed at kind $\knd$} + \begin{mathpar} + \inferrule{\PrincipalKind{\typ}}{\WellFormedAtKind{\typ}} + \and + \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ}} + \and + \inferrule{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} \\ - \inferrule{\ConsistentSubKind{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]]}{\ConsistentSubKind{\SKind[\knd[1]][\typ[1]]}{\SKind[\knd[2]][\typ[2]]}} + \inferrule{\WellFormedAtKind{\typ[2]}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} + \and + \inferrule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[3]]] \\ \WellFormedAtKind{\typ[3]}[\SKind[\knd][\typ[2]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} + \end{mathpar} + \end{minipage} +\subsection*{} + \begin{minipage}{\textwidth} + \judgbox{\MatchedPiKind{\knd}}{$\knd$ has matched $\Pi$-kind $\DepFunKind$} + \begin{mathpar} + \inferrule{ }{\MatchedPiKind{\KHole}[\DepFunKind[][\KHole][\KHole]]} + \and + \inferrule{\KindEquiv{\knd}{\DepFunKind}}{\MatchedPiKind{\knd}} \end{mathpar} \end{minipage} \subsection*{} @@ -119,6 +144,26 @@ \subsection*{} \inferrule{\KindEquiv{\knd[1]}{\knd[2]} \\ \KindEquiv[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\knd[3]}{\knd[4]}}{\KindEquiv{\DepFunKind[][\knd[1]][\knd[2]]}{\DepFunKind[][\knd[3]][\knd[4]]}} \end{mathpar} \end{minipage} +\subsection*{} + \begin{minipage}{\textwidth} + \judgbox{\ConsistentSubKind{\knd[1]}{\knd[2]}}{$\knd[1]$ is a consistent subkind of $\knd[2]$} + \begin{mathpar} + \inferrule{\KindWellFormed}{\ConsistentSubKind{\KHole}{\knd}} + \and + \inferrule{\KindWellFormed}{\ConsistentSubKind{\knd}{\KHole}} + \\ + \inferrule{\KindEquiv{\knd[1]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} + \and + \inferrule{\KindEquiv{\knd[1]}{\knd[3]} \\ \ConsistentSubKind{\knd[3]}{\knd[4]} \\ \KindEquiv{\knd[4]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} + \and + \inferrule{\WellFormedAtKind{\typ}}{\ConsistentSubKind{\SKind}{\knd}} + \and + \inferrule{\ConsistentSubKind{\knd[3]}{\knd[1]} \\ \ConsistentSubKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[3]]{\knd[2]}{\knd[4]}}{\ConsistentSubKind{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}} + \mprset{myfraction=\inferruledotfrac} + \\ + \inferrule{\ConsistentSubKind{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]]}{\ConsistentSubKind{\SKind[\knd[1]][\typ[1]]}{\SKind[\knd[2]][\typ[2]]}} + \end{mathpar} + \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{$\typ[1]$ is equivalent to $\typ[2]$ at kind $\knd$} @@ -143,49 +188,4 @@ \subsection*{} \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \end{mathpar} \end{minipage} -\subsection*{} - \begin{minipage}{\textwidth} - \judgbox{\PrincipalKind{\typ}}{$\typ$ has principal kind $\knd$} - \begin{mathpar} - \inferrule{ }{\PrincipalKind{\bse}[\SKind[\Type][\bse]]} - \and - \inferrule{\typvar\KndAssump\knd\in\TypVarCtx[]}{\PrincipalKind{\typvar}[\SKind[\knd][\typvar]]} - \and - \inferrule{\WellFormedAtKind{\typ[1]}[\Type] \\ \WellFormedAtKind{\typ[2]}[\Type]}{\PrincipalKind{\binop}[\SKind[\Type][\binop]]} - \and - \inferrule{\hole\KndAssump\knd\in\HoleCtx[]}{\PrincipalKind{\ETypeHole}} - \and - \inferrule{\hole\KndAssump\knd\in\HoleCtx[] \\ \WellFormedAtKind{\typ}[\knd[1]]}{\PrincipalKind{\NETypeHole}} - \and - \inferrule{\hole\KndAssump\knd\in\HoleCtx[] \\ \typvar\notin\dom{\TypVarCtx[]}}{\PrincipalKind{\UnboundTypeVar}} - \and - \inferrule{\PrincipalKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} - \and - \inferrule{\PrincipalKind{\typ[1]} \\ \MatchedPiKind{\knd} \\ \WellFormedAtKind{\typ[2]}[\knd[1]]}{\PrincipalKind{\TypeAp}[\subst{\typ[2]}{\typvar}{\knd[2]}]} - \end{mathpar} - \end{minipage} -\subsection*{} - \begin{minipage}{\textwidth} - \judgbox{\MatchedPiKind{\knd}}{$\knd$ has matched $\Pi$-kind $\DepFunKind$} - \begin{mathpar} - \inferrule{ }{\MatchedPiKind{\KHole}[\DepFunKind[][\KHole][\KHole]]} - \and - \inferrule{\KindEquiv{\knd}{\DepFunKind}}{\MatchedPiKind{\knd}} - \end{mathpar} - \end{minipage} -\subsection*{} - \begin{minipage}{\textwidth} - \judgbox{\WellFormedAtKind{\typ}}{$\typ$ is well formed at kind $\knd$} - \begin{mathpar} - \inferrule{\PrincipalKind{\typ}}{\WellFormedAtKind{\typ}} - \and - \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ}} - \and - \inferrule{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} - \\ - \inferrule{\WellFormedAtKind{\typ[2]}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} - \and - \inferrule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[3]]] \\ \WellFormedAtKind{\typ[3]}[\SKind[\knd][\typ[2]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} - \end{mathpar} - \end{minipage} \end{document} From f9491db0fbe587de1c019466fcfcffb991c74ba3 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Thu, 15 Jul 2021 19:56:06 +0000 Subject: [PATCH 033/181] KindWellFormed --- .../latex/kind-judgements.pdf | Bin 179856 -> 187436 bytes .../latex/kind-judgements.tex | 15 ++++++++++++++- 2 files changed, 14 insertions(+), 1 deletion(-) diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index bcc528b7917326b4c854e659e019f8d52a874dc5..f105bcfd008e4b7e184da3509a5a4b95ce38d4a2 100644 GIT binary patch delta 17346 zcmV)GK)%0_{R*tO3$Tz012-`d629wK@F*Py1{w`C;8f+1sNs}d`lzrPL6i{K1F@HOPfbCM-JNLrz+EpFcxZ?1olQhjr?Ql$8wO5Xf>`~8boPE|eyQh#`_ zSY-M9Z{Z1khZIICZ}`uDhS%+X{pmHo*2#(hC)ED)zetT65!>O)osjLvqfw(U#%w?S z@@4$t-D}q(D!(3%!w3cYVL&{Lf81XEaiz*ns)A5}fr69~hIR4w)76__W$}*x?|XiS z_OAH5eDYIalt5OgEFOxVuYQb0pnu^XQVGx|C4!$uq0t_QP+$al7$jpmlXelcnWR^^ z#1_j{N_fLRS-VQ3Ag4=o!5&?;)qe3xtLmq`K!@8!`S|5#rNLhReFJoT|M0N>?fv7Q 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\inferrule{\PrincipalKind{\typ}}{\WellFormedAtKind{\typ}} \and \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ}} - \and + \\ \inferrule{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} \\ \inferrule{\WellFormedAtKind{\typ[2]}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} @@ -188,4 +188,17 @@ \subsection*{} \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \end{mathpar} \end{minipage} +\subsection*{} + \begin{minipage}{\textwidth} + \judgbox{\KindWellFormed}{$\knd$ is well formed} + \begin{mathpar} + \inferrule{ }{\KindWellFormed[\Type]} + \and + \inferrule{ }{\KindWellFormed[\KHole]} + \and + \inferrule{\WellFormedAtKind{\typ}}{\KindWellFormed[\SKind]} + \and + \inferrule{\KindWellFormed[{\knd[1]}] \\ \KindWellFormed[{\knd[2]}]}{\KindWellFormed[\DepFunKind]} + \end{mathpar} + \end{minipage} \end{document} From 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give actual names sometime) + \newcounter{counter} + \refstepcounter{counter} + \renewcommand*{\rule}[1][\thecounter\refstepcounter{counter}]{\inferrule*[right=\texttt{#1}]} % get rid of 10-modules stuff \renewcommand*{\TypVarCtx}[1][]{\Phi} \renewcommand*{\ModVarCtx}[1][]{} @@ -82,123 +87,123 @@ \section{syntax} \section{Declaratives} \subsection*{} \begin{minipage}{\textwidth} - \judgbox{\PrincipalKind{\typ}}{$\typ$ has principal kind $\knd$} + \judgbox{\PrincipalKind{\typ}}{$\typ$ has principal (well formed) kind $\knd$} \begin{mathpar} - \inferrule{ }{\PrincipalKind{\bse}[\SKind[\Type][\bse]]} + \rule{ }{\PrincipalKind{\bse}[\SKind[\Type][\bse]]} \and - \inferrule{\typvar\KndAssump\knd\in\TypVarCtx[]}{\PrincipalKind{\typvar}[\SKind[\knd][\typvar]]} + \rule{\typvar\KndAssump\knd\in\TypVarCtx[]}{\PrincipalKind{\typvar}[\SKind[\knd][\typvar]]} \and - \inferrule{\WellFormedAtKind{\typ[1]}[\Type] \\ \WellFormedAtKind{\typ[2]}[\Type]}{\PrincipalKind{\binop}[\SKind[\Type][\binop]]} + \rule{\WellFormedAtKind{\typ[1]}[\Type] \\ \WellFormedAtKind{\typ[2]}[\Type]}{\PrincipalKind{\binop}[\SKind[\Type][\binop]]} \and - \inferrule{\hole\KndAssump\knd\in\HoleCtx[]}{\PrincipalKind{\ETypeHole}} + \rule{\hole\KndAssump\knd\in\HoleCtx[]}{\PrincipalKind{\ETypeHole}} \and - \inferrule{\hole\KndAssump\knd\in\HoleCtx[] \\ \WellFormedAtKind{\typ}[\knd[1]]}{\PrincipalKind{\NETypeHole}} + \rule{\hole\KndAssump\knd\in\HoleCtx[] \\ \WellFormedAtKind{\typ}[\knd[1]]}{\PrincipalKind{\NETypeHole}} \and - \inferrule{\hole\KndAssump\knd\in\HoleCtx[] \\ \typvar\notin\dom{\TypVarCtx[]}}{\PrincipalKind{\UnboundTypeVar}} + \rule{\hole\KndAssump\knd\in\HoleCtx[] \\ \typvar\notin\dom{\TypVarCtx[]}}{\PrincipalKind{\UnboundTypeVar}} \and - \inferrule{\PrincipalKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} + \rule{\PrincipalKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} \and - \inferrule{\PrincipalKind{\typ[1]} \\ \MatchedPiKind{\knd} \\ \WellFormedAtKind{\typ[2]}[\knd[1]]}{\PrincipalKind{\TypeAp}[\subst{\typ[2]}{\typvar}{\knd[2]}]} + \rule{\PrincipalKind{\typ[1]} \\ \MatchedPiKind{\knd} \\ \WellFormedAtKind{\typ[2]}[\knd[1]]}{\PrincipalKind{\TypeAp}[\subst{\typ[2]}{\typvar}{\knd[2]}]} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\WellFormedAtKind{\typ}}{$\typ$ is well formed at kind $\knd$} \begin{mathpar} - \inferrule{\PrincipalKind{\typ}}{\WellFormedAtKind{\typ}} + \rule{\PrincipalKind{\typ}}{\WellFormedAtKind{\typ}} \and - \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ}} + \rule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ}} \\ - \inferrule{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} + \rule{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} \\ - \inferrule{\WellFormedAtKind{\typ[2]}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} + \rule{\WellFormedAtKind{\typ[2]}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} \and - \inferrule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[3]]] \\ \WellFormedAtKind{\typ[3]}[\SKind[\knd][\typ[2]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} + \rule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[3]]] \\ \WellFormedAtKind{\typ[3]}[\SKind[\knd][\typ[2]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\MatchedPiKind{\knd}}{$\knd$ has matched $\Pi$-kind $\DepFunKind$} \begin{mathpar} - \inferrule{ }{\MatchedPiKind{\KHole}[\DepFunKind[][\KHole][\KHole]]} + \rule{ }{\MatchedPiKind{\KHole}[\DepFunKind[][\KHole][\KHole]]} \and - \inferrule{\KindEquiv{\knd}{\DepFunKind}}{\MatchedPiKind{\knd}} + \rule{\KindEquiv{\knd}{\DepFunKind}}{\MatchedPiKind{\knd}} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\KindEquiv{\knd[1]}{\knd[2]}}{$\knd[1]$ is equivalent to $\knd[2]$} \begin{mathpar} - \inferrule{\KindWellFormed}{\KindEquiv{\knd}{\knd}} + \rule{\KindWellFormed}{\KindEquiv{\knd}{\knd}} \and - \inferrule{\KindEquiv{\knd[2]}{\knd[1]}}{\KindEquiv{\knd[1]}{\knd[2]}} + \rule{\KindEquiv{\knd[2]}{\knd[1]}}{\KindEquiv{\knd[1]}{\knd[2]}} \and - \inferrule{\KindEquiv{\knd[1]}{\knd[3]} \\ \KindEquiv{\knd[3]}{\knd[2]}}{\KindEquiv{\knd[1]}{\knd[2]}} + \rule{\KindEquiv{\knd[1]}{\knd[3]} \\ \KindEquiv{\knd[3]}{\knd[2]}}{\KindEquiv{\knd[1]}{\knd[2]}} \and - \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{\KindEquiv{\SKind[\knd][\typ[1]]}{\SKind[\knd][\typ[2]]}} + \rule{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{\KindEquiv{\SKind[\knd][\typ[1]]}{\SKind[\knd][\typ[2]]}} \and - \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\KindEquiv{\SKind[\SKind[\knd][\typ[1]]][\typ]}{\SKind[\knd][\typ[1]]}} + \rule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\KindEquiv{\SKind[\SKind[\knd][\typ[1]]][\typ]}{\SKind[\knd][\typ[1]]}} \and - \inferrule{\WellFormedAtKind{\typ}[\DepFunKind]}{\KindEquiv{\SKind[\DepFunKind][\typ]}{\DepFunKind[][\knd[1]][\SKind[\knd[2]][\TypeAp[\typ][\typvar]]]}} + \rule{\WellFormedAtKind{\typ}[\DepFunKind]}{\KindEquiv{\SKind[\DepFunKind][\typ]}{\DepFunKind[][\knd[1]][\SKind[\knd[2]][\TypeAp[\typ][\typvar]]]}} \and - \inferrule{\KindEquiv{\knd[1]}{\knd[2]} \\ \KindEquiv[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\knd[3]}{\knd[4]}}{\KindEquiv{\DepFunKind[][\knd[1]][\knd[2]]}{\DepFunKind[][\knd[3]][\knd[4]]}} + \rule{\KindEquiv{\knd[1]}{\knd[2]} \\ \KindEquiv[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\knd[3]}{\knd[4]}}{\KindEquiv{\DepFunKind[][\knd[1]][\knd[2]]}{\DepFunKind[][\knd[3]][\knd[4]]}} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\ConsistentSubKind{\knd[1]}{\knd[2]}}{$\knd[1]$ is a consistent subkind of $\knd[2]$} \begin{mathpar} - \inferrule{\KindWellFormed}{\ConsistentSubKind{\KHole}{\knd}} + \rule{\KindWellFormed}{\ConsistentSubKind{\KHole}{\knd}} \and - \inferrule{\KindWellFormed}{\ConsistentSubKind{\knd}{\KHole}} + \rule{\KindWellFormed}{\ConsistentSubKind{\knd}{\KHole}} \\ - \inferrule{\KindEquiv{\knd[1]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} + \rule{\KindEquiv{\knd[1]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} \and - \inferrule{\KindEquiv{\knd[1]}{\knd[3]} \\ \ConsistentSubKind{\knd[3]}{\knd[4]} \\ \KindEquiv{\knd[4]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} + \rule{\KindEquiv{\knd[1]}{\knd[3]} \\ \ConsistentSubKind{\knd[3]}{\knd[4]} \\ \KindEquiv{\knd[4]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} \and - \inferrule{\WellFormedAtKind{\typ}}{\ConsistentSubKind{\SKind}{\knd}} + \rule{\WellFormedAtKind{\typ}}{\ConsistentSubKind{\SKind}{\knd}} \and - \inferrule{\ConsistentSubKind{\knd[3]}{\knd[1]} \\ \ConsistentSubKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[3]]{\knd[2]}{\knd[4]}}{\ConsistentSubKind{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}} - \mprset{myfraction=\inferruledotfrac} + \rule{\ConsistentSubKind{\knd[3]}{\knd[1]} \\ \ConsistentSubKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[3]]{\knd[2]}{\knd[4]}}{\ConsistentSubKind{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}} \\ - \inferrule{\ConsistentSubKind{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]]}{\ConsistentSubKind{\SKind[\knd[1]][\typ[1]]}{\SKind[\knd[2]][\typ[2]]}} + \mprset{myfraction=\inferruledotfrac} + \rule{\ConsistentSubKind{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]]}{\ConsistentSubKind{\SKind[\knd[1]][\typ[1]]}{\SKind[\knd[2]][\typ[2]]}} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{$\typ[1]$ is equivalent to $\typ[2]$ at kind $\knd$} \begin{mathpar} - \inferrule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd]} + \rule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd]} \\ \mprset{myfraction=\inferruledotfrac} - \inferrule{\WellFormedAtKind{\typ}}{\TypeEquivAtKind{\typ}{\typ}} + \rule{\WellFormedAtKind{\typ}}{\TypeEquivAtKind{\typ}{\typ}} \and - \inferrule{\TypeEquivAtKind{\typ[2]}{\typ[1]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} + \rule{\TypeEquivAtKind{\typ[2]}{\typ[1]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \and - \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[3]} \\ \TypeEquivAtKind{\typ[3]}{\typ[1]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} + \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]} \\ \TypeEquivAtKind{\typ[3]}{\typ[1]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \and - \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\Type] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\Type]}{\TypeEquivAtKind{\binop}{\binop[\typ[3]][\typ[4]]}[\Type]} + \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\Type] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\Type]}{\TypeEquivAtKind{\binop}{\binop[\typ[3]][\typ[4]]}[\Type]} \and - \inferrule{\KindEquiv{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ[1]}{\typ[2]}}{\TypeEquivAtKind{\TypeFun[][\knd[1]][\typ[1]]}{\TypeFun[][\knd[2]][\typ[2]]}[\DepFunKind[][\knd[1]][\knd]]} + \rule{\KindEquiv{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ[1]}{\typ[2]}}{\TypeEquivAtKind{\TypeFun[][\knd[1]][\typ[1]]}{\TypeFun[][\knd[2]][\typ[2]]}[\DepFunKind[][\knd[1]][\knd]]} \and - \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[1]}{\typvar}{\knd[2]}]} + \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[1]}{\typvar}{\knd[2]}]} \and - \inferrule{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} + \rule{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} \and - \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} + \rule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\KindWellFormed}{$\knd$ is well formed} \begin{mathpar} - \inferrule{ }{\KindWellFormed[\Type]} + \rule{ }{\KindWellFormed[\Type]} \and - \inferrule{ }{\KindWellFormed[\KHole]} + \rule{ }{\KindWellFormed[\KHole]} \and - \inferrule{\WellFormedAtKind{\typ}}{\KindWellFormed[\SKind]} + \rule{\WellFormedAtKind{\typ}}{\KindWellFormed[\SKind]} \and - \inferrule{\KindWellFormed[{\knd[1]}] \\ \KindWellFormed[{\knd[2]}]}{\KindWellFormed[\DepFunKind]} + \rule{\KindWellFormed[{\knd[1]}] \\ \KindWellFormed[{\knd[2]}]}{\KindWellFormed[\DepFunKind]} \end{mathpar} \end{minipage} \end{document} From ea84685ab2428c070ec976790adceffc4c0c4efb Mon Sep 17 00:00:00 2001 From: Johnson He Date: 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zPlJ$gmMzj70{i^oJllvlFE^G@^O$&&zTE`;(HczZe^-V)tZ~%REdyTAK2N&}(d0751R(E&~>I4E;QR^KiC7j%YRcbD;b0uz^|1p{6~L^(k)MnN(%GdV*>G&C?mL`60- zG%z?cLqat+GBq?aJ|H|qIYBT+K{7HkIYUM?G%!L$MK&@tFgP?rLNzurH8eAqMg{{n ze;w1wZ;eqL#_`{C?mZn$HS|(S6{UunimI_xsd=ia#u~a4E2~;r+B8_j&t-0!pbXnbzk4Gh{-ua~2oA#$2~jl%`?ZyK25`V_UtKEZa_0XtzA?1p~W1AAc~2dYs6upb8D033xOe>eum zVHi%p2%LmdFbb#P44j2?a2_teMYsf);R;-ZF}MbGMc0JC!ga;SQ`fj+^1JJX;@NxG zO~vdN*Db}{->%z=xw`9))cMkNR~npgO-SQCu6xqdBiDUtrnnwRug6>urO!WHlhWUJ zR0*XA1f85bE!F3RcZR8tMPN6?R7ONy?NqlR{HwE)uQxk)aCyN zlP=7+FyF#_3-c|^x9|%qeZXi7qb-cKFxtXsOTW^e=b=hWN%_>3UhQ}a&&ldWo8O7IX5>qG72RnMNdWwDGq8s diff --git a/9-type-aliases-redux/latex/kind-judgements.tex b/9-type-aliases-redux/latex/kind-judgements.tex index e1a4fdb..a358834 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -54,7 +54,7 @@ % (will give actual names sometime) \newcounter{counter} \refstepcounter{counter} - \renewcommand*{\rule}[1][\thecounter\refstepcounter{counter}]{\inferrule*[right=\texttt{#1}]} + \renewcommand*{\rule}[1][(\thecounter)\refstepcounter{counter}]{\inferrule*[vcenter,rightstyle=\texttt,right=#1]} % get rid of 10-modules stuff \renewcommand*{\TypVarCtx}[1][]{\Phi} \renewcommand*{\ModVarCtx}[1][]{} From 5a39d2fae823ac5ae4034071f4e0e1d73c08db46 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Fri, 16 Jul 2021 21:04:50 +0000 Subject: [PATCH 036/181] probably should've named the branch 11-parameterized-types --- 11-type-constructors/11-type-constructors.md | 18 ++ 11-type-constructors/11-type-constructors.tex | 19 ++ .../latex/.gitignore | 0 .../latex/etal.sty | 0 .../latex/hejohns-colors.sty | 0 .../latex/hejohns-hazel.sty | 0 .../latex/joshuadunfield.sty | 0 .../latex/kind-judgements.pdf | Bin .../latex/kind-judgements.tex | 0 11-type-constructors/latex2md | 12 + 9-type-aliases-redux/9-type-aliases-redux.md | 231 ------------------ 11 files changed, 49 insertions(+), 231 deletions(-) create mode 100644 11-type-constructors/11-type-constructors.md create mode 100644 11-type-constructors/11-type-constructors.tex rename {9-type-aliases-redux => 11-type-constructors}/latex/.gitignore (100%) rename {9-type-aliases-redux => 11-type-constructors}/latex/etal.sty (100%) rename {9-type-aliases-redux => 11-type-constructors}/latex/hejohns-colors.sty (100%) rename {9-type-aliases-redux => 11-type-constructors}/latex/hejohns-hazel.sty (100%) rename {9-type-aliases-redux => 11-type-constructors}/latex/joshuadunfield.sty (100%) rename {9-type-aliases-redux => 11-type-constructors}/latex/kind-judgements.pdf (100%) rename {9-type-aliases-redux => 11-type-constructors}/latex/kind-judgements.tex (100%) create mode 100755 11-type-constructors/latex2md delete mode 100644 9-type-aliases-redux/9-type-aliases-redux.md diff --git a/11-type-constructors/11-type-constructors.md b/11-type-constructors/11-type-constructors.md new file mode 100644 index 0000000..fa431fb --- /dev/null +++ b/11-type-constructors/11-type-constructors.md @@ -0,0 +1,18 @@ +10-modules +========== + +Introduction +------------ + +something + +Notes +----- + +- test + +- test + + + + module M = {} in diff --git a/11-type-constructors/11-type-constructors.tex b/11-type-constructors/11-type-constructors.tex new file mode 100644 index 0000000..caca8ee --- /dev/null +++ b/11-type-constructors/11-type-constructors.tex @@ -0,0 +1,19 @@ +\documentclass[12pt]{article} +\usepackage{enumitem} +\usepackage{verbatim} + +\begin{document} +\section*{10-modules} +\today +\subsection*{Introduction} +something +\subsection*{Notes} +\begin{itemize} + \item test + \item test +\end{itemize} +\begin{verbatim} + module M = {} in +\end{verbatim} + +\end{document} diff --git a/9-type-aliases-redux/latex/.gitignore b/11-type-constructors/latex/.gitignore similarity index 100% rename from 9-type-aliases-redux/latex/.gitignore rename to 11-type-constructors/latex/.gitignore diff --git a/9-type-aliases-redux/latex/etal.sty b/11-type-constructors/latex/etal.sty similarity index 100% rename from 9-type-aliases-redux/latex/etal.sty rename to 11-type-constructors/latex/etal.sty diff --git a/9-type-aliases-redux/latex/hejohns-colors.sty b/11-type-constructors/latex/hejohns-colors.sty similarity index 100% rename from 9-type-aliases-redux/latex/hejohns-colors.sty rename to 11-type-constructors/latex/hejohns-colors.sty diff --git a/9-type-aliases-redux/latex/hejohns-hazel.sty b/11-type-constructors/latex/hejohns-hazel.sty similarity index 100% rename from 9-type-aliases-redux/latex/hejohns-hazel.sty rename to 11-type-constructors/latex/hejohns-hazel.sty diff --git a/9-type-aliases-redux/latex/joshuadunfield.sty b/11-type-constructors/latex/joshuadunfield.sty similarity index 100% rename from 9-type-aliases-redux/latex/joshuadunfield.sty rename to 11-type-constructors/latex/joshuadunfield.sty diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/11-type-constructors/latex/kind-judgements.pdf similarity index 100% rename from 9-type-aliases-redux/latex/kind-judgements.pdf rename to 11-type-constructors/latex/kind-judgements.pdf diff --git a/9-type-aliases-redux/latex/kind-judgements.tex b/11-type-constructors/latex/kind-judgements.tex similarity index 100% rename from 9-type-aliases-redux/latex/kind-judgements.tex rename to 11-type-constructors/latex/kind-judgements.tex diff --git a/11-type-constructors/latex2md b/11-type-constructors/latex2md new file mode 100755 index 0000000..5f2b62b --- /dev/null +++ b/11-type-constructors/latex2md @@ -0,0 +1,12 @@ +#!/usr/bin/env perl + +use v5.28; +use utf8; +use strict; +use warnings FATAL => qw(uninitialized); + +my $TODAY = `date '+%Y-%m-%d'`; $? >> 8 and die $!; +my $i = $ARGV[0]; +$i =~ m/(.+)\.tex/ or die "missing '.tex' extension. $!"; +my $o = $1 . '.md'; +`cat $i | perl -p -e 's/\\today/$TODAY/' | pandoc --from=latex --to=markdown_strict -s -o $o`; $? >> 8 and die $!; diff --git a/9-type-aliases-redux/9-type-aliases-redux.md b/9-type-aliases-redux/9-type-aliases-redux.md deleted file mode 100644 index f1721f6..0000000 --- a/9-type-aliases-redux/9-type-aliases-redux.md +++ /dev/null @@ -1,231 +0,0 @@ -# Introduction - -Several future Hazel extensions will require support for type variables, a -prerequisite for type aliases including PHI 3 (8?) (Explicit Polymorphism). -This PHI adds support for type variables and type aliases to Hazel. - -# Type Language - -## Type Identifiers - -We define a type `TyId.t` as follows: - -``` -type TyId.t = string -``` - -The syntax of type variable identifiers is initially the same as expression -variable identifiers, but this may change in the future. We restrict type -variable identifiers from using keywords in the same way (to support -extraction). - -## Type Pattern - -Type patterns are the part of the language between `type` and `=` in `type a = in e` - -It is represented as greek letter `rho` in the attached -[latex document](./latex/kind-judgements.pdf). - -``` -module VarPatErrStatus = { - type t = - | Keyword(ExpandingKeyword.t) -} - -type TPat.t = - | EmptyHole - | TyVar(option(VarPatErrStatus.t), TyId.t) -``` - -## Unexpanded Types - -We extend the syntax of `UHTyp.t` as follows: - -``` -type UHTyp.t = ... | TyVar(VarErrStatus.t, TyId.t) -``` - -## Expanded Types - -In expanded types, we will use a de Bruijn indexed representation. This requires -defining a type abbreviation `HTyp.idx` as follows: - -``` -type HTyp.idx = int -``` - -Then, we extend the syntax of `HTyp.t` as follows: - -``` -type HTyp.t = ... - | TyVar(idx, TyId.t) - | TyVarHole(MetaVar.t, Var.t) -``` - -The `TyVar` constructor represents a bound type variable. We retain the original -identifier for the purposes of supporting contraction back to a `UHTyp.t`. - -The `TyVarHole` constructor represents a free type variable, which is understood -as a type variable hole. Like other holes, each type variable hole has a -metavariable. - -## Kinds - -We will need kinds to distinguish types from type-level holes, such as `TyVarHole`. -We define a new type, `Kind.t`, as follows: -``` -type Kind.t = KHole - | Type - | Singleton(HTyp.t) -``` - -## Unexpanded Expressions - -To support type aliases, we need to add a new line item to the language and the proposed syntax is - -`TyAliasLine(TPat.t, UHTyp.t)` for abstract syntax - -`type t = in e` for GUI syntax - -### `ZExp.re` : - -By considering the cursor positions of `TyAlias`, we need to add two more zline items which are - -`TyAliasLineP(ZTPat.t, UHTyp.t)` (when cursor is on the type pattern) - -`TyAliasLineT(TPat.t, ZTyp.t)` (when cursor is on the type) - -## Expanded Expressions - -So that we can still assign a type properly to the resulting `DExp.t` we add the following new syntax: - -`TyAlias(TPat.t, HTyp.t, Kind.t, DHExp.t)` for abstract syntax - -# Static Semantics - -This PHI introduces a gradual kind system supporting singletons into Hazel. - -The addition of type variables demands we have a type variable context to store them. - -We can define bidirectional type elaboration judgements which will replace the existing `UHTyp.expand` function. - -Type aliases are given meaning with singleton kinds. A singleton kind is kind dependent on a type. - -Singleton kinds to the language demand a subkinding relation and in the context of gradual typing, subkinding is replaced with consistent subkinding relation derived from PFPL chapter 43 and the [Consistent Subtyping For All](https://i.cs.hku.hk/~bruno/papers/gradual-esop18.pdf) paper. In addition to the subkinding relation, we also add the following mutually recursive auxilliary judgements in order to properly assign singleton kinds in the type system: Kind Formation, Kind Equivalence, Kind Assignment, Kind Constructor Equivalence, (along with Consistent Subkinding). - -We extend bidirectional expression elaboration to support the new type alias form using these auxilliary judgements, and add a new type pattern analysis judgement that yields new bindings. - -There are several accompanying theorems presented in the attached -[latex document](./latex/kind-judgements.pdf). - -## Type Variable Contexts - -Type variable contexts are values of type `TyVarCtx.t`, which is defined -as follows: - -``` -type TyVarCtx.t = list((TyId.t, Kind.t)) -``` - -The position within a type variable context determines the de Bruijn index -(most recent bindings at the head). For this reason, type identifiers *can* -be duplicated. - -## Consistent Subkinding - -Consistent Subkinding is defined by a judgement of the form `Kind.t <~ Kind.t` -with the rules presented in the attached -[latex document](./latex/kind-judgements.pdf). - -This judgement handles both consistency and subkinding at the same time. - -## Kind Formation - -Kind Formation is defined by a judgement of the form -`Delta.t; TyVarCtx.t |- Kind.t kind` there is a single rule that says if -`HTyp.t : kind Ty` then we can form the kind `Singleton(Kind.t)`. It is -presented in the attached [latex document](./latex/kind-judgements.pdf). - -## Kind Equivalence - -Kind Equivalence is a reflexive, symmetric, and transitive relation that caries -kind constructor equivalence through to singleton kinds. -`Delta.t; TyVarCtx.t |- Kind.t_1 === Kind.t_2` It is presented in the attached -[latex document](./latex/kind-judgements.pdf). - -## Kind Synthesis/Analysis - -Kind synthesis and analysis is defined by bidirectional judgements -`Delta.t ; TyVarCtx.t |- HTyp.t => Kind.t` and `Delta.t ; TyVarCtx.t |- HTyp.t <= Kind.t` with rules presented in the attached [latex document](./latex/kind-judgements.pdf). Kind synthesis derives the more precise singletons over Ty wherever possible. - -## Kind Constructor Equivalence - -Currently, Kind Constructor Equivalence is exactly type equivalence because -Hazel (even after applying this proposal) doesn't support higher kinds. Kind -Constructor Equivalence is a reflexive, symmetric, and transitive relation -where the "type aliasing" using the singleton kind occurs. The judgement -`Delta.t; TyVarCtx.t |- HTyp.t_1 === HTyp.t_2` is defined in the attached -[latex document](./latex/kind-judgements.pdf). - -## Type Elaboration - -Expansion from unexpanded types to expanded types is defined by -bidirectional judgements `TyVarCtx.t |- UHTyp.t => Kind.t ~> HTyp.t -| Delta` -and `TyVarCtx.t |- UHtyp.t <= Kind.t ~> HTyp.t -| Delta`. These judgements are shown in the attached -[latex document](./latex/kind-judgements.pdf). Type elaboration synthesis has an affinity for singletons over Ty. - -`Delta.t`'s `hole_sort` will be changed to a `Hole.t` so we can track type -holes there too. `Hole.t` is an ADT that captures type holes' different `Kind.t` -and `TyVarCtx.t` instead of `HTyp.t` and `VarCtx.t` and we'll still have the -same keyspace (the `u`s are consistent and incrementing across all the holes): - -```reasonml -module Hole = { -type t = - | Expression(HTyp.t, VarCtx.t) - | Type(Kind.t, TyVarCtx.t) - | Pattern(HTyp.t, VarCtx.t); - -type t = MetaVarMap.t(Hole.t); -``` - -## Type Pattern Analysis - -Type patterns analyze with a type to create new tyvar bindings and hole bindings -`Delta.t_1 ; TyVarCtx.t_1 |- TPat.t <- HTyp.t : Kind.t -| TyVarCtx.t_2 ; Delta.t_2`. This -judgement is in the attached [latex document](./latex/kind-judgements.pdf). - -## Expression Elaboration for Type Aliases - -The expanded type alias expression also contains an explicit ascription for the kind associated with the internal type. See the attached [latex document](./latex/kind-judgements.pdf). - -# Action Semantics - -## Type Variables - -Adding support for type variables requires adding support for character-level -editing at the type-level. Currently, types are bound to individual uppercase -characters, e.g. `I` inserts the `Int` type immediately. This needs to be -changed to support natural text editing. We will follow the approach taken -with variables in expressions and patterns. - -We will need to change the action semantics for types to support kinds: - -Previously the type-level action semantics was defined by a judgement of -the form: - -`ZTyp.t --Action.t--> ZTyp.t` - -Now, we will need to define a bidirectional action semantics as with -expressions by judgements of the following form: - -1. `TVarCtx.t |- ZTyp.t => Kind.t --Action.t--> ZTyp.t => Kind.t` -2. `TVarCtx.t |- ZTyp.t --Action.t--> ZTyp.t <= Kind.t` - -## Construction of type aliases - -TODO: Adapt from polymorphism PHI - -## Backspace Actions - -TODO: Port from polymorphism PHI From 7e80c008cd29c8f208915c412c2f0e25b298e5ce Mon Sep 17 00:00:00 2001 From: Johnson He Date: Fri, 16 Jul 2021 21:25:15 +0000 Subject: [PATCH 037/181] rename things --- 11-type-constructors/11-type-constructors.md | 4 ++-- 11-type-constructors/11-type-constructors.tex | 2 +- 11-type-constructors/README.md | 18 ++++++++++++++++++ 11-type-constructors/gen | 4 ++++ 11-type-constructors/latex/.gitignore | 4 ++++ ...udgements.pdf => 11-type-constructors.pdf} | Bin 198765 -> 198765 bytes ...udgements.tex => 11-type-constructors.tex} | 0 11-type-constructors/latex/hejohns-hazel.sty | 2 +- 8 files changed, 30 insertions(+), 4 deletions(-) create mode 100644 11-type-constructors/README.md create mode 100755 11-type-constructors/gen rename 11-type-constructors/latex/{kind-judgements.pdf => 11-type-constructors.pdf} (99%) rename 11-type-constructors/latex/{kind-judgements.tex => 11-type-constructors.tex} (100%) diff --git a/11-type-constructors/11-type-constructors.md b/11-type-constructors/11-type-constructors.md index fa431fb..95111e2 100644 --- a/11-type-constructors/11-type-constructors.md +++ b/11-type-constructors/11-type-constructors.md @@ -1,5 +1,5 @@ -10-modules -========== +11-type-constructors +==================== Introduction ------------ diff --git a/11-type-constructors/11-type-constructors.tex b/11-type-constructors/11-type-constructors.tex index caca8ee..0406db2 100644 --- a/11-type-constructors/11-type-constructors.tex +++ b/11-type-constructors/11-type-constructors.tex @@ -3,7 +3,7 @@ \usepackage{verbatim} \begin{document} -\section*{10-modules} +\section*{11-type-constructors} \today \subsection*{Introduction} something diff --git a/11-type-constructors/README.md b/11-type-constructors/README.md new file mode 100644 index 0000000..95111e2 --- /dev/null +++ b/11-type-constructors/README.md @@ -0,0 +1,18 @@ +11-type-constructors +==================== + +Introduction +------------ + +something + +Notes +----- + +- test + +- test + + + + module M = {} in diff --git a/11-type-constructors/gen b/11-type-constructors/gen new file mode 100755 index 0000000..76bc373 --- /dev/null +++ b/11-type-constructors/gen @@ -0,0 +1,4 @@ +#!/usr/bin/env bash + +./latex2md 11-type-constructors.tex +cp 11-type-constructors.md README.md diff --git a/11-type-constructors/latex/.gitignore b/11-type-constructors/latex/.gitignore index 0e98f99..02d864a 100644 --- a/11-type-constructors/latex/.gitignore +++ b/11-type-constructors/latex/.gitignore @@ -1,2 +1,6 @@ *.log *.aux +*.fdb_latexmk +*.fls +*.out +*.gz diff --git a/11-type-constructors/latex/kind-judgements.pdf b/11-type-constructors/latex/11-type-constructors.pdf similarity index 99% rename from 11-type-constructors/latex/kind-judgements.pdf rename to 11-type-constructors/latex/11-type-constructors.pdf index 708da8e5201b610d2ed180b14ee6029f29c93f7e..f383d657bae81558d9d1ef77a72d86c795f72c73 100644 GIT binary patch delta 112 zcmaDmf#>Z6o`x32EliSs*^CU0jEv2v%l~B(hqAZZ{AD`M;cRN?YUbu*Wa(z?Vq#=y hX<%+{VB+dzV&rCF=4NQ&=xC>4LrBSXH#TNZ6o`x32EliSs*$j*f%!~}C%l~B(hqAZZ{AD`M;p}E$Xz66+V&ddzY-(xb hXkcPs;OOdNXliQi?C56T=whc}LrBSXH#TN Date: Tue, 20 Jul 2021 19:10:04 +0000 Subject: [PATCH 038/181] does 21 make sense?? --- .../latex/11-type-constructors.pdf | Bin 198765 -> 222379 bytes .../latex/11-type-constructors.tex | 286 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\RequirePackage{joshuadunfield} % ? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % syntax \newcommand*{\knd}[1][]{\redit{\kappa_{#1}}} From 9f6442b9cac76eebfdda7d9c956bffa0999952f4 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Tue, 20 Jul 2021 22:44:41 +0000 Subject: [PATCH 039/181] a bunch of well formedness stuff --- .../latex/11-type-constructors.pdf | Bin 222379 -> 223355 bytes .../latex/11-type-constructors.tex | 76 ++++++++++++------ 11-type-constructors/latex/hejohns-hazel.sty | 10 ++- 3 files changed, 59 insertions(+), 27 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index 9e871886af464118954a9f7ef5b8fc51556582d1..fe3fa855c2fa5e14df5af8083cc26c9b50ab3af6 100644 GIT binary patch delta 69772 zcmV)2K+M0Z$qoC^4Ui-QHZn4ovCRS~kxvVM-lWrDy)%2akNe=0sIi@%B#ruHJ14Bg zBx-3%mFOn!ug@$9lGw#{K~SO+J&C4?!R}nX`DSMTp8cHh>~BZB=kJTNqbH|=XULh> 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ziur8ioaMctP|5J=lW2i?UBQbCgSml|OochHSlOVha-~zU9{bv^@{3{@uRuZeyieELfztAN#J@j7wCvlMO4{;1crz`r2O zN#=I4$TtbH2pU@!wDVfZ)kQl|6vFW4WghBSCjz#7pa)~jsO4v diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index 84fb5f7..f68b90f 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -57,7 +57,7 @@ \renewcommand*{\rule}[1][(\thecounter)\refstepcounter{counter}]{\inferrule*[vcenter,rightstyle=\texttt,right=#1]} % get rid of 10-modules stuff \RenewDocumentCommand{\TypVarCtx}{oO{}}{\Phi_{#2}} - \RenewDocumentCommand{\InTypVarCtx}{ooO{\knd}}{\TypVarCtx[][1],\typvar[#2]\KndAssump{#3},\TypVarCtx[][2]} + \RenewDocumentCommand{\InTypVarCtx}{oO{}O{\knd}}{\TypVarCtx[][1],\typvar[#2]\KndAssump{#3},\TypVarCtx[][2]} \RenewDocumentCommand{\ExtendTypVarCtx}{ooO{\knd}}{\TypVarCtx[],\typvar[#2]\KndAssump{#3}#1} \renewcommand*{\ModVarCtx}[1][]{} \renewcommand*{\SigVarCtx}[1][]{} @@ -95,7 +95,7 @@ \section{Declaratives} \and \rule{\ContextWellFormed[\HoleCtx\InTypVarCtx]}{\PrincipalKind{\typvar}[\SKind[\knd][\typvar]]} \and - \rule{\ContextWellFormed \\ \WellFormedAtKind{\typ[1]}[\Type] \\ \WellFormedAtKind{\typ[2]}[\Type]}{\PrincipalKind{\binop}[\SKind[\Type][\binop]]} + \rule{\WellFormedAtKind{\typ[1]}[\Type] \\ \WellFormedAtKind{\typ[2]}[\Type]}{\PrincipalKind{\binop}[\SKind[\Type][\binop]]} \and \rule{\ContextWellFormed[\InTypHoleCtx\TypVarCtx]}{\PrincipalKind{\ETypeHole}[\SKind[\knd][\ETypeHole]]} \and @@ -103,7 +103,7 @@ \section{Declaratives} \and \rule{\ContextWellFormed[\InTypHoleCtx\TypVarCtx] \\ \typvar\notin\dom{\TypVarCtx[]}}{\PrincipalKind{\UnboundTypeVar}[\SKind[\knd][\UnboundTypeVar]]} \and - \rule{\KindWellFormed[{\knd[1]}] \\ \PrincipalKind[\HoleCtx\ExtendTypVarCtx[][][{\knd[1]}]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} + \rule{\PrincipalKind[\HoleCtx\ExtendTypVarCtx[][][{\knd[1]}]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} \and \rule{\PrincipalKind{\typ[1]} \\ \MatchedPiKind{\knd} \\ \WellFormedAtKind{\typ[2]}[\knd[1]]}{\PrincipalKind{\TypeAp}[\subst{\typ[2]}{\typvar}{\knd[2]}]} \end{mathpar} @@ -268,6 +268,10 @@ \section{Metatheory} \vphantom{.}\\ If $\WellFormedAtKind{\typ[L]}[\knd[L1]]$ and $\KindWellFormed[{\knd[L2]}][\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]$, then $\ContextWellFormed$ and $\KindWellFormed[\subst{\typ[L]}{\typvar[L]}{\knd[L2]}]$ \\ (induction on $\KindWellFormed[{\knd[L2]}][\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]$) \end{lemma} + %\begin{lemma}[PK-Substitution] + % \vphantom{.}\\ If $\PrincipalKind{\typ[L1]}[\knd[L1]]$ and $\PrincipalKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$, then $\PrincipalKind{\subst{\typ[L1]}{\typvar[L]}{\typ[L2]}}[\subst{\typ[L1]}{\typvar[L]}{\knd[L2]}]$ \\ + % (induction on $\PrincipalKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$) + %\end{lemma} \begin{lemma}[K-Substitution] \vphantom{.}\\ If $\WellFormedAtKind{\typ[L1]}[\knd[L1]]$ and $\WellFormedAtKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$, then $\WellFormedAtKind{\subst{\typ[L1]}{\typvar[L]}{\typ[L2]}}[\subst{\typ[L1]}{\typvar[L]}{\knd[L2]}]$ \\ (induction on $\WellFormedAtKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$) @@ -301,6 +305,9 @@ \section{Metatheory} \end{tabular*} \end{proof} \vphantom{.} + \begin{lemma}[PK-Unicity] + If $\PrincipalKind{\typ[L]}[\knd[L1]]$ and $\PrincipalKind{\typ[L]}[\knd[L2]]$ then ${\knd[L1]}$ is ${\knd[L2]}$ + \end{lemma} \begin{lemma} If $\PrincipalKind{\typ}[\knd[1]]$ and $\WellFormedAtKind{\typ}[\knd[2]]$, then $\ConsistentSubKind{\knd[1]}{\knd[2]}$ \end{lemma} diff --git a/11-type-constructors/latex/hejohns-hazel.sty b/11-type-constructors/latex/hejohns-hazel.sty index 69b2c37..3d5d230 100644 --- a/11-type-constructors/latex/hejohns-hazel.sty +++ b/11-type-constructors/latex/hejohns-hazel.sty @@ -130,7 +130,7 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % misc \newcommand*{\ExpVarCtx}[1][;]{\Gamma#1} - \NewDocumentCommand{\TypVarCtx}{O{;}O{}}{\Phi#1_{#2}} + \NewDocumentCommand{\TypVarCtx}{O{;}o}{\Phi#1_{#2}} \NewDocumentCommand{\InTypVarCtx}{O{;}oO{\knd}}{\TypVarCtx[][1],\typvar[#2]\KndAssump{#3},\TypVarCtx[][2]#1} \NewDocumentCommand{\ExtendTypVarCtx}{O{;}oO{\knd}}{\TypVarCtx[],\typvar[#2]\KndAssump{#3}#1} \newcommand*{\ModVarCtx}[1][;]{\Xi#1} From b6dbc6ef2789b25e1d917984bd46e7190ebb0b14 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Fri, 23 Jul 2021 06:34:00 +0000 Subject: [PATCH 044/181] ?! looks strange. 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\section{Declaratives} \begin{mathpar} \rule{\PrincipalKind{\typ}}{\WellFormedAtKind{\typ}} \and + \rule{\PrincipalKind{\typ}[\SKind[\knd][\typ]]}{\WellFormedAtKind{\typ}} + \and \rule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]] \\ \WellFormedAtKind{\typ[1]}}{\WellFormedAtKind{\typ}} \\ \rule{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} \\ - \rule{\WellFormedAtKind{\typ[2]}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} - \and - \rule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[3]]] \\ \WellFormedAtKind{\typ[3]}[\SKind[\knd][\typ[2]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} - \\ \mprset{myfraction=\inferruledotfrac} \rule{\WellFormedAtKind{\typ}}{\WellFormedAtKind{\typ}[\SKind[\knd][\typ]]} \end{mathpar} @@ -130,7 +128,9 @@ \section{Declaratives} \begin{minipage}{\textwidth} \judgbox{\MatchedPiKind{\knd}}{$\knd$ has matched $\Pi$-kind $\DepFunKind$} \begin{mathpar} - \rule{\KindEquiv{\knd}{\KHole}}{\MatchedPiKind{\knd}[\DepFunKind[][\KHole][\KHole]]} + \rule{ }{\MatchedPiKind{\KHole}[\DepFunKind[][\KHole][\KHole]]} + \and + \rule{\KindEquiv{\knd}{\SKind[\KHole][\typ[1]]}}{\MatchedPiKind{\knd}[\DepFunKind[][\knd][\knd]]} \and \rule{\KindEquiv{\knd}{\DepFunKind}}{\MatchedPiKind{\knd}} \end{mathpar} @@ -147,8 +147,6 @@ \section{Declaratives} \and \rule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\KindEquiv{\SKind[\SKind[\knd][\typ[1]]][\typ]}{\SKind[\knd][\typ[1]]}} \and - \rule{\WellFormedAtKind{\typ}[\SKind[\KHole][\typ[1]]]}{\KindEquiv{\SKind[\KHole]}{\KHole}} - \and \rule{\WellFormedAtKind{\typ}[\DepFunKind]}{\KindEquiv{\SKind[\DepFunKind][\typ]}{\DepFunKind[][\knd[1]][\SKind[\knd[2]][\TypeAp[\typ][\typvar]]]}} \and \rule{\KindEquiv{\knd[1]}{\knd[2]} \\ \KindEquiv[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\knd[3]}{\knd[4]}}{\KindEquiv{\DepFunKind[][\knd[1]][\knd[2]]}{\DepFunKind[][\knd[3]][\knd[4]]}} @@ -164,12 +162,16 @@ \section{Declaratives} \rule{\KindWellFormed}{\ConsistentSubKind{\KHole}{\knd}} \and \rule{\KindWellFormed}{\ConsistentSubKind{\knd}{\KHole}} + \and + \rule{\KindWellFormed[{\SKind[\KHole]}] \\ \KindWellFormed}{\ConsistentSubKind{\SKind[\KHole]}{\knd}} + \and + \rule{\KindWellFormed \\ \KindWellFormed[{\SKind[\KHole]}]}{\ConsistentSubKind{\knd}{\SKind[\KHole]}} \\ \rule{\KindEquiv{\knd[1]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} \and \rule{\KindEquiv{\knd[1]}{\knd[3]} \\ \ConsistentSubKind{\knd[3]}{\knd[4]} \\ \KindEquiv{\knd[4]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} \and - \rule{\WellFormedAtKind{\typ}}{\ConsistentSubKind{\SKind}{\knd}} + \rule{\KindWellFormed[{\SKind}]}{\ConsistentSubKind{\SKind}{\knd}} \and \rule{\ConsistentSubKind{\knd[3]}{\knd[1]} \\ \ConsistentSubKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[3]]{\knd[2]}{\knd[4]}}{\ConsistentSubKind{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}} \\ @@ -179,9 +181,16 @@ \section{Declaratives} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} - \judgbox{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{$\typ[1]$ is equivalent to $\typ[2]$ at kind $\knd$} + \judgbox{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{$\typ[1]$ is provably equivalent to $\typ[2]$ at kind $\knd$} \begin{mathpar} - \rule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd]} + \rule{\PrincipalKind{\typ[1]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\SKind[\knd][\typ[2]]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd]} + \and + \rule{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} + \mprset{myfraction=\inferruledotfrac} + \and + \rule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\SKind]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} + \and + \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[2]}{\typvar}{\knd[2]}]} \\ \mprset{myfraction=\inferruledotfrac} \rule{\WellFormedAtKind{\typ}}{\TypeEquivAtKind{\typ}{\typ}} @@ -194,10 +203,6 @@ \section{Declaratives} \and \rule{\KindEquiv{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ[1]}{\typ[2]}}{\TypeEquivAtKind{\TypeFun[][\knd[1]][\typ[1]]}{\TypeFun[][\knd[2]][\typ[2]]}[\DepFunKind[][\knd[1]][\knd]]} \and - \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[1]}{\typvar}{\knd[2]}]} - \and - \rule{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} - \and \rule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \end{mathpar} \end{minipage} From aa481afe6345f7e2718b9df02ede84eae95f3d83 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Sat, 24 Jul 2021 06:20:01 +0000 Subject: [PATCH 046/181] title --- .../latex/11-type-constructors.pdf | Bin 239795 -> 240540 bytes .../latex/11-type-constructors.tex | 2 +- 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index 9fc858c60388d3f8d1f9bb1d0b0579456c170219..a6491360c0761d53bd6abb6232616f48dbee4a6b 100644 GIT binary patch delta 18782 zcmV(tK(GHx~4v-}RH!_oP3@njT8wq7bqGQGs$&jm&cQ$_oPBW=ZrJZEPs_Nu@ 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zTbwRYs6rsq_BTY8g5m3s-87-6y;MkS0;d6ge2@_ONLMo?-VGVD0@$xSjaR*-<4t{Z0)jA(J~e-i2H3 z$cOUEM|?B<;S?Vxh!tpv6`+U}dT!+KIFNN0;E0AexSqr9i>j9lI+6aIQJ@qccb>JvRqfnyrBlHLZqQY!C3DmIXsewy-Mp78 zRo_`(13RXUFo7M@1OSCIChsrd^3{t)_hSA={aWDL^yf+~yhfs?@`R?+zXD)u(SwNP zTw=0VF|oAXr!r2 Date: Tue, 27 Jul 2021 01:30:22 +0000 Subject: [PATCH 049/181] actually need this rule after all for when eg type function is a variable, so can't look inside body with normalization/reduction can probably prove transativity etc of type equality as a consequence of transativity etc of kind equality, but will be some work for sure --- .../latex/11-type-constructors.pdf | Bin 241252 -> 241235 bytes .../latex/11-type-constructors.tex | 4 ++-- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index 68c57bc5922c97ddb399dc393fb6db0a6185887f..19b63322108b482cf1c4ce7366e03eafee0230e8 100644 GIT binary patch delta 13366 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a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index 7c5fe21..a91f057 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -188,11 +188,11 @@ \section{Declaratives} \rule{\PrincipalKind{\typ[1]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\SKind[\knd][\typ[2]]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd]} \and \rule{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} + \and + \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[2]}{\typvar}{\knd[2]}]} \mprset{myfraction=\inferruledotfrac} \and \rule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\SKind]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} - \and - \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[2]}{\typvar}{\knd[2]}]} \\ \mprset{myfraction=\inferruledotfrac} \rule{\WellFormedAtKind{\typ}}{\TypeEquivAtKind{\typ}{\typ}} From 311380e84a1772eaacfebe1b413c23d112b8ee04 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Tue, 27 Jul 2021 03:15:24 +0000 Subject: [PATCH 050/181] yeah ok we do need refl,symm,trans for type equality otherwise we wouldn't be able to show the two ways to reduce a singleton of a singleton are equivalent --- .../latex/11-type-constructors.pdf | Bin 241235 -> 241370 bytes .../latex/11-type-constructors.tex | 13 ++++++++----- 2 files changed, 8 insertions(+), 5 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index 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z0M0%jxdM^(I|NW-h)oQTp^^a@(%Jq(XcXadb=G)+u6l{_MN>w+jGF(X@m6RmlaGG-T8yDPTNNiD;me=&2vmyQ-)kGyUl49y30}Wwjep)|#>@s^e1L7-$q|5Vcgks*_s| zz7ps1Ddggk;Zl8HO4&_EmkneGQa`*3*p01f-0f%QB0OZ33sW_$iy5`w^4+h}i>JG( zW5D0tu+Hcgw96WlEPhXk=7jLV{||v;LbgZj{gl%d&s1XC|JrKry2}SLPp7?Hoqx4I d|Mj7IbKhpG=^NKE%X1l9Sa7MTy863u0RSsJ0Db@f diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index a91f057..d00d7ac 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -152,7 +152,7 @@ \section{Declaratives} \rule{\WellFormedAtKind{\typ}[\DepFunKind]}{\KindEquiv{\SKind[\DepFunKind][\typ]}{\DepFunKind[1][\knd[1]][\SKind[\subst{\typvar[1]}{\typvar}{\knd[2]}][\TypeAp[\typ][\typvar[1]]]]}} \and \rule{\KindEquiv{\knd[1]}{\knd[2]} \\ \KindEquiv[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\knd[3]}{\knd[4]}}{\KindEquiv{\DepFunKind[][\knd[1]][\knd[2]]}{\DepFunKind[][\knd[3]][\knd[4]]}} - \mprset{myfraction=\inferruledotfrac} + %\mprset{myfraction=\inferruledotfrac} \and \rule{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{\KindEquiv{\SKind[\knd][\typ[1]]}{\SKind[\knd][\typ[2]]}} \end{mathpar} @@ -190,17 +190,17 @@ \section{Declaratives} \rule{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} \and \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[2]}{\typvar}{\knd[2]}]} - \mprset{myfraction=\inferruledotfrac} - \and - \rule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\SKind]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \\ - \mprset{myfraction=\inferruledotfrac} \rule{\WellFormedAtKind{\typ}}{\TypeEquivAtKind{\typ}{\typ}} \and \rule{\TypeEquivAtKind{\typ[2]}{\typ[1]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \and \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]} \\ \TypeEquivAtKind{\typ[3]}{\typ[1]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} + \mprset{myfraction=\inferruledotfrac} \and + \rule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\SKind]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} + \\ + \mprset{myfraction=\inferruledotfrac} \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\Type] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\Type]}{\TypeEquivAtKind{\binop}{\binop[\typ[3]][\typ[4]]}[\Type]} \and \rule{\KindEquiv{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ[1]}{\typ[2]}}{\TypeEquivAtKind{\TypeFun[][\knd[1]][\typ[1]]}{\TypeFun[][\knd[2]][\typ[2]]}[\DepFunKind[][\knd[1]][\knd]]} @@ -321,4 +321,7 @@ \section{Metatheory} \begin{lemma} If $\PrincipalKind{\typ}[\knd[1]]$ and $\WellFormedAtKind{\typ}[\knd[2]]$, then $\ConsistentSubKind{\knd[1]}{\knd[2]}$ \end{lemma} + \begin{lemma} + If $\ConsistentSubKind{\knd[1]}{\SKind[\knd[2]]}$, then $\ConsistentSubKind{\knd[1]}{\knd[2]}$ + \end{lemma} \end{document} From 2e1a28c0edfaf99f92660ca4420a2b910b234070 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Wed, 28 Jul 2021 05:27:08 +0000 Subject: [PATCH 051/181] naming rules --- .../latex/11-type-constructors.pdf | Bin 241370 -> 252157 bytes .../latex/11-type-constructors.tex | 55 +++++++++--------- 2 files changed, 27 insertions(+), 28 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index ef6eb1ffc3cff8f52b9c7a9dd8e0d796e95d9d71..b13264af1355845b1a207a62f2c079d8ca3e138c 100644 GIT binary patch delta 56194 zcmZU)Q*bX#^eh_Nww>(Q*s*Qfw(*N?+qP}n#*S?}x!?bsQ}^Dg^EUl3^Dxtw z+(GzmK#V5^V`gV%OjB$Dp#og%*gE63A^+6sJARAFdNX3!!>%VyFDrLN?6BJv9YtNs zU`JRkPZkO(%4X~5jo^FoZu7*7D?5D+7)N$*Y16so`C>$Y%Ks$Y9o12GpZ=DaQ^w^= z=K?VUYmuAtlkLGNL!cg_30EaGMIAj~MZMQWRV}GFF?l~99v?2LI09HeCDzIYr~mC{ 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\lambda]{\PrincipalKind[\HoleCtx\ExtendTypVarCtx[][][{\knd[1]}]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} + \rule[PK\hyph \lambda]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][{\knd[1]}]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} \and \rule[PK\hyph Ap]{\PrincipalKind{\typ[1]} \\ \MatchedPiKind{\knd} \\ \WellFormedAtKind{\typ[2]}[\knd[1]]}{\PrincipalKind{\TypeAp}[\subst{\typ[2]}{\typvar}{\knd[2]}]} \end{mathpar} @@ -152,7 +152,7 @@ \section{Declaratives} \and \rule[KEquiv\hyph SKind_{\Pi}]{\WellFormedAtKind{\typ}[\DepFunKind]}{\KindEquiv{\SKind[\DepFunKind][\typ]}{\DepFunKind[1][\knd[1]][\SKind[\subst{\typvar[1]}{\typvar}{\knd[2]}][\TypeAp[\typ][\typvar[1]]]]}} \and - \rule[KEquiv\hyph \Pi]{\KindEquiv{\knd[1]}{\knd[2]} \\ \KindEquiv[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\knd[3]}{\knd[4]}}{\KindEquiv{\DepFunKind[][\knd[1]][\knd[2]]}{\DepFunKind[][\knd[3]][\knd[4]]}} + \rule[KEquiv\hyph \Pi]{\KindEquiv{\knd[1]}{\knd[2]} \\ \KindEquiv[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\knd[3]}{\knd[4]}}{\KindEquiv{\DepFunKind[][\knd[1]][\knd[2]]}{\DepFunKind[][\knd[3]][\knd[4]]}} \and \rule[KEquiv\hyph SKind]{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd[2]}}{\KindEquiv{\SKind[\knd[1]][\typ[1]]}{\SKind[\knd[2]][\typ[2]]}} \end{mathpar} @@ -175,7 +175,7 @@ \section{Declaratives} \and \rule[CSK\hyph SKind]{\KindWellFormed[{\SKind}]}{\ConsistentSubKind{\SKind}{\knd}} \and - \rule[CSK\hyph \Pi]{\ConsistentSubKind{\knd[3]}{\knd[1]} \\ \ConsistentSubKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[3]]{\knd[2]}{\knd[4]}}{\ConsistentSubKind{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}} + \rule[CSK\hyph \Pi]{\ConsistentSubKind{\knd[3]}{\knd[1]} \\ \ConsistentSubKind[\HoleCtx\xExtendTypVarCtx[][][\knd[3]]]{\knd[2]}{\knd[4]}}{\ConsistentSubKind{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}} \\ \mprset{myfraction=\inferruledotfrac} \rule[CSK-?]{\ConsistentSubKind{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]]}{\ConsistentSubKind{\SKind[\knd[1]][\typ[1]]}{\SKind[\knd[2]][\typ[2]]}} @@ -193,7 +193,7 @@ \section{Declaratives} \\ \rule[EquivAK\hyph SKind]{\PrincipalKind{\typ[1]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\SKind[\knd][\typ[2]]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd]} \and - \rule[EquivAK\hyph\Pi]{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} + \rule[EquivAK\hyph\Pi]{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\xTypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} \and \rule[EquivAK\hyph Ap]{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[2]}{\typvar}{\knd[2]}]} \\ @@ -204,7 +204,7 @@ \section{Declaratives} \mprset{myfraction=\inferruledotfrac} \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\Type] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\Type]}{\TypeEquivAtKind{\binop}{\binop[\typ[3]][\typ[4]]}[\Type]} \and - \rule{\KindEquiv{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ[1]}{\typ[2]}}{\TypeEquivAtKind{\TypeFun[][\knd[1]][\typ[1]]}{\TypeFun[][\knd[2]][\typ[2]]}[\DepFunKind[][\knd[1]][\knd]]} + \rule{\KindEquiv{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ[1]}{\typ[2]}}{\TypeEquivAtKind{\TypeFun[][\knd[1]][\typ[1]]}{\TypeFun[][\knd[2]][\typ[2]]}[\DepFunKind[][\knd[1]][\knd]]} \and \rule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \end{mathpar} @@ -219,7 +219,7 @@ \section{Declaratives} \and \rule[KWF\hyph SKind]{\WellFormedAtKind{\typ}}{\KindWellFormed[\SKind]} \and - \rule[KWF\hyph\Pi]{\KindWellFormed[{\knd[1]}] \\ \KindWellFormed[{\knd[2]}][\HoleCtx\ExtendTypVarCtx[][][{\knd[1]}]]}{\KindWellFormed[\DepFunKind]} + \rule[KWF\hyph\Pi]{\KindWellFormed[{\knd[2]}][\HoleCtx\xExtendTypVarCtx[][][{\knd[1]}]]}{\KindWellFormed[\DepFunKind]} \end{mathpar} \end{minipage} \subsection*{} @@ -228,15 +228,20 @@ \section{Declaratives} \begin{mathpar} \rule[CWF\hyph Nil]{ }{\ContextWellFormed[\cdot;\cdot]} \and - \rule[CWF\hyph TypVar]{\KindWellFormed}{\ContextWellFormed[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd]} + \rule[CWF\hyph TypVar]{\typvar\notin\dom{\TypVarCtx[]} \\ \KindWellFormed}{\ContextWellFormed[\HoleCtx\xTypVarCtx[],\typvar\KndAssump\knd]} \and - \rule[CWF\hyph Hole]{\KindWellFormed}{\ContextWellFormed[\HoleCtx[],\hole\KndAssump\knd;\TypVarCtx[]]} + \rule[CWF\hyph Hole]{\hole\notin\dom{\HoleCtx[]} \\ \KindWellFormed}{\ContextWellFormed[\xHoleCtx[],\hole\KndAssump\knd;\TypVarCtx[]]} \end{mathpar} - *Variables implicitly assumed to be fresh as necessary* \\ - (eg $\typvar[L]$ is fresh in $\mathcal{J}$ in Weakening) \end{minipage} \section{Metatheory} \subsection*{} + \begin{lemma}[COK] + If $\J$, then $\ContextWellFormed$% in a subderivation (where $\J \neq \ContextWellFormed$) + \end{lemma} + \begin{proof} + By simultaneous induction on derivations. \\ + No interesting cases. + \end{proof} \begin{lemma}[Exchange] \vphantom{.}\\ If $\J[\HoleCtx\TypVarCtx[][1],\typvar[L1]\KndAssump\knd[L1],\typvar[L2]\KndAssump\knd[L2],\TypVarCtx[][2]]$ and $\ContextWellFormed[\HoleCtx\TypVarCtx[][1],\typvar[L2]\KndAssump\knd[L2],\typvar[L1]\KndAssump\knd[L1],\TypVarCtx[][2]]$, then $\J[\HoleCtx\TypVarCtx[][1],\typvar[L2]\KndAssump\knd[L2],\typvar[L1]\KndAssump\knd[L1],\TypVarCtx[][2]]$ @@ -248,29 +253,26 @@ \section{Metatheory} \end{proof} \begin{lemma}[Weakening] \vphantom{.}\\ - If $\J$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][L][\knd[L]]]$, then $\J[\HoleCtx\ExtendTypVarCtx[][L][\knd[L]]]$ + If $\J$ and $\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$, then $\J[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ %If $\J$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][L][\knd[L]]\TypVarCtx[]]$, then $\J[\ExtendTypHoleCtx[;][L][\knd[L]]\TypVarCtx[]]$ \end{lemma} \begin{lemma}[OK-PK] - If $\PrincipalKind{\typ}$, then $\ContextWellFormed$ and $\KindWellFormed$ + If $\PrincipalKind{\typ}$, then $\KindWellFormed$ \end{lemma} \begin{lemma}[OK-WFaK] - If $\WellFormedAtKind{\typ}$, then $\ContextWellFormed$ and $\KindWellFormed$ + If $\WellFormedAtKind{\typ}$, then $\KindWellFormed$ \end{lemma} \begin{lemma}[OK-MatchPi] - If $\MatchedPiKind{\knd}$, then $\ContextWellFormed$ and $\KindWellFormed$ and $\KindWellFormed[\DepFunKind]$ + If $\MatchedPiKind{\knd}$, then $\KindWellFormed$ and $\KindWellFormed[\DepFunKind]$ \end{lemma} \begin{lemma}[OK-KEquiv] - If $\KindEquiv{\knd[1]}{\knd[2]}$, then $\ContextWellFormed$ and $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ + If $\KindEquiv{\knd[1]}{\knd[2]}$, then $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ \end{lemma} \begin{lemma}[OK-CSK] - If $\ConsistentSubKind{\knd[1]}{\knd[2]}$, then $\ContextWellFormed$ and $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ + If $\ConsistentSubKind{\knd[1]}{\knd[2]}$, then $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ \end{lemma} \begin{lemma}[OK-EquivAK] - If $\TypeEquivAtKind{\typ[1]}{\typ[2]}$, then $\ContextWellFormed$ and $\WellFormedAtKind{\typ[1]}$ and $\WellFormedAtKind{\typ[2]}$ and $\KindWellFormed$ - \end{lemma} - \begin{lemma}[OK-KWF] - If $\KindWellFormed$, then $\ContextWellFormed$ + If $\TypeEquivAtKind{\typ[1]}{\typ[2]}$, then $\WellFormedAtKind{\typ[1]}$ and $\WellFormedAtKind{\typ[2]}$ and $\KindWellFormed$ \end{lemma} \begin{lemma}[OK-Substitution] \vphantom{.}\\ @@ -292,10 +294,12 @@ \section{Metatheory} \begin{tabular*}{\textwidth}{lc@{\extracolsep{\fill}}ll} Weakening. &PK-$\lambda$& $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][L][\knd[2]]]$ & by IH \\ - & & $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[1]]]$ & by OK-PK on premiss \\ - & & $\KindWellFormed[\knd[1]]$ & by CWF-TypVar \\ - & & $\KindWellFormed[\knd[1]][\HoleCtx\ExtendTypVarCtx[][L][\knd[L]]]$ & by Weakening \\ - & & $\PrincipalKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ}[\knd[2]]$ & by premiss \\ + & & $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[1]]]$ & by COK on premiss \\ + && $\KindWellFormed[\knd[1]]$ & by premiss \\ + && $\KindWellFormed[\knd[1]][\HoleCtx\ExtendTypVarCtx[][L][\knd[L]]]$ & by Weakening on premiss \\ + && $\ContextWellFormed[\HoleCtx\TypVarCtx[],\typvar[L]\KndAssump\knd[L],\typvar\KndAssump\knd[1]]$ & \\ + && $\PrincipalKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1],\typvar[L]\KndAssump\knd[L]]{\typ}[\knd[2]]$ & \\ + && $\PrincipalKind[\HoleCtx\TypVarCtx[],\typvar[L]\KndAssump\knd[L],\typvar\KndAssump\knd[1]]{\typ}[\knd[2]]$ & \\ OK-PK. &PK-Base& $\WellFormedAtKind{\bse}[\SKind[\Type][\bse]]$ & by (9) \\ & & $\WellFormedAtKind{\bse}[\Type]$ & by (10) \\ diff --git a/11-type-constructors/latex/hejohns-hazel.sty b/11-type-constructors/latex/hejohns-hazel.sty index 6824e28..7799956 100644 --- a/11-type-constructors/latex/hejohns-hazel.sty +++ b/11-type-constructors/latex/hejohns-hazel.sty @@ -130,15 +130,20 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % misc \newcommand*{\ExpVarCtx}[1][;]{\Gamma#1} - \NewDocumentCommand{\TypVarCtx}{O{;}O{}}{\Phi#1_{#2}} + \NewDocumentCommand{\TypVarCtx}{O{;}O{}}{\Phi_{#2}#1} + \NewDocumentCommand{\xTypVarCtx}{O{;}O{}}{\underline{\Phi_{#2}}#1} \NewDocumentCommand{\InTypVarCtx}{O{;}O{}O{\knd}}{\TypVarCtx[][1],\typvar[#2]\KndAssump{#3},\TypVarCtx[][2]#1} \NewDocumentCommand{\ExtendTypVarCtx}{O{;}oO{\knd}}{\TypVarCtx[],\typvar[#2]\KndAssump{#3}#1} + \NewDocumentCommand{\xExtendTypVarCtx}{O{;}oO{\knd}}{\xTypVarCtx[],\typvar[#2]\KndAssump{#3}#1} \newcommand*{\ModVarCtx}[1][;]{\Xi#1} \newcommand*{\SigVarCtx}[1][;]{\Psi#1} \NewDocumentCommand{\HoleCtx}{O{;}O{}}{\Delta_{#2}#1} + \NewDocumentCommand{\xHoleCtx}{O{;}O{}}{\underline{\Delta_{#2}}#1} \NewDocumentCommand{\InTypHoleCtx}{O{;}O{}O{\knd}}{\HoleCtx[][1],\hole[#2]\KndAssump{#3},\HoleCtx[][2]#1} \NewDocumentCommand{\ExtendTypHoleCtx}{O{;}oO{\knd}}{\HoleCtx[],\hole[#2]\KndAssump{#3}#1} + \NewDocumentCommand{\xExtendTypHoleCtx}{O{;}oO{\knd}}{\xHoleCtx[],\hole[#2]\KndAssump{#3}#1} \newcommand*{\dom}[1]{\mathsf{dom(}#1\mathsf{)}} + \renewcommand*{\dom}[1]{#1} \newcommand*{\val}[1]{\mathsf{val(}#1\mathsf{)}} \newcommand*{\type}[2]{\mathsf{type(}#1, #2\mathsf{)}} \newcommand*{\submodule}[1]{\mathsf{submodule(}#1\mathsf{)}} From 92f4950aa719237b47d96afef420a1ca0e9edfe0 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Thu, 29 Jul 2021 18:50:55 +0000 Subject: [PATCH 060/181] fix context marking --- .../latex/11-type-constructors.pdf | Bin 244549 -> 244518 bytes .../latex/11-type-constructors.tex | 6 +++--- 11-type-constructors/latex/hejohns-hazel.sty | 4 ++-- 3 files changed, 5 insertions(+), 5 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index 65319884c8a90b39b2ec206f5d9db55a2f155db6..50723337c4e4daf3ab6ab46a82236ff35a277178 100644 GIT binary patch delta 16553 zcmaf(Lv$`ou&!fc$F^x~lr=clZUN z#iV!8MF!XA~jR4a82&M*%q^avB|eRmL*3w%^ora45k?zpb4>y(sAme>MM7n+deZQYiN=u{&kPT2D zzej}T98B&X;vAm4IC>L#aKB%^09-bjQMYS$z285bGplU@Xn_{4mD7hyC(WdPs~3}^ 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z`9LrJ@tO9=#v@(F$K3nfi}D8y5U(NWBg?#ZL%3R*JN~Eo4`lE`g;3BQEJh@+2`*5} z*2fCKC-mrsp$9PH1AqX0kBgdcI0C>2g*;M;CN5CH$r}FHLI1at1{jI*^Ycl|NP$4o z0s_1OFd0cnUOuR#BtI`73 zw>^491I1zJX~^AW5>=;L#O7{qV)wz)+_Jq5#vIcDV-z|k!7FAqQQ{yj+Ar>Gerus~ z;^!#tz0uzVB{09XAU7}WtbB!nl|BJHYo~+75t{N$l`TS zs4K+5Eaqi{j_w_;itYNTn}^HaGm{~n-IVBu>&di)r0+0_FN}d-FF{@@ydRJ{AjQbI` diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index dd740f5..cbea534 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -247,10 +247,14 @@ \section{Metatheory} If $\J[\HoleCtx\TypVarCtx[][1],\typvar[L1]\KndAssump\knd[L1],\typvar[L2]\KndAssump\knd[L2],\TypVarCtx[][2]]$ and $\ContextWellFormed[\HoleCtx\TypVarCtx[][1],\typvar[L2]\KndAssump\knd[L2],\typvar[L1]\KndAssump\knd[L1],\TypVarCtx[][2]]$, then $\J[\HoleCtx\TypVarCtx[][1],\typvar[L2]\KndAssump\knd[L2],\typvar[L1]\KndAssump\knd[L1],\TypVarCtx[][2]]$ \end{lemma} \begin{proof} - By simultaneous induction on derivations. \\ + By induction on derivations. \\ No interesting cases. \\ (Only rules with $\TypVarCtx$ extended in the consequent are interesting, which is only CWF-TypVar, but when $\mathcal{J}$ is CWF, Exchange is identity) \end{proof} + \begin{corollary}[Marked-Exchange] + \vphantom{.}\\ + If $\J[\HoleCtx\xxExtendTypVarCtx[][L1]{\knd[L1]}[L2]{\knd[L2]}]$ and $\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L2]{\knd[L2]}[L1]{\knd[L1]}]$, then $\J[\HoleCtx\xxExtendTypVarCtx[][L2]{\knd[L2]}[L1]{\knd[L1]}]$ + \end{corollary} \begin{lemma}[Weakening] \vphantom{.}\\ If $\J$ and $\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$, then $\J[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ @@ -291,15 +295,30 @@ \section{Metatheory} \begin{proof} By simultaneous induction on derivations. \\ The interesting cases per lemma: \\ + \subsubsection*{Weakening} + \begin{mathpar} + \infer[PK-$\lambda$] + {\infer[Marked-Exchange] + {\infer[Weakening] + { } + {\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]} + \\\infer[CWF-TypVar] + { } + {\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]}} + {\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]{\typ}[\knd[2]]}} + {\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} + \end{mathpar} \begin{tabular*}{\textwidth}{lc@{\extracolsep{\fill}}ll} Weakening. - &PK-$\lambda$& $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][L][\knd[2]]]$ & by IH \\ - & & $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[1]]]$ & by COK on premiss \\ - && $\KindWellFormed[\knd[1]]$ & by premiss \\ - && $\KindWellFormed[\knd[1]][\HoleCtx\ExtendTypVarCtx[][L][\knd[L]]]$ & by Weakening on premiss \\ - && $\ContextWellFormed[\HoleCtx\TypVarCtx[],\typvar[L]\KndAssump\knd[L],\typvar\KndAssump\knd[1]]$ & \\ - && $\PrincipalKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1],\typvar[L]\KndAssump\knd[L]]{\typ}[\knd[2]]$ & \\ - && $\PrincipalKind[\HoleCtx\TypVarCtx[],\typvar[L]\KndAssump\knd[L],\typvar\KndAssump\knd[1]]{\typ}[\knd[2]]$ & \\ + %&PK-$\lambda$& $\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]$ & by Weakening on premiss \\ + && $\KindWellFormed[\knd[1]][\HoleCtx\TypVarCtx[]]$ & by subderivation premiss \\ + && $\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ & by IH \\ + && $\KindWellFormed[\knd[1]][\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ & by Weakening on subderivation premiss \\ + && $\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]$ & by CWF-TypVar \\ + && $\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]$ & by ? \\ + && $\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]$ & by Weakening on premiss \\ + && $\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]{\typ}[\knd[2]]$ & by Marked-Exchange \\ + && $\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]$ & by PK-$\lambda$ \\ OK-PK. &PK-Base& $\WellFormedAtKind{\bse}[\SKind[\Type][\bse]]$ & by (9) \\ & & $\WellFormedAtKind{\bse}[\Type]$ & by (10) \\ diff --git a/11-type-constructors/latex/hejohns-hazel.sty b/11-type-constructors/latex/hejohns-hazel.sty index 4623322..a7e6f8b 100644 --- a/11-type-constructors/latex/hejohns-hazel.sty +++ b/11-type-constructors/latex/hejohns-hazel.sty @@ -10,6 +10,7 @@ \RequirePackage{thmtools} \declaretheorem{theorem} \declaretheorem[sibling=theorem]{lemma} + \declaretheorem[sibling=theorem]{corollary} \declaretheorem{definition} \RequirePackage{tikz} % https://tex.stackexchange.com/questions/12678/squiggly-arrows-in-tikz @@ -133,15 +134,17 @@ \NewDocumentCommand{\TypVarCtx}{O{;}O{}}{\Phi_{#2}#1} \NewDocumentCommand{\xTypVarCtx}{O{;}O{}}{\underline{\Phi_{#2}}#1} \NewDocumentCommand{\InTypVarCtx}{O{;}O{}O{\knd}}{\TypVarCtx[][1],\typvar[#2]\KndAssump{#3},\TypVarCtx[][2]#1} - \NewDocumentCommand{\ExtendTypVarCtx}{O{;}oO{\knd}}{\TypVarCtx[],\typvar[#2]\KndAssump{#3}#1} - \NewDocumentCommand{\xExtendTypVarCtx}{O{;}oO{\knd}}{\underline{\TypVarCtx[],\typvar[#2]\KndAssump{#3}}#1} + \NewDocumentCommand{\ExtendTypVarCtx}{O{;}O{}O{\knd}}{\TypVarCtx[],\typvar[#2]\KndAssump{#3}#1} + \NewDocumentCommand{\xExtendTypVarCtx}{O{;}O{}O{\knd}}{\underline{\TypVarCtx[],\typvar[#2]\KndAssump{#3}}#1} + \NewDocumentCommand{\xxExtendTypVarCtx}{O{;}O{}mO{}m}{\underline{\underline{\TypVarCtx[],\typvar[#2]\KndAssump{#3}},\typvar[#4]\KndAssump{#5}}#1} \newcommand*{\ModVarCtx}[1][;]{\Xi#1} \newcommand*{\SigVarCtx}[1][;]{\Psi#1} \NewDocumentCommand{\HoleCtx}{O{;}O{}}{\Delta_{#2}#1} \NewDocumentCommand{\xHoleCtx}{O{;}O{}}{\underline{\Delta_{#2}}#1} \NewDocumentCommand{\InTypHoleCtx}{O{;}O{}O{\knd}}{\HoleCtx[][1],\hole[#2]\KndAssump{#3},\HoleCtx[][2]#1} - \NewDocumentCommand{\ExtendTypHoleCtx}{O{;}oO{\knd}}{\HoleCtx[],\hole[#2]\KndAssump{#3}#1} - \NewDocumentCommand{\xExtendTypHoleCtx}{O{;}oO{\knd}}{\underline{\HoleCtx[],\hole[#2]\KndAssump{#3}}#1} + \NewDocumentCommand{\ExtendTypHoleCtx}{O{;}O{}O{\knd}}{\HoleCtx[],\hole[#2]\KndAssump{#3}#1} + \NewDocumentCommand{\xExtendTypHoleCtx}{O{;}O{}O{\knd}}{\underline{\HoleCtx[],\hole[#2]\KndAssump{#3}}#1} + \NewDocumentCommand{\xxExtendTypHoleCtx}{O{;}O{}mO{}m}{\underline{\underline{\TypHoleCtx[],\hole[#2]\KndAssump{#3}},\hole[#4]\KndAssump{#5}}#1} \newcommand*{\dom}[1]{\mathsf{dom(}#1\mathsf{)}} \renewcommand*{\dom}[1]{#1} \newcommand*{\val}[1]{\mathsf{val(}#1\mathsf{)}} @@ -164,6 +167,7 @@ \newcommand*{\inferruledotfrac}[2]{ \hbox{$\dotfrac{#1}{#2}$} } + \RenewDocumentCommand{\infer}{o}{\inferrule*[right=#1]} \newcommand{\hintpagebreak}{\pagebreak[3]} % 0-4; 2 too weak, 4 too strong 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\newcommand*{\vt}{~\vert~} % automatically label rules @@ -295,12 +296,23 @@ \section{Metatheory} \begin{proof} By simultaneous induction on derivations. \\ The interesting cases per lemma: \\ + \begin{landscape} \subsubsection*{Weakening} \begin{mathpar} \infer[PK-$\lambda$] {\infer[Marked-Exchange] {\infer[Weakening] - { } + {\infer[premiss] + {d} + {\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]} + \\\infer[CWF-TypVar] + {\infer[Weakening] + {a} + {\KindWellFormed[\knd[2]]} + \\\infer[] + {b} + {c}} + {\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]}} {\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]} \\\infer[CWF-TypVar] { } @@ -308,6 +320,7 @@ \section{Metatheory} {\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]{\typ}[\knd[2]]}} {\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} \end{mathpar} + \end{landscape} \begin{tabular*}{\textwidth}{lc@{\extracolsep{\fill}}ll} Weakening. %&PK-$\lambda$& $\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]$ & by Weakening on premiss \\ diff --git a/11-type-constructors/latex/hejohns-hazel.sty b/11-type-constructors/latex/hejohns-hazel.sty index a7e6f8b..39ba637 100644 --- a/11-type-constructors/latex/hejohns-hazel.sty +++ b/11-type-constructors/latex/hejohns-hazel.sty @@ -167,7 +167,7 @@ \newcommand*{\inferruledotfrac}[2]{ \hbox{$\dotfrac{#1}{#2}$} } - \RenewDocumentCommand{\infer}{o}{\inferrule*[right=#1]} + \RenewDocumentCommand{\infer}{oO{}}{\inferrule*[right=#1#2]} \newcommand{\hintpagebreak}{\pagebreak[3]} % 0-4; 2 too weak, 4 too strong %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % judgments From 86dd78e5db339bec75e5f9bfca53eba41c5725b4 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Thu, 29 Jul 2021 23:54:54 +0000 Subject: [PATCH 063/181] :( --- .../latex/11-type-constructors.pdf | Bin 250667 -> 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standard header \documentclass[11pt]{article} - \usepackage[margin=1.5cm]{geometry} + \usepackage[a2paper,margin=2cm]{geometry} \usepackage[utf8x]{inputenc} % for utf8 \usepackage{titlesec} % see below \usepackage{xcolor} % \color \textcolor @@ -303,15 +303,24 @@ \section{Metatheory} {\infer[Marked-Exchange] {\infer[Weakening] {\infer[premiss] - {d} + { } {\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]} \\\infer[CWF-TypVar] {\infer[Weakening] - {a} - {\KindWellFormed[\knd[2]]} + {\infer[premiss of subderivation] + {\infer[IH] + { } + {\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}} + {\KindWellFormed[\knd[L]]} + \\\infer[COK] + {\infer[premiss] + { } + {\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}} + {\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}} + {\KindWellFormed[\knd[L]][\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]} \\\infer[] - {b} - {c}} + { } + {\typvar[L]\notin\xExtendTypVarCtx[][][\knd[1]]}} {\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]}} {\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]} \\\infer[CWF-TypVar] From 66ec9c93b5df0c967183a3d12c1e075c48063a38 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Fri, 30 Jul 2021 01:16:17 +0000 Subject: [PATCH 064/181] :-() --- .../latex/11-type-constructors.pdf | Bin 248450 -> 249114 bytes .../latex/11-type-constructors.tex | 62 ++++++++++++++++-- 2 files changed, 57 insertions(+), 5 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index 36ca772e3cd063eab3c62f7af37d48287eb502c7..ac3a346a3c99d6db69fa7deda386af34104dd6d8 100644 GIT binary patch delta 60342 zcmYhiQ*fYN&@CL>wrx8TOl;e>ttXrq6Wg|J+s zUaLDe1vxVlIe`M4hnqW1qz{}1xYXD7#2-ruc&_X3{%si}a~bV=!P_vVVCRD`8%azs z2TLJ>Wg4X>#nO7bHrKPF)#+1FYcOvW=+z)ysa#&$Ipyie_iip>348@|eS9yS5P$5! zi&##h%qC|G0)p(p=mpa;u0xTD4o@E|7iH0X3}24kuDsaEhJN42Z$(o;NAX*5!|Pe% 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typearea + \usepackage[paper=A4,pagesize,DIV=18]{typearea} \usepackage[utf8x]{inputenc} % for utf8 \usepackage{titlesec} % see below \usepackage{xcolor} % \color \textcolor @@ -48,7 +49,6 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % packages \usepackage{hejohns-hazel} - \usepackage{pdflscape} % https://tex.stackexchange.com/questions/150394/how-to-switch-from-portrait-to-landscape-page-orientation?noredirect=1&lq=1 % commands \newcommand*{\vt}{~\vert~} % automatically label rules @@ -67,6 +67,7 @@ \pagenumbering{gobble} \nonfrenchspacing %\frenchspacing % when monospaced +\recalctypearea \begin{document} \title{Hazel Phi: 11-type-constructors} \author{} @@ -265,6 +266,9 @@ \section{Metatheory} \begin{proof} Exchange when $\TypVarCtx[][2]=\cdot$ \end{proof} + \newpage + \KOMAoptions{paper=A1,paper=landscape} + \recalctypearea \begin{lemma}[Weakening] \vphantom{.}\\ If $\J$ and $\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ and $\typvar[L]\notin\mathcal{J}$ and $\forall\typvar\in\knd[L],\typvar\notin\mathcal{J}$, then $\J[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ @@ -274,7 +278,6 @@ \section{Metatheory} By induction on derivations. \\ When applying Weakening in the induction, check that the left premiss is always a subderivation, and check variable exclusion conditions are satisfied (usually checked elsewhere in the derivation). \\ - \begin{landscape} \subsubsection*{Weakening} \begin{mathpar} \rinfer[PK-$\lambda$]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]}{ @@ -430,8 +433,10 @@ \section{Metatheory} \rinfer[KWF-$\Pi$]{sim}{ } \end{mathpar} - \end{landscape} \end{proof} + \newpage + \KOMAoptions{paper=A4,paper=portrait} + \recalctypearea \begin{lemma}[OK-PK] If $\PrincipalKind{\typ}$, then $\KindWellFormed$ \end{lemma} @@ -467,8 +472,7 @@ \section{Metatheory} \begin{proof} By simultaneous induction on derivations. \\ The interesting cases per lemma: \\ - (PoS = premiss of subderivation) - \begin{tabular*}{.5\textwidth}{lc@{\extracolsep{\fill}}ll} + \begin{tabular*}{.6\textwidth}{lc@{\extracolsep{\fill}}ll} OK-PK. &PK-Base& $\WellFormedAtKind{\bse}[\SKind[\Type][\bse]]$ & by (9) \\ & & $\WellFormedAtKind{\bse}[\Type]$ & by (10) \\ From 5277d18b741347c6c41f29c827c650d9a89637ee Mon Sep 17 00:00:00 2001 From: Johnson He Date: Sat, 31 Jul 2021 03:05:13 +0000 Subject: [PATCH 073/181] I think we need induction on type size for K-subst --- .../latex/11-type-constructors.pdf | Bin 255066 -> 259366 bytes .../latex/11-type-constructors.tex | 106 ++++++++++++------ 2 files changed, 69 insertions(+), 37 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index bba303312cf61936b0b76c85d9b51578a76ec9e8..c36c748c2301f13817b8bb89df5bf7bac3f7e5c2 100644 GIT binary patch delta 67262 zcmZU4Q*fXSux#v&ZQHgswr$(i7u(#}*x26K=EnBMc5?r7t8U$gGfy+mQ(ZIDJ-t4L 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z?#MuDm`q$7v%LQ+lr?rPvcj;swQlZvrk8=66iaYN(AzB}F(N1XCj*j-N=ii<;XeRC C@Htoj diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index 76098e6..6ef9fa9 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -266,19 +266,54 @@ \section{Metatheory} \begin{proof} Exchange when $\TypVarCtx[][2]=\cdot$ \end{proof} - \newpage - \KOMAoptions{paper=A1,paper=landscape} - \recalctypearea \begin{lemma}[Weakening] \vphantom{.}\\ If $\J$ and $\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ and $\typvar[L]\notin\mathcal{J}$ and $\forall\typvar\in\knd[L],\typvar\notin\mathcal{J}$, then $\J[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ %If $\J$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][L][\knd[L]]\TypVarCtx[]]$, then $\J[\ExtendTypHoleCtx[;][L][\knd[L]]\TypVarCtx[]]$ \end{lemma} \begin{proof} + see addendum + \end{proof} + \begin{lemma}[K-Substitution] + \vphantom{.}\\ + If $\WellFormedAtKind{\typ[L1]}[\knd[L1]]$ and $\WellFormedAtKind[\HoleCtx\xExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$, then $\WellFormedAtKind{\subst{\typ[L1]}{\typvar[L]}{\typ[L2]}}[\subst{\typ[L1]}{\typvar[L]}{\knd[L2]}]$ \\ + (induction on $\WellFormedAtKind[\HoleCtx\xExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$) + \end{lemma} + \begin{lemma}[OK-Substitution] + \vphantom{.}\\ + If $\WellFormedAtKind{\typ[L]}[\knd[L1]]$ and $\KindWellFormed[{\knd[L2]}][\HoleCtx\xExtendTypVarCtx[][L][{\knd[L1]}]]$, then $\KindWellFormed[\subst{\typ[L]}{\typvar[L]}{\knd[L2]}]$ \\ + (induction on $\KindWellFormed[{\knd[L2]}][\HoleCtx\xExtendTypVarCtx[][L][{\knd[L1]}]]$) + \end{lemma} + \begin{lemma}[OK-PK] + If $\PrincipalKind{\typ}$, then $\KindWellFormed$ + \end{lemma} + \begin{lemma}[OK-WFaK] + If $\WellFormedAtKind{\typ}$, then $\KindWellFormed$ + \end{lemma} + \begin{lemma}[OK-MatchPi] + If $\MatchedPiKind{\knd}$, then $\KindWellFormed$ and $\KindWellFormed[\DepFunKind]$ + \end{lemma} + \begin{lemma}[OK-KEquiv] + If $\KindEquiv{\knd[1]}{\knd[2]}$, then $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ + \end{lemma} + \begin{lemma}[OK-CSK] + If $\ConsistentSubKind{\knd[1]}{\knd[2]}$, then $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ + \end{lemma} + \begin{lemma}[OK-EquivAK] + If $\TypeEquivAtKind{\typ[1]}{\typ[2]}$, then $\WellFormedAtKind{\typ[1]}$ and $\WellFormedAtKind{\typ[2]}$ and $\KindWellFormed$ + \end{lemma} + \begin{proof} + see addendum + \end{proof} + \newpage + \KOMAoptions{paper=A1,paper=landscape} + \recalctypearea + \begin{proof} + \vphantom{.} \\ + \subsubsection*{Weakening} By induction on derivations. \\ - When applying Weakening in the induction, check that the left premiss is always a subderivation, + \par Note: When applying Weakening in the induction, check that the left premiss is always a subderivation, and check variable exclusion conditions are satisfied (usually checked elsewhere in the derivation). \\ - \subsubsection*{Weakening} \begin{mathpar} \rinfer[PK-$\lambda$]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]}{ \rinfer[Marked-Exchange]{\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]{\typ}[\knd[2]]}{ @@ -433,42 +468,37 @@ \section{Metatheory} \rinfer[KWF-$\Pi$]{sim}{ } \end{mathpar} + \subsubsection*{O?K-.*} + By simultaneous induction on derivations. \\ + \par The interesting cases per lemma: \\ + \subparagraph{K-Substitution} + by type size?? + \begin{mathpar} + \end{mathpar} + \subparagraph{OK-Substitution} + \begin{mathpar} + \rinfer[]{}{ + } + \end{mathpar} + \subparagraph{OK-PK} + \begin{mathpar} + \rinfer[KWF-SKind]{\KindWellFormed[\SKind[\Type][\bse]]}{ + \rinfer[WFaK-1]{\WellFormedAtKind{\bse}[\Type]}{ + \rinfer[premiss]{\PrincipalKind{\bse}[\SKind[\Type][\bse]]}{ + } + } + } + \and + \rinfer[OK-Substitution]{\KindWellFormed[\subst{\typ[2]}{\typvar}{\knd[2]}]}{ + \rinfer[]{}{ + } + } + \end{mathpar} + \subparagraph{OK-WFaK} \end{proof} \newpage \KOMAoptions{paper=A4,paper=portrait} \recalctypearea - \begin{lemma}[OK-PK] - If $\PrincipalKind{\typ}$, then $\KindWellFormed$ - \end{lemma} - \begin{lemma}[OK-WFaK] - If $\WellFormedAtKind{\typ}$, then $\KindWellFormed$ - \end{lemma} - \begin{lemma}[OK-MatchPi] - If $\MatchedPiKind{\knd}$, then $\KindWellFormed$ and $\KindWellFormed[\DepFunKind]$ - \end{lemma} - \begin{lemma}[OK-KEquiv] - If $\KindEquiv{\knd[1]}{\knd[2]}$, then $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ - \end{lemma} - \begin{lemma}[OK-CSK] - If $\ConsistentSubKind{\knd[1]}{\knd[2]}$, then $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ - \end{lemma} - \begin{lemma}[OK-EquivAK] - If $\TypeEquivAtKind{\typ[1]}{\typ[2]}$, then $\WellFormedAtKind{\typ[1]}$ and $\WellFormedAtKind{\typ[2]}$ and $\KindWellFormed$ - \end{lemma} - \begin{lemma}[OK-Substitution] - \vphantom{.}\\ - If $\WellFormedAtKind{\typ[L]}[\knd[L1]]$ and $\KindWellFormed[{\knd[L2]}][\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]$, then $\KindWellFormed[\subst{\typ[L]}{\typvar[L]}{\knd[L2]}]$ \\ - (induction on $\KindWellFormed[{\knd[L2]}][\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]$) - \end{lemma} - %\begin{lemma}[PK-Substitution] - % \vphantom{.}\\ If $\PrincipalKind{\typ[L1]}[\knd[L1]]$ and $\PrincipalKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$, then $\PrincipalKind{\subst{\typ[L1]}{\typvar[L]}{\typ[L2]}}[\subst{\typ[L1]}{\typvar[L]}{\knd[L2]}]$ \\ - % (induction on $\PrincipalKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$) - %\end{lemma} - \begin{lemma}[K-Substitution] - \vphantom{.}\\ - If $\WellFormedAtKind{\typ[L1]}[\knd[L1]]$ and $\WellFormedAtKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$, then $\WellFormedAtKind{\subst{\typ[L1]}{\typvar[L]}{\typ[L2]}}[\subst{\typ[L1]}{\typvar[L]}{\knd[L2]}]$ \\ - (induction on $\WellFormedAtKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$) - \end{lemma} \begin{proof} By simultaneous induction on derivations. \\ The interesting cases per lemma: \\ @@ -507,4 +537,6 @@ \section{Metatheory} \begin{lemma} If $\ConsistentSubKind{\knd[1]}{\SKind[\knd[2]]}$, then $\ConsistentSubKind{\knd[1]}{\knd[2]}$ \end{lemma} +\section*{Elaboration} + \subsection*{todo} \end{document} From 8be7879002c48b76a68a2060f9d7bcdbc488757d Mon Sep 17 00:00:00 2001 From: Johnson He Date: Thu, 12 Aug 2021 18:57:52 +0000 Subject: [PATCH 074/181] lemma -> theorem, OK-PK --- .../latex/11-type-constructors.pdf | Bin 259366 -> 257318 bytes .../latex/11-type-constructors.tex | 69 ++++++------------ 2 files changed, 22 insertions(+), 47 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf 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+279,32 @@ \section{Metatheory} If $\WellFormedAtKind{\typ[L1]}[\knd[L1]]$ and $\WellFormedAtKind[\HoleCtx\xExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$, then $\WellFormedAtKind{\subst{\typ[L1]}{\typvar[L]}{\typ[L2]}}[\subst{\typ[L1]}{\typvar[L]}{\knd[L2]}]$ \\ (induction on $\WellFormedAtKind[\HoleCtx\xExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$) \end{lemma} + \begin{lemma}[PK-Substitution] + If $\WellFormedAtKind{\typ[L1]}[\knd[L1]]$ and $\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][L][\knd[L1]]]{\typ[L2]}[\knd[L2]]$ and $\PrincipalKind{\subst{\typ[L1]}{\typvar[L]}{\typ[L2]}}[\knd[L3]]$, then $\KindEquiv{\subst{\typ[L2]}{\typvar[L]}{\knd[L2]}}{\knd[L3]}$ + \end{lemma} \begin{lemma}[OK-Substitution] \vphantom{.}\\ If $\WellFormedAtKind{\typ[L]}[\knd[L1]]$ and $\KindWellFormed[{\knd[L2]}][\HoleCtx\xExtendTypVarCtx[][L][{\knd[L1]}]]$, then $\KindWellFormed[\subst{\typ[L]}{\typvar[L]}{\knd[L2]}]$ \\ (induction on $\KindWellFormed[{\knd[L2]}][\HoleCtx\xExtendTypVarCtx[][L][{\knd[L1]}]]$) \end{lemma} - \begin{lemma}[OK-PK] + \begin{theorem}[OK-PK] If $\PrincipalKind{\typ}$, then $\KindWellFormed$ - \end{lemma} - \begin{lemma}[OK-WFaK] + \end{theorem} + \begin{theorem}[OK-WFaK] If $\WellFormedAtKind{\typ}$, then $\KindWellFormed$ - \end{lemma} - \begin{lemma}[OK-MatchPi] + \end{theorem} + \begin{theorem}[OK-MatchPi] If $\MatchedPiKind{\knd}$, then $\KindWellFormed$ and $\KindWellFormed[\DepFunKind]$ - \end{lemma} - \begin{lemma}[OK-KEquiv] + \end{theorem} + \begin{theorem}[OK-KEquiv] If $\KindEquiv{\knd[1]}{\knd[2]}$, then $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ - \end{lemma} - \begin{lemma}[OK-CSK] + \end{theorem} + \begin{theorem}[OK-CSK] If $\ConsistentSubKind{\knd[1]}{\knd[2]}$, then $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ - \end{lemma} - \begin{lemma}[OK-EquivAK] + \end{theorem} + \begin{theorem}[OK-EquivAK] If $\TypeEquivAtKind{\typ[1]}{\typ[2]}$, then $\WellFormedAtKind{\typ[1]}$ and $\WellFormedAtKind{\typ[2]}$ and $\KindWellFormed$ - \end{lemma} + \end{theorem} \begin{proof} see addendum \end{proof} @@ -499,44 +502,16 @@ \section{Metatheory} \newpage \KOMAoptions{paper=A4,paper=portrait} \recalctypearea - \begin{proof} - By simultaneous induction on derivations. \\ - The interesting cases per lemma: \\ - \begin{tabular*}{.6\textwidth}{lc@{\extracolsep{\fill}}ll} - OK-PK. - &PK-Base& $\WellFormedAtKind{\bse}[\SKind[\Type][\bse]]$ & by (9) \\ - & & $\WellFormedAtKind{\bse}[\Type]$ & by (10) \\ - &*& $\KindWellFormed[{\SKind[\Type][\bse]}]$ & by (43) \\ - &*& $\ContextWellFormed$ & by premiss \\ - &PK-Ap& & bad \\ - OK-WFaK. - &(12)& $\WellFormedAtKind{\typ[2]}$ & by (10) \\ - &*& $\KindWellFormed[{\SKind[\knd][\typ[2]]}]$ & by (43) \\ - OK-KEquiv. - &(22)& $\PrincipalKind{\TypeAp[\typ][\typvar]}$ & \\ - OK-Substitution. - &(41)& $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]$ & premiss (41) \\ - & & $\KindWellFormed[{\knd[L1]}]$ & by subderivation premiss (46) \\ - &*& $\ContextWellFormed$ & by OK-KWF \\ - &*& $\KindWellFormed[\subst{\typ[L]}{\typvar[L]}{\Type}]$ & by (41) and degenerate subst \\ - &(43)& $\WellFormedAtKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ}[\knd]$ & premiss (43) \\ - & & $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]$ & by OK-WFaK \\ - & & $\KindWellFormed[{\knd[L1]}]$ & by subderivation premiss (46) \\ - &*& $\ContextWellFormed$ & by OK-KWF \\ - & & $\WellFormedAtKind{\subst{\typ[L1]}{\typvar[L]}{\typ}}[\subst{\typ[L1]}{\typvar[L]}{\knd}]$ & by K-Substitution on premiss \\ - &*& $\KindWellFormed[\subst{\typ[L]}{\typvar[L]}{\SKind}]$ & by (43) \\ - \end{tabular*} - \end{proof} \vphantom{.} - \begin{lemma}[PK-Unicity] - If $\PrincipalKind{\typ[L]}[\knd[L1]]$ and $\PrincipalKind{\typ[L]}[\knd[L2]]$ then ${\knd[L1]}$ is ${\knd[L2]}$ - \end{lemma} - \begin{lemma} + \begin{theorem}[PK-Unicity] + If $\PrincipalKind{\typ[L]}[\knd[L1]]$ and $\PrincipalKind{\typ[L]}[\knd[L2]]$ then ${\knd[L1]} = {\knd[L2]}$ + \end{theorem} + \begin{theorem} If $\PrincipalKind{\typ}[\knd[1]]$ and $\WellFormedAtKind{\typ}[\knd[2]]$, then $\ConsistentSubKind{\knd[1]}{\knd[2]}$ - \end{lemma} - \begin{lemma} + \end{theorem} + \begin{theorem} If $\ConsistentSubKind{\knd[1]}{\SKind[\knd[2]]}$, then $\ConsistentSubKind{\knd[1]}{\knd[2]}$ - \end{lemma} + \end{theorem} \section*{Elaboration} \subsection*{todo} \end{document} From 3a0627808448c110506d39d26e9552cc4555f471 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Thu, 12 Aug 2021 19:47:57 +0000 Subject: [PATCH 075/181] must've changed DIV on accident --- 11-type-constructors/11-type-constructors.md | 5 +++++ 11-type-constructors/11-type-constructors.tex | 5 ++++- 11-type-constructors/README.md | 5 +++++ .../latex/11-type-constructors.pdf | Bin 257318 -> 257314 bytes .../latex/11-type-constructors.tex | 2 +- 5 files changed, 15 insertions(+), 2 deletions(-) diff --git a/11-type-constructors/11-type-constructors.md b/11-type-constructors/11-type-constructors.md index 95111e2..da0b718 100644 --- a/11-type-constructors/11-type-constructors.md +++ b/11-type-constructors/11-type-constructors.md @@ -6,6 +6,11 @@ Introduction something +TODO +---- + +- clean derivations w/ abbreviations + Notes ----- diff --git a/11-type-constructors/11-type-constructors.tex b/11-type-constructors/11-type-constructors.tex index 0406db2..be3f7ce 100644 --- a/11-type-constructors/11-type-constructors.tex +++ b/11-type-constructors/11-type-constructors.tex @@ -7,6 +7,10 @@ \section*{11-type-constructors} \today \subsection*{Introduction} something +\subsection*{TODO} +\begin{itemize} + \item clean derivations w/ abbreviations +\end{itemize} \subsection*{Notes} \begin{itemize} \item test @@ -15,5 +19,4 @@ \subsection*{Notes} \begin{verbatim} module M = {} in \end{verbatim} - \end{document} diff --git a/11-type-constructors/README.md b/11-type-constructors/README.md index 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a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -309,7 +309,7 @@ \section{Metatheory} see addendum \end{proof} \newpage - \KOMAoptions{paper=A1,paper=landscape,DIV=40} + \KOMAoptions{paper=A0,paper=landscape,DIV=40} \recalctypearea \begin{proof} \vphantom{.} \\ From 0fed84e5a95849dbf50780bbf051f797bf94ad86 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Tue, 17 Aug 2021 03:05:55 +0000 Subject: [PATCH 077/181] sending it out to cyrus- --- 11-type-constructors/11-type-constructors.md | 18 +++++++------ 11-type-constructors/11-type-constructors.tex | 14 +++++----- 11-type-constructors/README.md | 18 +++++++------ 11-type-constructors/SURE.pdf | Bin 0 -> 1087273 bytes .../latex/11-type-constructors.pdf | Bin 257340 -> 261855 bytes .../latex/11-type-constructors.tex | 25 ++++++++++++++++-- 6 files changed, 50 insertions(+), 25 deletions(-) create mode 100644 11-type-constructors/SURE.pdf diff --git a/11-type-constructors/11-type-constructors.md b/11-type-constructors/11-type-constructors.md index da0b718..ab0e455 100644 --- a/11-type-constructors/11-type-constructors.md +++ b/11-type-constructors/11-type-constructors.md @@ -4,20 +4,22 @@ Introduction ------------ -something +see my SURE poster for a motivating example. +(fill in) a slightly misleading title, the “intended" application is +parameterized types TODO ---- -- clean derivations w/ abbreviations +- clean derivations w/ abbreviations (looking at you A0 page) -Notes ------ +- PK unicity doesn’t actually hold rn (see red markup) -- test +- Sigma kinds -- test +- some other stuff in my notebook that’s in a box - +Notes +----- - module M = {} in +- elaboration rules should be straightforward from bkase’s phi diff --git a/11-type-constructors/11-type-constructors.tex b/11-type-constructors/11-type-constructors.tex index be3f7ce..a47297c 100644 --- a/11-type-constructors/11-type-constructors.tex +++ b/11-type-constructors/11-type-constructors.tex @@ -6,17 +6,17 @@ \section*{11-type-constructors} \today \subsection*{Introduction} -something +see my SURE poster for a motivating example. \\ +(fill in) a slightly misleading title, the ``intended" application is parameterized types \subsection*{TODO} \begin{itemize} - \item clean derivations w/ abbreviations + \item clean derivations w/ abbreviations (looking at you A0 page) + \item PK unicity doesn't actually hold rn (see red markup) + \item Sigma kinds + \item some other stuff in my notebook that's in a box \end{itemize} \subsection*{Notes} \begin{itemize} - \item test - \item test + \item elaboration rules should be straightforward from bkase's phi \end{itemize} -\begin{verbatim} - module M = {} in -\end{verbatim} \end{document} diff --git a/11-type-constructors/README.md b/11-type-constructors/README.md index da0b718..ab0e455 100644 --- a/11-type-constructors/README.md +++ b/11-type-constructors/README.md @@ -4,20 +4,22 @@ 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$\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$, then $\J[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ + If $\J$ and $\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ and $\typvar[L]\notin\mathcal{J}$ and $\forall\typvar\in\knd[L],\typvar\notin\mathcal{J}$, then $\J[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ %If $\J$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][L][\knd[L]]\TypVarCtx[]]$, then $\J[\ExtendTypHoleCtx[;][L][\knd[L]]\TypVarCtx[]]$ \end{lemma} \begin{lemma}[OK-PK] @@ -296,6 +296,7 @@ \section{Metatheory} \begin{proof} By simultaneous induction on derivations. \\ The interesting cases per lemma: \\ + (PoS = premiss of subderivation) \begin{landscape} \subsubsection*{Weakening} \begin{mathpar} @@ -307,7 +308,7 @@ \section{Metatheory} {\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]} \\\infer[CWF-TypVar] {\infer[Weakening] - {\infer[premiss of subderivation] + {\infer[PoS] {\infer[IH] { } {\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}} @@ -319,12 +320,63 @@ \section{Metatheory} {\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}} {\KindWellFormed[\knd[L]][\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]} \\\infer[] - { } + {\infer[PoS] + {\infer[IH] + { } + {\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}} + {\typvar[L]\notin\TypVarCtx} + \\\infer[] + {\infer[IH] + { } + {\typvar[L]\notin\mathcal{J}} + \\\infer[] + { } + {\typvar\in\mathcal{J}}} + {\typvar[L]\neq\typvar} + \\\infer[] + {\infer[IH] + { } + {\typvar[L]\notin\mathcal{J}}} + {\typvar[L]\notin\knd[1]}} {\typvar[L]\notin\xExtendTypVarCtx[][][\knd[1]]}} {\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]}} {\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]} \\\infer[CWF-TypVar] - { } + {\infer[Weakening] + {\infer[PoS] + {\infer[premiss] + { } + {\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}} + {\KindWellFormed[\knd[1]]} + \\\infer[premiss] + { } + {\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}} + {\KindWellFormed[\knd[1]][\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]} + \\\infer[] + {\infer[PoS] + {\infer[COK] + {\infer[premiss] + { } + {\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}} + {\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}} + {\typvar\notin\TypVarCtx} + \\\infer[] + {\infer[IH] + { } + {\typvar[L]\notin\mathcal{J}} + \\\infer[] + { } + {\typvar\in\mathcal{J}}} + {\typvar\neq\typvar[L]} + \\\infer[] + {\infer[IH] + { } + {\forall\dot{\typvar}\in\knd[L],\dot{\typvar}\notin\mathcal{J}} + \\\infer[] + { } + {\typvar\in\mathcal{J}}} + {\typvar\notin\knd[L]}} + {\typvar\notin\xExtendTypVarCtx[][L][\knd[L]]}} {\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]}} {\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]{\typ}[\knd[2]]}} 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--- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -349,6 +349,22 @@ \section{Metatheory} {\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]{\typ}[\knd[2]]}} {\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} \end{mathpar} + \begin{mathpar} + \rinfer[KEquiv-$\Pi$]{\KindEquiv[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}}{ + \rinfer[Weakening]{\KindEquiv[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\knd[1]}{\knd[2]}}{ + \rinfer[premiss]{\KindEquiv{\knd[1]}{\knd[2]}}{ + } + \\\rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ + } + } + \\\rinfer[Marked-Exchange]{\KindEquiv[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}{\knd[1]}]{\knd[3]}{\knd[4]}}{ + \rinfer[]{\KindEquiv[\HoleCtx\xxExtendTypVarCtx[]{\knd[1]}[L]{\knd[L]}]{\knd[3]}{\knd[4]}}{ + } + \\\rinfer[]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}{\knd[1]}]}{ + } + } + } + \end{mathpar} \end{landscape} \end{proof} \begin{lemma}[OK-PK] @@ -387,7 +403,7 @@ \section{Metatheory} By simultaneous induction on derivations. \\ The interesting cases per lemma: \\ (PoS = premiss of subderivation) - \begin{tabular*}{\textwidth}{lc@{\extracolsep{\fill}}ll} + \begin{tabular*}{.5\textwidth}{lc@{\extracolsep{\fill}}ll} OK-PK. &PK-Base& $\WellFormedAtKind{\bse}[\SKind[\Type][\bse]]$ & by (9) \\ & & $\WellFormedAtKind{\bse}[\Type]$ & by (10) \\ diff --git a/11-type-constructors/latex/hejohns-hazel.sty b/11-type-constructors/latex/hejohns-hazel.sty index 39ba637..7018419 100644 --- a/11-type-constructors/latex/hejohns-hazel.sty +++ b/11-type-constructors/latex/hejohns-hazel.sty @@ -168,6 +168,7 @@ \hbox{$\dotfrac{#1}{#2}$} } \RenewDocumentCommand{\infer}{oO{}}{\inferrule*[right=#1#2]} + \NewDocumentCommand{\rinfer}{oO{}mm}{\inferrule*[right=#1#2]{#4}{#3}} \newcommand{\hintpagebreak}{\pagebreak[3]} % 0-4; 2 too weak, 4 too strong %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % judgments From d6f78f07004fc1e2f2c5ff2e625a3bcae991d65d Mon Sep 17 00:00:00 2001 From: Johnson He Date: Fri, 30 Jul 2021 02:26:31 +0000 Subject: [PATCH 067/181] got vim macro'd to hell. We'll fix the spacing another time --- .../latex/11-type-constructors.pdf | Bin 248989 -> 248989 bytes .../latex/11-type-constructors.tex | 84 +----------------- 2 files changed, 1 insertion(+), 83 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index 5ff6739bce406972cf806064988d6223d71423be..f9bad803d2dc22704777a11555c207d19451cc25 100644 GIT binary patch delta 110 zcmbO`lYj0^{)QID7N#xCya!o~%nVGX3m#;a1hcm59c2E);p}2!Zt3J`=xE?#>Eh_< gYHVx<1dc`qt}Ye^CMHgXb_zCxlx)vE#caR?0H})`@c;k- delta 110 zcmbO`lYj0^{)QID7N#xCya!nf3{B0a3m#;a1hcm59c2E);p}AQ=IUf(=Im@@XkuYz hX=v Date: Fri, 30 Jul 2021 21:58:53 +0000 Subject: [PATCH 068/181] we've gone back/forth 100 times but COK does need subderivation --- .../latex/11-type-constructors.pdf | Bin 248989 -> 250297 bytes .../latex/11-type-constructors.tex | 23 +++++++++++++++--- 2 files changed, 19 insertions(+), 4 deletions(-) diff --git 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a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -237,7 +237,7 @@ \section{Declaratives} \section{Metatheory} \subsection*{} \begin{lemma}[COK] - If $\J$, then $\ContextWellFormed$% in a subderivation (where $\J \neq \ContextWellFormed$) + If $\J$, then $\ContextWellFormed$ in a subderivation (where $\J\neq\ContextWellFormed$) \end{lemma} \begin{proof} By simultaneous induction on derivations. \\ @@ -263,11 +263,14 @@ \section{Metatheory} \end{lemma} \begin{proof} By induction on derivations. \\ - When applying Weakening in the induction, check that the left premiss is always a subderivation. \\ + When applying Weakening in the induction, check that the left premiss is always a subderivation, + and check variable exclusion conditions are satisfied. \\ (PoS = premiss of subderivation) \begin{landscape} \subsubsection*{Weakening} -\begin{mathpar}\rinfer[PK-$\lambda$]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]}{\rinfer[Marked-Exchange]{\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]{\typ}[\knd[2]]}{\rinfer[Weakening]{\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]}{\rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ }\\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]}{\rinfer[Weakening]{\KindWellFormed[\knd[L]][\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{\rinfer[PoS]{\KindWellFormed[\knd[L]]}{\rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ }}\\\rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{\rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ }}}\\\rinfer[]{\typvar[L]\notin\xExtendTypVarCtx[][][\knd[1]]}{\rinfer[PoS]{\typvar[L]\notin\TypVarCtx}{\rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ }}\\\rinfer[]{\typvar[L]\neq\typvar}{\rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ }\\\rinfer[]{\typvar\in\mathcal{J}}{ }}\\\rinfer[]{\typvar[L]\notin\knd[1]}{\rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ }}}}}\\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]}{\rinfer[]{\typvar\notin\xExtendTypVarCtx[][L][\knd[L]]}{\rinfer[PoS]{\typvar\notin\TypVarCtx}{\rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{\rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ }}}\\\rinfer[]{\typvar\neq\typvar[L]}{\rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ }\\\rinfer[]{\typvar\in\mathcal{J}}{ }}\\\rinfer[]{\typvar\notin\knd[L]}{\rinfer[IH]{\forall\dot{\typvar}\in\knd[L],\dot{\typvar}\notin\mathcal{J}}{ }\\\rinfer[]{\typvar\in\mathcal{J}}{ }}}\\\rinfer[Weakening]{\KindWellFormed[\knd[1]][\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{\rinfer[PoS]{\KindWellFormed[\knd[1]]}{\rinfer[premiss]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ }}\\\rinfer[premiss]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ }}}}}\end{mathpar}\begin{mathpar} + \begin{mathpar} + \rinfer[PK-$\lambda$]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]}{\rinfer[Marked-Exchange]{\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]{\typ}[\knd[2]]}{\rinfer[Weakening]{\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]}{\rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ }\\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]}{\rinfer[Weakening]{\KindWellFormed[\knd[L]][\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{\rinfer[PoS]{\KindWellFormed[\knd[L]]}{\rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ }}\\\rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{\rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ }}}\\\rinfer[]{\typvar[L]\notin\xExtendTypVarCtx[][][\knd[1]]}{\rinfer[PoS]{\typvar[L]\notin\TypVarCtx}{\rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ }}\\\rinfer[]{\typvar[L]\neq\typvar}{\rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ }\\\rinfer[]{\typvar\in\mathcal{J}}{ }}\\\rinfer[]{\typvar[L]\notin\knd[1]}{\rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ }}}}}\\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]}{\rinfer[]{\typvar\notin\xExtendTypVarCtx[][L][\knd[L]]}{\rinfer[PoS]{\typvar\notin\TypVarCtx}{\rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{\rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ }}}\\\rinfer[]{\typvar\neq\typvar[L]}{\rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ }\\\rinfer[]{\typvar\in\mathcal{J}}{ }}\\\rinfer[]{\typvar\notin\knd[L]}{\rinfer[IH]{\forall\dot{\typvar}\in\knd[L],\dot{\typvar}\notin\mathcal{J}}{ }\\\rinfer[]{\typvar\in\mathcal{J}}{ }}}\\\rinfer[Weakening]{\KindWellFormed[\knd[1]][\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{\rinfer[PoS]{\KindWellFormed[\knd[1]]}{\rinfer[premiss]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ }}\\\rinfer[premiss]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ }}}}} + \and \rinfer[KEquiv-$\Pi$]{\KindEquiv[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}}{ \rinfer[Weakening]{\KindEquiv[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\knd[1]}{\knd[2]}}{ \rinfer[premiss]{\KindEquiv{\knd[1]}{\knd[2]}}{ @@ -278,7 +281,19 @@ \section{Metatheory} \\\rinfer[Marked-Exchange]{\KindEquiv[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}{\knd[1]}]{\knd[3]}{\knd[4]}}{ \rinfer[]{\KindEquiv[\HoleCtx\xxExtendTypVarCtx[]{\knd[1]}[L]{\knd[L]}]{\knd[3]}{\knd[4]}}{ } - \\\rinfer[]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}{\knd[1]}]}{ + \\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}{\knd[1]}]}{ + \rinfer[]{}{ + } + \\\rinfer[Weakening]{\KindWellFormed[\knd[1]][\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ + \rinfer[PoS]{\KindWellFormed[\knd[1]]}{ + \rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + \rinfer[premiss]{\KindEquiv[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\knd[3]}{\knd[4]}}{ + } + } + } + \\\rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ + } + } } } } From 2928d32198102f79d5dcc5eba0da78d5c9bc91d2 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Fri, 30 Jul 2021 22:20:52 +0000 Subject: [PATCH 069/181] ended up just fixing weakening manually. had to fix an error --- .../latex/11-type-constructors.pdf | Bin 250297 -> 250297 bytes .../latex/11-type-constructors.tex | 67 +++++++++++++++++- 2 files changed, 66 insertions(+), 1 deletion(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index c3db0797bcb0f25b17883356f3943a0f34cd0ccd..00efdafa5b6aa3dd838bf0fa9976a83eece2b8c3 100644 GIT binary patch delta 128 zcmdnFn1APD{)QID7N#xC3dfX;3@uCyqBObmee+XX5=&AQG+eBV42%p7%#96>?I zjxlpOxfz)oJ2@Jd8M;{*n;E!RIGbBISsJ)F8JM~_Sr{4`*eTc$P_n({JhLei0LEG% A`2YX_ delta 128 zcmdnFn1APD{)QID7N#xC3dfWTO)U&8qcpknee+XX5=&AQG+eBV42%p7%#DF0NOHN; zF=kFD3uiMUM<*8xV@Fpf7guLfBTGXwOCw`5GiN6YCsQK>I|Um8O18J0XEtR50NgDh ATL1t6 diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index e3809b1..fd6850c 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -269,7 +269,72 @@ \section{Metatheory} \begin{landscape} \subsubsection*{Weakening} \begin{mathpar} - \rinfer[PK-$\lambda$]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]}{\rinfer[Marked-Exchange]{\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]{\typ}[\knd[2]]}{\rinfer[Weakening]{\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]}{\rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ }\\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]}{\rinfer[Weakening]{\KindWellFormed[\knd[L]][\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{\rinfer[PoS]{\KindWellFormed[\knd[L]]}{\rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ }}\\\rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{\rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ }}}\\\rinfer[]{\typvar[L]\notin\xExtendTypVarCtx[][][\knd[1]]}{\rinfer[PoS]{\typvar[L]\notin\TypVarCtx}{\rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ }}\\\rinfer[]{\typvar[L]\neq\typvar}{\rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ }\\\rinfer[]{\typvar\in\mathcal{J}}{ }}\\\rinfer[]{\typvar[L]\notin\knd[1]}{\rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ }}}}}\\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]}{\rinfer[]{\typvar\notin\xExtendTypVarCtx[][L][\knd[L]]}{\rinfer[PoS]{\typvar\notin\TypVarCtx}{\rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{\rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ }}}\\\rinfer[]{\typvar\neq\typvar[L]}{\rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ }\\\rinfer[]{\typvar\in\mathcal{J}}{ }}\\\rinfer[]{\typvar\notin\knd[L]}{\rinfer[IH]{\forall\dot{\typvar}\in\knd[L],\dot{\typvar}\notin\mathcal{J}}{ }\\\rinfer[]{\typvar\in\mathcal{J}}{ }}}\\\rinfer[Weakening]{\KindWellFormed[\knd[1]][\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{\rinfer[PoS]{\KindWellFormed[\knd[1]]}{\rinfer[premiss]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ }}\\\rinfer[premiss]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ }}}}} + \rinfer[PK-$\lambda$]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]}{ + \rinfer[Marked-Exchange]{\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]{\typ}[\knd[2]]}{ + \rinfer[Weakening]{\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]}{ + \rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ + } + \\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]}{ + \rinfer[Weakening]{\KindWellFormed[\knd[L]][\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + \rinfer[PoS]{\KindWellFormed[\knd[L]]}{ + \rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ + } + } + \\\rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + \rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ + } + } + } + \\\rinfer[]{\typvar[L]\notin\xExtendTypVarCtx[][][\knd[1]]}{ + \rinfer[PoS]{\typvar[L]\notin\TypVarCtx}{ + \rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ + } + } + \\\rinfer[]{\typvar[L]\neq\typvar}{ + \rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ + } + \\\rinfer[]{\typvar\in\mathcal{J}}{ + } + } + \\\rinfer[]{\typvar[L]\notin\knd[1]}{ + \rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ + } + } + } + } + } + \\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]}{ + \rinfer[]{\typvar\notin\xExtendTypVarCtx[][L][\knd[L]]}{ + \rinfer[PoS]{\typvar\notin\TypVarCtx}{ + \rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + \rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ + } + } + } + \\\rinfer[]{\typvar\neq\typvar[L]}{ + \rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ + } + \\\rinfer[]{\typvar\in\mathcal{J}}{ + } + } + \\\rinfer[]{\typvar\notin\knd[L]}{ + \rinfer[IH]{\forall\dot{\typvar}\in\knd[L],\dot{\typvar}\notin\mathcal{J}}{ + } + \\\rinfer[]{\typvar\in\mathcal{J}}{ + } + } + } + \\\rinfer[Weakening]{\KindWellFormed[\knd[1]][\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ + \rinfer[PoS]{\KindWellFormed[\knd[1]]}{ + \rinfer[premiss]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + } + } + \\\rinfer[premiss]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ + } + } + } + } + } \and \rinfer[KEquiv-$\Pi$]{\KindEquiv[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}}{ \rinfer[Weakening]{\KindEquiv[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]{\knd[1]}{\knd[2]}}{ From 642e010a3bac8f9578f2657e44931418d40a1f26 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Fri, 30 Jul 2021 22:23:12 +0000 Subject: [PATCH 070/181] this error --- .../latex/11-type-constructors.pdf | Bin 250297 -> 250313 bytes .../latex/11-type-constructors.tex | 4 +++- 2 files changed, 3 insertions(+), 1 deletion(-) diff 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} \\\rinfer[Weakening]{\KindWellFormed[\knd[1]][\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ \rinfer[PoS]{\KindWellFormed[\knd[1]]}{ - \rinfer[premiss]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + \rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + \rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ + } } } \\\rinfer[premiss]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ From 21e7cd4522e928e8de158c26794f675067affa93 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Fri, 30 Jul 2021 23:13:01 +0000 Subject: [PATCH 071/181] weaking looks good! --- .../latex/11-type-constructors.pdf | Bin 250313 -> 252778 bytes .../latex/11-type-constructors.tex | 99 +++++++++++++++--- 2 files changed, 82 insertions(+), 17 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index 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z*MINA-?wcMNG=0(YXE*b;l;kde0fa~G|}Itw(K-Q5aTAcm^30}Dmii_4NoPMDbTa- zZb+bU=p=W_QYak<8hUVb%V75@XqjIWPTgJ_2=%|fT1bf6@YT9m;jSf%eipg804m9& zCI!MM$Apvc6W_~|feADT>q1na@{aAoix7YJk&b>I`J5HX%m@|v%T6wXPQmq4Y-iLv zuoikemc9WD(5-?PYB1Xj0C(LkOAwy9*bkr{AFgkzT;ILzz9aRaa>2dZAwd5^iRy#p zf-T+xX86|IAx+TT`8>+#|7`gjHkaJq|1(?D`wCke4ySCJ!ASGwap<_x~OgN*pU!ZmjfQULniP{vt^hT7@7$0CKZ`?KBDsF$HnB{|^hA_D28! diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index 5040e15..537efca 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -236,11 +236,17 @@ \section{Declaratives} \end{minipage} \section{Metatheory} \subsection*{} + subderivation preserving inferences: + \begin{itemize} + \item premiss + \item COK (Context OK) + \item PoS (premiss of subderivation) + \end{itemize} \begin{lemma}[COK] If $\J$, then $\ContextWellFormed$ in a subderivation (where $\J\neq\ContextWellFormed$) \end{lemma} \begin{proof} - By simultaneous induction on derivations. \\ + By induction on derivations. \\ No interesting cases. \end{proof} \begin{lemma}[Exchange] @@ -256,6 +262,9 @@ \section{Metatheory} \vphantom{.}\\ If $\J[\HoleCtx\xxExtendTypVarCtx[][L1]{\knd[L1]}[L2]{\knd[L2]}]$ and $\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L2]{\knd[L2]}[L1]{\knd[L1]}]$, then $\J[\HoleCtx\xxExtendTypVarCtx[][L2]{\knd[L2]}[L1]{\knd[L1]}]$ \end{corollary} + \begin{proof} + Exchange when $\TypVarCtx[][2]=\cdot$ + \end{proof} \begin{lemma}[Weakening] \vphantom{.}\\ If $\J$ and $\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ and $\typvar[L]\notin\mathcal{J}$ and $\forall\typvar\in\knd[L],\typvar\notin\mathcal{J}$, then $\J[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]$ @@ -264,8 +273,7 @@ \section{Metatheory} \begin{proof} By induction on derivations. \\ When applying Weakening in the induction, check that the left premiss is always a subderivation, - and check variable exclusion conditions are satisfied. \\ - (PoS = premiss of subderivation) + and check variable exclusion conditions are satisfied (usually checked elsewhere in the derivation). \\ \begin{landscape} \subsubsection*{Weakening} \begin{mathpar} @@ -275,17 +283,7 @@ \section{Metatheory} \rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ } \\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]}{ - \rinfer[Weakening]{\KindWellFormed[\knd[L]][\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ - \rinfer[PoS]{\KindWellFormed[\knd[L]]}{ - \rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ - } - } - \\\rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ - \rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ - } - } - } - \\\rinfer[]{\typvar[L]\notin\xExtendTypVarCtx[][][\knd[1]]}{ + \rinfer[]{\typvar[L]\notin\xExtendTypVarCtx[][][\knd[1]]}{ \rinfer[PoS]{\typvar[L]\notin\TypVarCtx}{ \rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ } @@ -301,6 +299,16 @@ \section{Metatheory} } } } + \\\rinfer[Weakening]{\KindWellFormed[\knd[L]][\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + \rinfer[PoS]{\KindWellFormed[\knd[L]]}{ + \rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ + } + } + \\\rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + \rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ + } + } + } } } \\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}[]{\knd[1]}]}{ @@ -331,7 +339,7 @@ \section{Metatheory} } } } - \\\rinfer[premiss]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ + \\\rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ } } } @@ -346,10 +354,58 @@ \section{Metatheory} } } \\\rinfer[Marked-Exchange]{\KindEquiv[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}{\knd[1]}]{\knd[3]}{\knd[4]}}{ - \rinfer[]{\KindEquiv[\HoleCtx\xxExtendTypVarCtx[]{\knd[1]}[L]{\knd[L]}]{\knd[3]}{\knd[4]}}{ + \rinfer[Weakening]{\PrincipalKind[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]{\typ}[\knd[2]]}{ + \rinfer[premiss]{\PrincipalKind[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\typ}[\knd[2]]}{ + } + \\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][]{\knd[1]}[L]{\knd[L]}]}{ + \rinfer[]{\typvar[L]\notin\xExtendTypVarCtx[][][\knd[1]]}{ + \rinfer[PoS]{\typvar[L]\notin\TypVarCtx}{ + \rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ + } + } + \\\rinfer[]{\typvar[L]\neq\typvar}{ + \rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ + } + \\\rinfer[]{\typvar\in\mathcal{J}}{ + } + } + \\\rinfer[]{\typvar[L]\notin\knd[1]}{ + \rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ + } + } + } + \\\rinfer[Weakening]{\KindWellFormed[\knd[L]][\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + \rinfer[PoS]{\KindWellFormed[\knd[L]]}{ + \rinfer[IH]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ + } + } + \\\rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + \rinfer[premiss]{\KindEquiv[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\knd[3]}{\knd[4]}}{ + } + } + } + } } \\\rinfer[CWF-TypVar]{\ContextWellFormed[\HoleCtx\xxExtendTypVarCtx[][L]{\knd[L]}{\knd[1]}]}{ - \rinfer[]{}{ + \rinfer[]{\typvar\notin\xExtendTypVarCtx[][L][\knd[L]]}{ + \rinfer[PoS]{\typvar\notin\TypVarCtx}{ + \rinfer[COK]{\ContextWellFormed[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]}{ + \rinfer[premiss]{\KindEquiv[\HoleCtx\xExtendTypVarCtx[][][\knd[1]]]{\knd[3]}{\knd[4]}}{ + } + } + } + \\\rinfer[]{\typvar\neq\typvar[L]}{ + \rinfer[IH]{\typvar[L]\notin\mathcal{J}}{ + } + \\\rinfer[]{\typvar\in\mathcal{J}}{ + } + } + \\\rinfer[]{\typvar\notin\knd[L]}{ + \rinfer[IH]{\forall\dot{\typvar}\in\knd[L],\dot{\typvar}\notin\mathcal{J}}{ + } + \\\rinfer[]{\typvar\in\mathcal{J}}{ + } + } } \\\rinfer[Weakening]{\KindWellFormed[\knd[1]][\HoleCtx\xExtendTypVarCtx[][L][\knd[L]]]}{ \rinfer[PoS]{\KindWellFormed[\knd[1]]}{ @@ -364,6 +420,15 @@ \section{Metatheory} } } } + \and + \rinfer[CSK-$\Pi$]{sim}{ + } + \and + \rinfer[EquivAK-$\Pi$]{sim}{ + } + \and + \rinfer[KWF-$\Pi$]{sim}{ + } \end{mathpar} \end{landscape} \end{proof} From d981a97427f6b3f3ce19987ef63998cfe76c4f28 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Sat, 31 Jul 2021 00:34:51 +0000 Subject: [PATCH 072/181] finally figured out how to use typearea --- .../latex/11-type-constructors.pdf | Bin 252778 -> 255066 bytes .../latex/11-type-constructors.tex | 16 ++++++++++------ 2 files changed, 10 insertions(+), 6 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index 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f419565..634d692 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -54,7 +54,8 @@ % (will give actual names sometime) \newcounter{counter} \refstepcounter{counter} - \renewcommand*{\rule}[1][(\thecounter)\refstepcounter{counter}]{\inferrule*[vcenter,rightstyle=\texttt,right=#1]} + \newcommand*{\hyph}{\text{-}} + \newcommand*{\rule}[1][(\thecounter)\refstepcounter{counter}]{\inferrule*[vcenter,rightstyle=\texttt,right=$\mathtt{#1}$]} % get rid of 10-modules stuff \RenewDocumentCommand{\TypVarCtx}{oO{}}{\Phi_{#2}} \RenewDocumentCommand{\InTypVarCtx}{oO{}O{\knd}}{\TypVarCtx[][1],\typvar[#2]\KndAssump{#3},\TypVarCtx[][2]} @@ -160,13 +161,13 @@ \section{Declaratives} \begin{minipage}{\textwidth} \judgbox{\ConsistentSubKind{\knd[1]}{\knd[2]}}{$\knd[1]$ is a consistent subkind of $\knd[2]$} \begin{mathpar} - \rule{\KindWellFormed}{\ConsistentSubKind{\KHole}{\knd}} + \rule[CSK-KHoleL]{\KindWellFormed}{\ConsistentSubKind{\KHole}{\knd}} \and - \rule{\KindWellFormed}{\ConsistentSubKind{\knd}{\KHole}} + \rule[CSK-KHoleR]{\KindWellFormed}{\ConsistentSubKind{\knd}{\KHole}} \and - \rule{\KindWellFormed[{\SKind[\KHole]}] \\ \KindWellFormed}{\ConsistentSubKind{\SKind[\KHole]}{\knd}} + \rule[CSK-$\mathtt{SKind_{KHole}}$L]{\KindWellFormed[{\SKind[\KHole]}] \\ \KindWellFormed}{\ConsistentSubKind{\SKind[\KHole]}{\knd}} \and - \rule{\KindWellFormed \\ \KindWellFormed[{\SKind[\KHole]}]}{\ConsistentSubKind{\knd}{\SKind[\KHole]}} + \rule[CSK\hyph SKind_{KHole}R]{\KindWellFormed \\ \KindWellFormed[{\SKind[\KHole]}]}{\ConsistentSubKind{\knd}{\SKind[\KHole]}} \\ \rule{\KindEquiv{\knd[1]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} \and From 178ee30b4126c7e72ba8694eb652f6800faafd8f Mon Sep 17 00:00:00 2001 From: Johnson He Date: Wed, 28 Jul 2021 05:52:03 +0000 Subject: [PATCH 053/181] eh --- .../latex/11-type-constructors.tex | 56 +++++++++---------- 1 file changed, 28 insertions(+), 28 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index 634d692..95a7667 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -55,7 +55,7 @@ \newcounter{counter} \refstepcounter{counter} \newcommand*{\hyph}{\text{-}} - \newcommand*{\rule}[1][(\thecounter)\refstepcounter{counter}]{\inferrule*[vcenter,rightstyle=\texttt,right=$\mathtt{#1}$]} + \renewcommand*{\rule}[1][(\thecounter)\refstepcounter{counter}]{\inferrule*[vcenter,rightstyle=\texttt,right=$\mathtt{#1}$]} % get rid of 10-modules stuff \RenewDocumentCommand{\TypVarCtx}{oO{}}{\Phi_{#2}} \RenewDocumentCommand{\InTypVarCtx}{oO{}O{\knd}}{\TypVarCtx[][1],\typvar[#2]\KndAssump{#3},\TypVarCtx[][2]} @@ -92,80 +92,80 @@ \section{Declaratives} \begin{minipage}{\textwidth} \judgbox{\PrincipalKind{\typ}}{$\typ$ has principal (well formed) kind $\knd$} \begin{mathpar} - \rule[PK-Base]{\ContextWellFormed}{\PrincipalKind{\bse}[\SKind[\Type][\bse]]} + \rule[PK\hyph Base]{\ContextWellFormed}{\PrincipalKind{\bse}[\SKind[\Type][\bse]]} \and - \rule[PK-Var]{\ContextWellFormed[\HoleCtx\InTypVarCtx]}{\PrincipalKind{\typvar}[\SKind[\knd][\typvar]]} + \rule[PK\hyph Var]{\ContextWellFormed[\HoleCtx\InTypVarCtx]}{\PrincipalKind{\typvar}[\SKind[\knd][\typvar]]} \and - \rule[PK-$\oplus$]{\WellFormedAtKind{\typ[1]}[\Type] \\ \WellFormedAtKind{\typ[2]}[\Type]}{\PrincipalKind{\binop}[\SKind[\Type][\binop]]} + \rule[PK\hyph \oplus]{\WellFormedAtKind{\typ[1]}[\Type] \\ \WellFormedAtKind{\typ[2]}[\Type]}{\PrincipalKind{\binop}[\SKind[\Type][\binop]]} \and - \rule[PK-EHole]{\ContextWellFormed[\InTypHoleCtx\TypVarCtx]}{\PrincipalKind{\ETypeHole}[\SKind[\knd][\ETypeHole]]} + \rule[PK\hyph EHole]{\ContextWellFormed[\InTypHoleCtx\TypVarCtx]}{\PrincipalKind{\ETypeHole}[\SKind[\knd][\ETypeHole]]} \and - \rule[PK-NEHole]{\ContextWellFormed[\InTypHoleCtx\TypVarCtx] \\ \WellFormedAtKind{\typ}[\knd[1]]}{\PrincipalKind{\NETypeHole}[\SKind[\knd][\NETypeHole]]} + \rule[PK\hyph NEHole]{\ContextWellFormed[\InTypHoleCtx\TypVarCtx] \\ \WellFormedAtKind{\typ}[\knd[1]]}{\PrincipalKind{\NETypeHole}[\SKind[\knd][\NETypeHole]]} \and - \rule[PK-Unbound]{\ContextWellFormed[\InTypHoleCtx\TypVarCtx] \\ \typvar\notin\dom{\TypVarCtx[]}}{\PrincipalKind{\UnboundTypeVar}[\SKind[\knd][\UnboundTypeVar]]} + \rule[PK\hyph Unbound]{\ContextWellFormed[\InTypHoleCtx\TypVarCtx] \\ \typvar\notin\dom{\TypVarCtx[]}}{\PrincipalKind{\UnboundTypeVar}[\SKind[\knd][\UnboundTypeVar]]} \and - \rule[PK-$\lambda$]{\PrincipalKind[\HoleCtx\ExtendTypVarCtx[][][{\knd[1]}]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} + \rule[PK\hyph \lambda]{\PrincipalKind[\HoleCtx\ExtendTypVarCtx[][][{\knd[1]}]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} \and - \rule[PK-Ap]{\PrincipalKind{\typ[1]} \\ \MatchedPiKind{\knd} \\ \WellFormedAtKind{\typ[2]}[\knd[1]]}{\PrincipalKind{\TypeAp}[\subst{\typ[2]}{\typvar}{\knd[2]}]} + \rule[PK\hyph Ap]{\PrincipalKind{\typ[1]} \\ \MatchedPiKind{\knd} \\ \WellFormedAtKind{\typ[2]}[\knd[1]]}{\PrincipalKind{\TypeAp}[\subst{\typ[2]}{\typvar}{\knd[2]}]} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\WellFormedAtKind{\typ}}{$\typ$ is well formed at kind $\knd$} \begin{mathpar} - \rule[WFaK-1]{\PrincipalKind{\typ}[\SKind[\knd][\typ]]}{\WellFormedAtKind{\typ}} + \rule[WFaK\hyph 1]{\PrincipalKind{\typ}[\SKind[\knd][\typ]]}{\WellFormedAtKind{\typ}} \and - \rule[WFaK-Subsump]{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} + \rule[WFaK\hyph Subsump]{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} \\ \mprset{myfraction=\inferruledotfrac} - \rule[WFaK-Reit]{\PrincipalKind{\typ}}{\WellFormedAtKind{\typ}} + \rule[WFaK\hyph Reit]{\PrincipalKind{\typ}}{\WellFormedAtKind{\typ}} \and - \rule[WFaK-Self]{\WellFormedAtKind{\typ}}{\WellFormedAtKind{\typ}[\SKind[\knd][\typ]]} + \rule[WFaK\hyph Self]{\WellFormedAtKind{\typ}}{\WellFormedAtKind{\typ}[\SKind[\knd][\typ]]} \and - \rule[WFaK-$\Pi$CSKTrans]{\WellFormedAtKind{\typ}[\DepFunKind[][\knd[3]][\knd[4]]] \\ \ConsistentSubKind{\DepFunKind[][\knd[3]][\knd[4]]}{\DepFunKind}}{\WellFormedAtKind{\typ}[\DepFunKind]} + \rule[WFaK\hyph \Pi CSKTrans]{\WellFormedAtKind{\typ}[\DepFunKind[][\knd[3]][\knd[4]]] \\ \ConsistentSubKind{\DepFunKind[][\knd[3]][\knd[4]]}{\DepFunKind}}{\WellFormedAtKind{\typ}[\DepFunKind]} \and - \rule[WFaK-Flatten]{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]] \\ \WellFormedAtKind{\typ[1]}}{\WellFormedAtKind{\typ}} + \rule[WFaK\hyph Flatten]{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]] \\ \WellFormedAtKind{\typ[1]}}{\WellFormedAtKind{\typ}} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\MatchedPiKind{\knd}}{$\knd$ has matched $\Pi$-kind $\DepFunKind$} \begin{mathpar} - \rule[$\MatchedPiKindOp$-KHole]{\ContextWellFormed}{\MatchedPiKind{\KHole}[\DepFunKind[][\KHole][\KHole]]} + \rule[\MatchedPiKindOp\hyph KHole]{\ContextWellFormed}{\MatchedPiKind{\KHole}[\DepFunKind[][\KHole][\KHole]]} \and - \rule[$\MatchedPiKindOp$-SKHole]{\KindEquiv{\knd}{\SKind[\KHole]}}{\MatchedPiKind{\knd}[\DepFunKind[][\SKind[\KHole]][\SKind[\KHole][\TypeAp[\typ][\typvar]]]]} + \rule[\MatchedPiKindOp\hyph SKHole]{\KindEquiv{\knd}{\SKind[\KHole]}}{\MatchedPiKind{\knd}[\DepFunKind[][\SKind[\KHole]][\SKind[\KHole][\TypeAp[\typ][\typvar]]]]} \and - \rule[$\MatchedPiKindOp$-$\Pi$]{\KindEquiv{\knd}{\DepFunKind}}{\MatchedPiKind{\knd}} + \rule[\MatchedPiKindOp\hyph \Pi]{\KindEquiv{\knd}{\DepFunKind}}{\MatchedPiKind{\knd}} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\KindEquiv{\knd[1]}{\knd[2]}}{$\knd[1]$ is equivalent to $\knd[2]$} \begin{mathpar} - \rule[KEquiv-Refl]{\KindWellFormed}{\KindEquiv{\knd}{\knd}} + \rule[KEquiv\hyph Refl]{\KindWellFormed}{\KindEquiv{\knd}{\knd}} \and - \rule[KEquiv-Symm]{\KindEquiv{\knd[2]}{\knd[1]}}{\KindEquiv{\knd[1]}{\knd[2]}} + \rule[KEquiv\hyph Symm]{\KindEquiv{\knd[2]}{\knd[1]}}{\KindEquiv{\knd[1]}{\knd[2]}} \and - \rule[KEquiv-Trans]{\KindEquiv{\knd[1]}{\knd[3]} \\ \KindEquiv{\knd[3]}{\knd[2]}}{\KindEquiv{\knd[1]}{\knd[2]}} + \rule[KEquiv\hyph Trans]{\KindEquiv{\knd[1]}{\knd[3]} \\ \KindEquiv{\knd[3]}{\knd[2]}}{\KindEquiv{\knd[1]}{\knd[2]}} \\ - \rule[KEquiv-$\mathtt{SKind_{SKind}}$]{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\KindEquiv{\SKind[\SKind[\knd][\typ[1]]][\typ]}{\SKind[\knd][\typ[1]]}} + \rule[KEquiv\hyph SKind_{SKind}]{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\KindEquiv{\SKind[\SKind[\knd][\typ[1]]][\typ]}{\SKind[\knd][\typ[1]]}} \and - \rule[KEquiv-$\mathtt{SKind_{\Pi}}$]{\WellFormedAtKind{\typ}[\DepFunKind]}{\KindEquiv{\SKind[\DepFunKind][\typ]}{\DepFunKind[1][\knd[1]][\SKind[\subst{\typvar[1]}{\typvar}{\knd[2]}][\TypeAp[\typ][\typvar[1]]]]}} + \rule[KEquiv\hyph SKind_{\Pi}]{\WellFormedAtKind{\typ}[\DepFunKind]}{\KindEquiv{\SKind[\DepFunKind][\typ]}{\DepFunKind[1][\knd[1]][\SKind[\subst{\typvar[1]}{\typvar}{\knd[2]}][\TypeAp[\typ][\typvar[1]]]]}} \and - \rule[KEquiv-$\Pi$]{\KindEquiv{\knd[1]}{\knd[2]} \\ \KindEquiv[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\knd[3]}{\knd[4]}}{\KindEquiv{\DepFunKind[][\knd[1]][\knd[2]]}{\DepFunKind[][\knd[3]][\knd[4]]}} + \rule[KEquiv\hyph \Pi]{\KindEquiv{\knd[1]}{\knd[2]} \\ \KindEquiv[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\knd[3]}{\knd[4]}}{\KindEquiv{\DepFunKind[][\knd[1]][\knd[2]]}{\DepFunKind[][\knd[3]][\knd[4]]}} \and - \rule[KEquiv-SKind]{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd[2]}}{\KindEquiv{\SKind[\knd[1]][\typ[1]]}{\SKind[\knd[2]][\typ[2]]}} + \rule[KEquiv\hyph SKind]{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd[2]}}{\KindEquiv{\SKind[\knd[1]][\typ[1]]}{\SKind[\knd[2]][\typ[2]]}} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\ConsistentSubKind{\knd[1]}{\knd[2]}}{$\knd[1]$ is a consistent subkind of $\knd[2]$} \begin{mathpar} - \rule[CSK-KHoleL]{\KindWellFormed}{\ConsistentSubKind{\KHole}{\knd}} + \rule[CSK\hyph KHoleL]{\KindWellFormed}{\ConsistentSubKind{\KHole}{\knd}} \and - \rule[CSK-KHoleR]{\KindWellFormed}{\ConsistentSubKind{\knd}{\KHole}} + \rule[CSK\hyph KHoleR]{\KindWellFormed}{\ConsistentSubKind{\knd}{\KHole}} \and - \rule[CSK-$\mathtt{SKind_{KHole}}$L]{\KindWellFormed[{\SKind[\KHole]}] \\ \KindWellFormed}{\ConsistentSubKind{\SKind[\KHole]}{\knd}} + \rule[CSK\hyph SKind_{KHole}L]{\KindWellFormed[{\SKind[\KHole]}] \\ \KindWellFormed}{\ConsistentSubKind{\SKind[\KHole]}{\knd}} \and \rule[CSK\hyph SKind_{KHole}R]{\KindWellFormed \\ \KindWellFormed[{\SKind[\KHole]}]}{\ConsistentSubKind{\knd}{\SKind[\KHole]}} \\ From 6aa25fb401e17447117a749ed73d73ec8bd7bab9 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Wed, 28 Jul 2021 06:03:05 +0000 Subject: [PATCH 054/181] all rules named --- .../latex/11-type-constructors.tex | 39 ++++++++++--------- 1 file changed, 20 insertions(+), 19 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index 95a7667..acf2b10 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -169,33 +169,34 @@ \section{Declaratives} \and \rule[CSK\hyph SKind_{KHole}R]{\KindWellFormed \\ \KindWellFormed[{\SKind[\KHole]}]}{\ConsistentSubKind{\knd}{\SKind[\KHole]}} \\ - \rule{\KindEquiv{\knd[1]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} + \rule[CSK\hyph KEquiv]{\KindEquiv{\knd[1]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} \and - \rule{\KindEquiv{\knd[1]}{\knd[3]} \\ \ConsistentSubKind{\knd[3]}{\knd[4]} \\ \KindEquiv{\knd[4]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} + \rule[CSK\hyph Normal]{\KindEquiv{\knd[1]}{\knd[3]} \\ \ConsistentSubKind{\knd[3]}{\knd[4]} \\ \KindEquiv{\knd[4]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} \and - \rule{\KindWellFormed[{\SKind}]}{\ConsistentSubKind{\SKind}{\knd}} + \rule[CSK\hyph SKind]{\KindWellFormed[{\SKind}]}{\ConsistentSubKind{\SKind}{\knd}} \and - \rule{\ConsistentSubKind{\knd[3]}{\knd[1]} \\ \ConsistentSubKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[3]]{\knd[2]}{\knd[4]}}{\ConsistentSubKind{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}} + \rule[CSK\hyph \Pi]{\ConsistentSubKind{\knd[3]}{\knd[1]} \\ \ConsistentSubKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[3]]{\knd[2]}{\knd[4]}}{\ConsistentSubKind{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}} \\ \mprset{myfraction=\inferruledotfrac} - \rule{\ConsistentSubKind{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]]}{\ConsistentSubKind{\SKind[\knd[1]][\typ[1]]}{\SKind[\knd[2]][\typ[2]]}} + \rule[CSK-?]{\ConsistentSubKind{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]]}{\ConsistentSubKind{\SKind[\knd[1]][\typ[1]]}{\SKind[\knd[2]][\typ[2]]}} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{$\typ[1]$ is provably equivalent to $\typ[2]$ at kind $\knd$} \begin{mathpar} - \rule{\PrincipalKind{\typ[1]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\SKind[\knd][\typ[2]]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd]} + \rule[EquivAK\hyph Refl]{\WellFormedAtKind{\typ}}{\TypeEquivAtKind{\typ}{\typ}} \and - \rule{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} + \rule[EquivAK\hyph Symm]{\TypeEquivAtKind{\typ[2]}{\typ[1]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \and - \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[2]}{\typvar}{\knd[2]}]} + \rule[EquivAK\hyph Trans]{\TypeEquivAtKind{\typ[1]}{\typ[3]} \\ \TypeEquivAtKind{\typ[3]}{\typ[1]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \\ - \rule{\WellFormedAtKind{\typ}}{\TypeEquivAtKind{\typ}{\typ}} + \rule[EquivAK\hyph SKind]{\PrincipalKind{\typ[1]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\SKind[\knd][\typ[2]]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd]} \and - \rule{\TypeEquivAtKind{\typ[2]}{\typ[1]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} + \rule[EquivAK\hyph\Pi]{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} \and - \rule{\TypeEquivAtKind{\typ[1]}{\typ[3]} \\ \TypeEquivAtKind{\typ[3]}{\typ[1]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} + \rule[EquivAK\hyph Ap]{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[2]}{\typvar}{\knd[2]}]} + \\ \mprset{myfraction=\inferruledotfrac} \and \rule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\SKind]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} @@ -212,24 +213,24 @@ \section{Declaratives} \begin{minipage}{\textwidth} \judgbox{\KindWellFormed}{$\knd$ is well formed} \begin{mathpar} - \rule{\ContextWellFormed}{\KindWellFormed[\Type]} + \rule[KWF\hyph Type]{\ContextWellFormed}{\KindWellFormed[\Type]} \and - \rule{\ContextWellFormed}{\KindWellFormed[\KHole]} + \rule[KWF\hyph KHole]{\ContextWellFormed}{\KindWellFormed[\KHole]} \and - \rule{\WellFormedAtKind{\typ}}{\KindWellFormed[\SKind]} + \rule[KWF\hyph SKind]{\WellFormedAtKind{\typ}}{\KindWellFormed[\SKind]} \and - \rule{\KindWellFormed[{\knd[1]}] \\ \KindWellFormed[{\knd[2]}][\HoleCtx\ExtendTypVarCtx[][][{\knd[1]}]]}{\KindWellFormed[\DepFunKind]} + \rule[KWF\hyph\Pi]{\KindWellFormed[{\knd[1]}] \\ \KindWellFormed[{\knd[2]}][\HoleCtx\ExtendTypVarCtx[][][{\knd[1]}]]}{\KindWellFormed[\DepFunKind]} \end{mathpar} \end{minipage} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\ContextWellFormed}{Context is well formed} \begin{mathpar} - \rule{ }{\ContextWellFormed[;]} + \rule[CWF\hyph Nil]{ }{\ContextWellFormed[;]} \and - \rule{\KindWellFormed}{\ContextWellFormed[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd]} + \rule[CWF\hyph TypVar]{\KindWellFormed}{\ContextWellFormed[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd]} \and - \rule{\KindWellFormed}{\ContextWellFormed[\HoleCtx[],\hole\KndAssump\knd;\TypVarCtx[]]} + \rule[CWF\hyph Hole]{\KindWellFormed}{\ContextWellFormed[\HoleCtx[],\hole\KndAssump\knd;\TypVarCtx[]]} \end{mathpar} Variables implicitly assumed to be fresh as necessary \end{minipage} @@ -268,7 +269,7 @@ \section{Metatheory} \begin{lemma}[OK-CSK] If $\ConsistentSubKind{\knd[1]}{\knd[2]}$, then $\ContextWellFormed$ and $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ \end{lemma} - \begin{lemma}[OK-TEquivAK] + \begin{lemma}[OK-EquivAK] If $\TypeEquivAtKind{\typ[1]}{\typ[2]}$, then $\ContextWellFormed$ and $\WellFormedAtKind{\typ[1]}$ and $\WellFormedAtKind{\typ[2]}$ and $\KindWellFormed$ \end{lemma} \begin{lemma}[OK-KWF] From 123f6af1d52d6daeb789a83facadcbeca9a23f61 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Wed, 28 Jul 2021 20:13:17 +0000 Subject: [PATCH 055/181] weakening --- .../latex/11-type-constructors.tex | 41 ++++++++++++++----- 11-type-constructors/latex/hejohns-hazel.sty | 3 +- 2 files changed, 33 insertions(+), 11 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index acf2b10..122fa5f 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -237,22 +237,43 @@ \section{Declaratives} \section{Metatheory} \subsection*{} \begin{lemma}[Weakening] - If $\WellFormedAtKind{\typ}$, then $\WellFormedAtKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ}$ when $\ContextWellFormed[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]$ + \vphantom{.}\\ + + If $\PrincipalKind{\typ}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\PrincipalKind[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\typ}$ + + If $\PrincipalKind{\typ}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\PrincipalKind[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\typ}$ + + If $\WellFormedAtKind{\typ}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\WellFormedAtKind[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\typ}$ + + If $\WellFormedAtKind{\typ}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\WellFormedAtKind[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\typ}$ + + If $\MatchedPiKind{\knd}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\MatchedPiKind[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\knd}$ + + If $\MatchedPiKind{\knd}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\MatchedPiKind[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\knd}$ + + If $\KindEquiv{\knd[1]}{\knd[2]}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\KindEquiv[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\knd[1]}{\knd[2]}$ + + If $\KindEquiv{\knd[1]}{\knd[2]}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\KindEquiv[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\knd[1]}{\knd[2]}$ + + If $\ConsistentSubKind{\knd[1]}{\knd[2]}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\ConsistentSubKind[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\knd[1]}{\knd[2]}$ + + If $\ConsistentSubKind{\knd[1]}{\knd[2]}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\ConsistentSubKind[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\knd[1]}{\knd[2]}$ + + If $\TypeEquivAtKind{\typ[1]}{\typ[2]}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\TypeEquivAtKind[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\typ[1]}{\typ[2]}$ + + If $\TypeEquivAtKind{\typ[1]}{\typ[2]}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\TypeEquivAtKind[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\typ[1]}{\typ[2]}$ + + If $\KindWellFormed$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\KindWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$ + + If $\KindWellFormed$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\KindWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$ \end{lemma} \begin{proof} - By rule induction/length of proof. \\ + By simultaneous induction on derivations. \\ \begin{tabular*}{\textwidth}{lc@{\extracolsep{\fill}}ll} L1. &(9)& $ $ & \\ \end{tabular*} \end{proof} - \begin{proof} - By rule induction/length of proof. \\ - \begin{tabular*}{\textwidth}{lc@{\extracolsep{\fill}}ll} - L2. - &(9)& $ $ & \\ - \end{tabular*} - \end{proof} \vphantom{.}\\ \begin{lemma}[OK-PK] If $\PrincipalKind{\typ}$, then $\ContextWellFormed$ and $\KindWellFormed$ @@ -288,7 +309,7 @@ \section{Metatheory} (induction on $\WellFormedAtKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$) \end{lemma} \begin{proof} - By simultaneous rule induction/length of proof. \\ + By simultaneous induction on derivations. \\ The interesting cases per lemma: \\ \begin{tabular*}{\textwidth}{lc@{\extracolsep{\fill}}ll} OK-PK. diff --git a/11-type-constructors/latex/hejohns-hazel.sty b/11-type-constructors/latex/hejohns-hazel.sty index 8ed56b3..b95bee1 100644 --- a/11-type-constructors/latex/hejohns-hazel.sty +++ b/11-type-constructors/latex/hejohns-hazel.sty @@ -135,8 +135,9 @@ \NewDocumentCommand{\ExtendTypVarCtx}{O{;}oO{\knd}}{\TypVarCtx[],\typvar[#2]\KndAssump{#3}#1} \newcommand*{\ModVarCtx}[1][;]{\Xi#1} \newcommand*{\SigVarCtx}[1][;]{\Psi#1} - \NewDocumentCommand{\HoleCtx}{O{;}O{}}{\Delta#1_{#2}} + \NewDocumentCommand{\HoleCtx}{O{;}O{}}{\Delta_{#2}#1} \NewDocumentCommand{\InTypHoleCtx}{O{;}O{}O{\knd}}{\HoleCtx[][1],\hole[#2]\KndAssump{#3},\HoleCtx[][2]#1} + \NewDocumentCommand{\ExtendTypHoleCtx}{O{;}oO{\knd}}{\HoleCtx[],\hole[#2]\KndAssump{#3}#1} \newcommand*{\dom}[1]{\mathsf{dom(}#1\mathsf{)}} \newcommand*{\val}[1]{\mathsf{val(}#1\mathsf{)}} \newcommand*{\type}[2]{\mathsf{type(}#1, #2\mathsf{)}} From efb1075e8c7659f1145350c62bfeb3b28f63034c Mon Sep 17 00:00:00 2001 From: Johnson He Date: Wed, 28 Jul 2021 20:53:39 +0000 Subject: [PATCH 056/181] idk why I insisted on typing it all out last time --- .../latex/11-type-constructors.tex | 42 +++---------------- 11-type-constructors/latex/hejohns-hazel.sty | 1 + 2 files changed, 7 insertions(+), 36 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index 122fa5f..4435082 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -238,43 +238,13 @@ \section{Metatheory} \subsection*{} \begin{lemma}[Weakening] \vphantom{.}\\ - - If $\PrincipalKind{\typ}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\PrincipalKind[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\typ}$ - - If $\PrincipalKind{\typ}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\PrincipalKind[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\typ}$ - - If $\WellFormedAtKind{\typ}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\WellFormedAtKind[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\typ}$ - - If $\WellFormedAtKind{\typ}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\WellFormedAtKind[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\typ}$ - - If $\MatchedPiKind{\knd}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\MatchedPiKind[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\knd}$ - - If $\MatchedPiKind{\knd}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\MatchedPiKind[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\knd}$ - - If $\KindEquiv{\knd[1]}{\knd[2]}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\KindEquiv[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\knd[1]}{\knd[2]}$ - - If $\KindEquiv{\knd[1]}{\knd[2]}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\KindEquiv[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\knd[1]}{\knd[2]}$ - - If $\ConsistentSubKind{\knd[1]}{\knd[2]}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\ConsistentSubKind[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\knd[1]}{\knd[2]}$ - - If $\ConsistentSubKind{\knd[1]}{\knd[2]}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\ConsistentSubKind[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\knd[1]}{\knd[2]}$ - - If $\TypeEquivAtKind{\typ[1]}{\typ[2]}$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\TypeEquivAtKind[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]{\typ[1]}{\typ[2]}$ - - If $\TypeEquivAtKind{\typ[1]}{\typ[2]}$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\TypeEquivAtKind[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]{\typ[1]}{\typ[2]}$ - - If $\KindWellFormed$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$, then $\KindWellFormed[\ExtendTypHoleCtx[;][][\knd[L]]\TypVarCtx[]]$ - - If $\KindWellFormed$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$, then $\KindWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[L]]]$ + If $\J$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][L][\knd[L]]]$, then $\J[\HoleCtx\ExtendTypVarCtx[][L][\knd[L]]]$ + %If $\J$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][L][\knd[L]]\TypVarCtx[]]$, then $\J[\ExtendTypHoleCtx[;][L][\knd[L]]\TypVarCtx[]]$ + \end{lemma} + \begin{lemma}[Exchange] + \vphantom{.}\\ + If $\J[\HoleCtx\TypVarCtx[],\typvar[L1]\KndAssump\knd[L1],\typvar[L2]\KndAssump\knd[L2]]$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][L2][\knd[L2]]]$, then $\J[\HoleCtx\TypVarCtx[],\typvar[L2]\KndAssump\knd[L2]\typvar[L1]\KndAssump\knd[L1]]$ \end{lemma} - \begin{proof} - By simultaneous induction on derivations. \\ - \begin{tabular*}{\textwidth}{lc@{\extracolsep{\fill}}ll} - L1. - &(9)& $ $ & \\ - \end{tabular*} - \end{proof} - \vphantom{.}\\ \begin{lemma}[OK-PK] If $\PrincipalKind{\typ}$, then $\ContextWellFormed$ and $\KindWellFormed$ \end{lemma} diff --git a/11-type-constructors/latex/hejohns-hazel.sty b/11-type-constructors/latex/hejohns-hazel.sty index b95bee1..6824e28 100644 --- a/11-type-constructors/latex/hejohns-hazel.sty +++ b/11-type-constructors/latex/hejohns-hazel.sty @@ -162,6 +162,7 @@ \newcommand{\hintpagebreak}{\pagebreak[3]} % 0-4; 2 too weak, 4 too strong %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % judgments + \NewDocumentCommand{\J}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}}{\mathrm{#1 \vdash \mathcal{J}}} % declarative \newcommand*{\TypAsc}{\black{:}} \newcommand*{\TypAssump}{\TypAsc} From 815296376d4ca26830c392f866e31f6fc424864b Mon Sep 17 00:00:00 2001 From: Johnson He Date: Thu, 29 Jul 2021 00:34:25 +0000 Subject: [PATCH 057/181] fix exchange --- .../latex/11-type-constructors.tex | 16 +++++++++++----- 1 file changed, 11 insertions(+), 5 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index 4435082..5472347 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -243,7 +243,7 @@ \section{Metatheory} \end{lemma} \begin{lemma}[Exchange] \vphantom{.}\\ - If $\J[\HoleCtx\TypVarCtx[],\typvar[L1]\KndAssump\knd[L1],\typvar[L2]\KndAssump\knd[L2]]$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][L2][\knd[L2]]]$, then $\J[\HoleCtx\TypVarCtx[],\typvar[L2]\KndAssump\knd[L2]\typvar[L1]\KndAssump\knd[L1]]$ + If $\J[\HoleCtx\TypVarCtx[][1],\typvar[L1]\KndAssump\knd[L1],\typvar[L2]\KndAssump\knd[L2],\TypVarCtx[][2]]$ and $\ContextWellFormed[\HoleCtx\TypVarCtx[][1],\typvar[L2]\KndAssump\knd[L2],\typvar[L1]\KndAssump\knd[L1],\TypVarCtx[][2]]$, then $\J[\HoleCtx\TypVarCtx[][1],\typvar[L2]\KndAssump\knd[L2],\typvar[L1]\KndAssump\knd[L1],\TypVarCtx[][2]]$ \end{lemma} \begin{lemma}[OK-PK] If $\PrincipalKind{\typ}$, then $\ContextWellFormed$ and $\KindWellFormed$ @@ -267,7 +267,8 @@ \section{Metatheory} If $\KindWellFormed$, then $\ContextWellFormed$ \end{lemma} \begin{lemma}[OK-Substitution] - \vphantom{.}\\ If $\WellFormedAtKind{\typ[L]}[\knd[L1]]$ and $\KindWellFormed[{\knd[L2]}][\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]$, then $\ContextWellFormed$ and $\KindWellFormed[\subst{\typ[L]}{\typvar[L]}{\knd[L2]}]$ \\ + \vphantom{.}\\ + If $\WellFormedAtKind{\typ[L]}[\knd[L1]]$ and $\KindWellFormed[{\knd[L2]}][\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]$, then $\KindWellFormed[\subst{\typ[L]}{\typvar[L]}{\knd[L2]}]$ \\ (induction on $\KindWellFormed[{\knd[L2]}][\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]$) \end{lemma} %\begin{lemma}[PK-Substitution] @@ -275,19 +276,24 @@ \section{Metatheory} % (induction on $\PrincipalKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$) %\end{lemma} \begin{lemma}[K-Substitution] - \vphantom{.}\\ If $\WellFormedAtKind{\typ[L1]}[\knd[L1]]$ and $\WellFormedAtKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$, then $\WellFormedAtKind{\subst{\typ[L1]}{\typvar[L]}{\typ[L2]}}[\subst{\typ[L1]}{\typvar[L]}{\knd[L2]}]$ \\ + \vphantom{.}\\ + If $\WellFormedAtKind{\typ[L1]}[\knd[L1]]$ and $\WellFormedAtKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$, then $\WellFormedAtKind{\subst{\typ[L1]}{\typvar[L]}{\typ[L2]}}[\subst{\typ[L1]}{\typvar[L]}{\knd[L2]}]$ \\ (induction on $\WellFormedAtKind[\HoleCtx\ExtendTypVarCtx[][L][{\knd[L1]}]]{\typ[L2]}[\knd[L2]]$) \end{lemma} \begin{proof} By simultaneous induction on derivations. \\ The interesting cases per lemma: \\ \begin{tabular*}{\textwidth}{lc@{\extracolsep{\fill}}ll} + Weakening. + && $ $ & \\ + Exchange. + && $ $ & \\ OK-PK. - &(1)& $\WellFormedAtKind{\bse}[\SKind[\Type][\bse]]$ & by (9) \\ + &PK-Base& $\WellFormedAtKind{\bse}[\SKind[\Type][\bse]]$ & by (9) \\ & & $\WellFormedAtKind{\bse}[\Type]$ & by (10) \\ &*& $\KindWellFormed[{\SKind[\Type][\bse]}]$ & by (43) \\ &*& $\ContextWellFormed$ & by premiss \\ - &(8)& & bad \\ + &PK-Ap& & bad \\ OK-WFaK. &(12)& $\WellFormedAtKind{\typ[2]}$ & by (10) \\ &*& $\KindWellFormed[{\SKind[\knd][\typ[2]]}]$ & by (43) \\ From ffbd8c4bbe50bc6646921639a9833816c3bbdbda Mon Sep 17 00:00:00 2001 From: Johnson He Date: Thu, 29 Jul 2021 02:35:17 +0000 Subject: [PATCH 058/181] I screwed up the layout of these lemmas --- .../latex/11-type-constructors.tex | 26 ++++++++++++------- 1 file changed, 17 insertions(+), 9 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index 5472347..f8cd9f6 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -226,25 +226,31 @@ \section{Declaratives} \begin{minipage}{\textwidth} \judgbox{\ContextWellFormed}{Context is well formed} \begin{mathpar} - \rule[CWF\hyph Nil]{ }{\ContextWellFormed[;]} + \rule[CWF\hyph Nil]{ }{\ContextWellFormed[\cdot;\cdot]} \and \rule[CWF\hyph TypVar]{\KindWellFormed}{\ContextWellFormed[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd]} \and \rule[CWF\hyph Hole]{\KindWellFormed}{\ContextWellFormed[\HoleCtx[],\hole\KndAssump\knd;\TypVarCtx[]]} \end{mathpar} - Variables implicitly assumed to be fresh as necessary + *Variables implicitly assumed to be fresh as necessary* \\ + (eg $\typvar[L]$ is fresh in $\mathcal{J}$ in Weakening) \end{minipage} \section{Metatheory} \subsection*{} + \begin{lemma}[Exchange] + \vphantom{.}\\ + If $\J[\HoleCtx\TypVarCtx[][1],\typvar[L1]\KndAssump\knd[L1],\typvar[L2]\KndAssump\knd[L2],\TypVarCtx[][2]]$ and $\ContextWellFormed[\HoleCtx\TypVarCtx[][1],\typvar[L2]\KndAssump\knd[L2],\typvar[L1]\KndAssump\knd[L1],\TypVarCtx[][2]]$, then $\J[\HoleCtx\TypVarCtx[][1],\typvar[L2]\KndAssump\knd[L2],\typvar[L1]\KndAssump\knd[L1],\TypVarCtx[][2]]$ + \end{lemma} + \begin{proof} + By simultaneous induction on derivations. \\ + No interesting cases. \\ + (Only rules with $\TypVarCtx$ extended in the consequent are interesting, which is only CWF-TypVar, but when $\mathcal{J}$ is CWF, Exchange is identity) + \end{proof} \begin{lemma}[Weakening] \vphantom{.}\\ If $\J$ and $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][L][\knd[L]]]$, then $\J[\HoleCtx\ExtendTypVarCtx[][L][\knd[L]]]$ %If $\J$ and $\ContextWellFormed[\ExtendTypHoleCtx[;][L][\knd[L]]\TypVarCtx[]]$, then $\J[\ExtendTypHoleCtx[;][L][\knd[L]]\TypVarCtx[]]$ \end{lemma} - \begin{lemma}[Exchange] - \vphantom{.}\\ - If $\J[\HoleCtx\TypVarCtx[][1],\typvar[L1]\KndAssump\knd[L1],\typvar[L2]\KndAssump\knd[L2],\TypVarCtx[][2]]$ and $\ContextWellFormed[\HoleCtx\TypVarCtx[][1],\typvar[L2]\KndAssump\knd[L2],\typvar[L1]\KndAssump\knd[L1],\TypVarCtx[][2]]$, then $\J[\HoleCtx\TypVarCtx[][1],\typvar[L2]\KndAssump\knd[L2],\typvar[L1]\KndAssump\knd[L1],\TypVarCtx[][2]]$ - \end{lemma} \begin{lemma}[OK-PK] If $\PrincipalKind{\typ}$, then $\ContextWellFormed$ and $\KindWellFormed$ \end{lemma} @@ -285,9 +291,11 @@ \section{Metatheory} The interesting cases per lemma: \\ \begin{tabular*}{\textwidth}{lc@{\extracolsep{\fill}}ll} Weakening. - && $ $ & \\ - Exchange. - && $ $ & \\ + &PK-$\lambda$& $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][L][\knd[2]]]$ & by IH \\ + & & $\ContextWellFormed[\HoleCtx\ExtendTypVarCtx[][][\knd[1]]]$ & by OK-PK on premiss \\ + & & $\KindWellFormed[\knd[1]]$ & by CWF-TypVar \\ + & & $\KindWellFormed[\knd[1]][\HoleCtx\ExtendTypVarCtx[][L][\knd[L]]]$ & by Weakening \\ + & & $\PrincipalKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ}[\knd[2]]$ & by premiss \\ OK-PK. &PK-Base& $\WellFormedAtKind{\bse}[\SKind[\Type][\bse]]$ & by (9) \\ & & $\WellFormedAtKind{\bse}[\Type]$ & by (10) \\ From 9325d5ce8e2b9fcd68661ff45a964dc9ded6be8f Mon Sep 17 00:00:00 2001 From: Johnson He Date: Thu, 29 Jul 2021 07:30:10 +0000 Subject: [PATCH 059/181] huge waste of a day. but at least it worked out in the end? --- .../latex/11-type-constructors.pdf | Bin 0 -> 244549 bytes .../latex/11-type-constructors.tex | 52 ++++++++++-------- 11-type-constructors/latex/hejohns-hazel.sty | 7 ++- 3 files changed, 34 insertions(+), 25 deletions(-) create mode 100644 11-type-constructors/latex/11-type-constructors.pdf diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf new file mode 100644 index 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zO{A--VVIB7>l^-ozHuHi26W{&L#%+((e&0NZk44}d@%JZP9 zIJMf-51q1LI%zF1vVi0W@@WrMaTw>fnq=k?DAQGQ^e!}=+LSMFTUajCpu7!OkgnSD z{9snxzYiSd!(;%5FfUU%3k^5(dk}R*MYZ}iv4~bSCyl4QS(L7!?Uv*3eA`81yaiyL z>U?kxjXh$u6Ri->#kp^guZmFr60$nFF`t9?;jG|R-@SVPoiDfO!adBV!O`@++eFve zg>r|qRS(Q14VTfLyF)hyZYP&#W74Apm=I_5y*P;c1gNn~LC>}1V&`z-`*+fkA+ZuS kA!O94UZ2+ge&ztIlRq}t-_4y3cJkLq7@N4bE=G^-UvX#I)c^nh diff --git a/11-type-constructors/latex/11-type-constructors.tex b/11-type-constructors/latex/11-type-constructors.tex index 8603f73..3bfe114 100644 --- a/11-type-constructors/latex/11-type-constructors.tex +++ b/11-type-constructors/latex/11-type-constructors.tex @@ -56,7 +56,8 @@ \refstepcounter{counter} \renewcommand*{\rule}[1][(\thecounter)\refstepcounter{counter}]{\inferrule*[vcenter,rightstyle=\texttt,right=#1]} % get rid of 10-modules stuff - \renewcommand*{\TypVarCtx}[1][]{\Phi} + \RenewDocumentCommand{\TypVarCtx}{oO{}}{\Phi_{#2}} + \RenewDocumentCommand{\InTypVarCtx}{oO{}O{\knd}}{\TypVarCtx[][1],\typvar[#2]\KndAssump{#3},\TypVarCtx[][2]} \renewcommand*{\ModVarCtx}[1][]{} \renewcommand*{\SigVarCtx}[1][]{} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -89,17 +90,17 @@ \section{Declaratives} \begin{minipage}{\textwidth} \judgbox{\PrincipalKind{\typ}}{$\typ$ has principal (well formed) kind $\knd$} \begin{mathpar} - \rule{ }{\PrincipalKind{\bse}[\SKind[\Type][\bse]]} + \rule{\ContextWellFormed}{\PrincipalKind{\bse}[\SKind[\Type][\bse]]} \and - \rule{\typvar\KndAssump\knd\in\TypVarCtx[]}{\PrincipalKind{\typvar}[\SKind[\knd][\typvar]]} + \rule{\ContextWellFormed[\HoleCtx\InTypVarCtx]}{\PrincipalKind{\typvar}[\SKind[\knd][\typvar]]} \and - \rule{\WellFormedAtKind{\typ[1]}[\Type] \\ \WellFormedAtKind{\typ[2]}[\Type]}{\PrincipalKind{\binop}[\SKind[\Type][\binop]]} + \rule{\ContextWellFormed \\ \WellFormedAtKind{\typ[1]}[\Type] \\ \WellFormedAtKind{\typ[2]}[\Type]}{\PrincipalKind{\binop}[\SKind[\Type][\binop]]} \and - \rule{\hole\KndAssump\knd\in\HoleCtx[]}{\PrincipalKind{\ETypeHole}[\SKind[\knd][\ETypeHole]]} + \rule{\ContextWellFormed[\InTypHoleCtx\TypVarCtx]}{\PrincipalKind{\ETypeHole}[\SKind[\knd][\ETypeHole]]} \and - \rule{\hole\KndAssump\knd\in\HoleCtx[] \\ \WellFormedAtKind{\typ}[\knd[1]]}{\PrincipalKind{\NETypeHole}[\SKind[\knd][\NETypeHole]]} + \rule{\ContextWellFormed[\InTypHoleCtx\TypVarCtx] \\ \WellFormedAtKind{\typ}[\knd[1]]}{\PrincipalKind{\NETypeHole}[\SKind[\knd][\NETypeHole]]} \and - \rule{\hole\KndAssump\knd\in\HoleCtx[] \\ \typvar\notin\dom{\TypVarCtx[]}}{\PrincipalKind{\UnboundTypeVar}[\SKind[\knd][\UnboundTypeVar]]} + \rule{\ContextWellFormed[\InTypHoleCtx\TypVarCtx] \\ \typvar\notin\dom{\TypVarCtx[]}}{\PrincipalKind{\UnboundTypeVar}[\SKind[\knd][\UnboundTypeVar]]} \and \rule{\KindWellFormed[{\knd[1]}] \\ \PrincipalKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} \and @@ -203,40 +204,67 @@ \section{Declaratives} \begin{minipage}{\textwidth} \judgbox{\KindWellFormed}{$\knd$ is well formed} \begin{mathpar} - \rule{ }{\KindWellFormed[\Type]} + \rule{\ContextWellFormed}{\KindWellFormed[\Type]} \and - \rule{ }{\KindWellFormed[\KHole]} + \rule{\ContextWellFormed}{\KindWellFormed[\KHole]} \and \rule{\WellFormedAtKind{\typ}}{\KindWellFormed[\SKind]} \and \rule{\KindWellFormed[{\knd[1]}] \\ \KindWellFormed[{\knd[2]}]}{\KindWellFormed[\DepFunKind]} \end{mathpar} \end{minipage} + \subsection*{} + \begin{minipage}{\textwidth} + \judgbox{\ContextWellFormed}{Context is well formed} + \begin{mathpar} + \rule{ }{\ContextWellFormed[;]} + \and + \rule{\KindWellFormed}{\ContextWellFormed[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd]} + \and + \rule{\KindWellFormed}{\ContextWellFormed[\HoleCtx[],\hole\KndAssump\knd;\TypVarCtx[]]} + \end{mathpar} + Variables implicitly assumed to be fresh as necessary + \end{minipage} \section{Metatheory} \subsection*{} - Contexts implicitly assumed to be wellformed in the usual way. - \begin{definition}[size of kinds a la Stone] - \vphantom{.} - \begin{longtable}{LCL} - size(\Type) &=& 1 \\ - size(\KHole) &=& 1 \\ - size(\SKind) &=& size(\knd) + 2 \\ - size(\DepFunKind) &=& size(\knd[1]) + size(\knd[2]) + 2 \\ - \end{longtable} - \end{definition} \begin{lemma} - If $\PrincipalKind{\typ}$, then $\KindWellFormed$ + If $\PrincipalKind{\typ}$, then $\ContextWellFormed$ and $\KindWellFormed$ + \end{lemma} + \begin{lemma} + If $\WellFormedAtKind{\typ}$, then $\ContextWellFormed$ and $\KindWellFormed$ + \end{lemma} + \begin{lemma} + If $\MatchedPiKind{\knd}$, then $\ContextWellFormed$ and $\KindWellFormed$ and $\KindWellFormed[\DepFunKind]$ \end{lemma} \begin{lemma} - If $\WellFormedAtKind{\typ}$, then $\KindWellFormed$ + If $\KindEquiv{\knd[1]}{\knd[2]}$, then $\ContextWellFormed$ and $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ \end{lemma} \begin{lemma} - If $\KindEquiv{\knd[1]}{\knd[2]}$, then $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ + If $\ConsistentSubKind{\knd[1]}{\knd[2]}$, then $\ContextWellFormed$ and $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ \end{lemma} \begin{lemma} - If $\ConsistentSubKind{\knd[1]}{\knd[2]}$, then $\KindWellFormed[{\knd[1]}]$ and $\KindWellFormed[{\knd[2]}]$ + If $\KindWellFormed$, then $\ContextWellFormed$ \end{lemma} - \vphantom{.}\\ + \begin{proof} + By simultaneous rule induction/length of proof. \\ + The interesting cases per lemma: + \begin{enumerate}[label=L\arabic*.] + \item + \vphantom{.} \\ + \begin{tabular*}{\textwidth}{l@{\extracolsep{\fill}}ll} + (1)& $\WellFormedAtKind{\bse}[\SKind[\Type][\bse]]$ & by (9) \\ + & $\WellFormedAtKind{\bse}[\Type]$ & by (10) \\ + & $\KindWellFormed[{\SKind[\Type][\bse]}]$ & by (43) \\ + (8)& & text \\ + \end{tabular*} + \item + \item + \item + \item (25): L6 + (42) + \item (43): L2 + \end{enumerate} + \end{proof} + \vphantom{.} \begin{lemma} If $\PrincipalKind{\typ}[\knd[1]]$ and $\WellFormedAtKind{\typ}[\knd[2]]$, then $\ConsistentSubKind{\knd[1]}{\knd[2]}$ \end{lemma} diff --git a/11-type-constructors/latex/hejohns-hazel.sty b/11-type-constructors/latex/hejohns-hazel.sty index bbb193c..91c05d2 100644 --- a/11-type-constructors/latex/hejohns-hazel.sty +++ b/11-type-constructors/latex/hejohns-hazel.sty @@ -130,15 +130,17 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % misc \newcommand*{\ExpVarCtx}[1][;]{\Gamma#1} - \newcommand*{\TypVarCtx}[1][;]{\Phi#1} + \NewDocumentCommand{\TypVarCtx}{O{;}O{}}{\Phi#1_{#2}} + \NewDocumentCommand{\InTypVarCtx}{O{;}O{}O{\knd}}{\TypVarCtx[][1],\typvar[#2]\KndAssump{#3},\TypVarCtx[][2]#1} \newcommand*{\ModVarCtx}[1][;]{\Xi#1} \newcommand*{\SigVarCtx}[1][;]{\Psi#1} - \newcommand*{\HoleCtx}[1][;]{\Delta#1} + \NewDocumentCommand{\HoleCtx}{O{;}O{}}{\Delta#1_{#2}} + \NewDocumentCommand{\InTypHoleCtx}{O{;}O{}O{\knd}}{\HoleCtx[][1],\hole[#2]\KndAssump{#3},\HoleCtx[][2]#1} \newcommand*{\dom}[1]{\mathsf{dom(}#1\mathsf{)}} \newcommand*{\val}[1]{\mathsf{val(}#1\mathsf{)}} \newcommand*{\type}[2]{\mathsf{type(}#1, #2\mathsf{)}} \newcommand*{\submodule}[1]{\mathsf{submodule(}#1\mathsf{)}} - \renewcommand*{\hole}[1][]{\blackrm{u#1}} + \renewcommand*{\hole}[1][]{\blackrm{u_{#1}}} \newcommand*{\subst}[3]{\mathsf{[#1/#2]#3}} % https://tex.stackexchange.com/questions/128991/frac-or-inference-rule-with-dotted-line @@ -209,6 +211,8 @@ % sdecs \newcommand{\SubSdecOp}{\le_{\sdec}} \newcommand{\SubSdec}[3]{\mathrm{#1 \vdash #2 \SubSdecOp #3}} + % contexts + \NewDocumentCommand{\ContextWellFormed}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}}{\mathrm{#1 \vdash \textsf{OK}}} % ??? \newcommand*{\WellFormedAtType}[3]{\mathrm{#1 \vdash #2 \TypAsc #3}} % algorithmic From 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\section{Metatheory} \end{lemma} \begin{proof} By simultaneous rule induction/length of proof. \\ - The interesting cases per lemma: - \begin{enumerate}[label=L\arabic*.] - \item - \vphantom{.} \\ - \begin{tabular*}{\textwidth}{l@{\extracolsep{\fill}}ll} - (1)& $\WellFormedAtKind{\bse}[\SKind[\Type][\bse]]$ & by (9) \\ - & $\WellFormedAtKind{\bse}[\Type]$ & by (10) \\ - & $\KindWellFormed[{\SKind[\Type][\bse]}]$ & by (43) \\ - (8)& & text \\ - \end{tabular*} - \item - \item - \item - \item (25): L6 + (42) - \item (43): L2 - \end{enumerate} + The interesting cases per lemma: \\ + \begin{tabular*}{\textwidth}{lc@{\extracolsep{\fill}}ll} + L1.&(1)& $\WellFormedAtKind{\bse}[\SKind[\Type][\bse]]$ & by (9) \\ + & & $\WellFormedAtKind{\bse}[\Type]$ & by (10) \\ + &*& $\KindWellFormed[{\SKind[\Type][\bse]}]$ & by (43) \\ + &*& $\ContextWellFormed$ & by premiss \\ + &(8)& & bad \\ + L2.&(10)& $\KindWellFormed[{\SKind[\knd][\typ[1]]}]$ & by L2 \\ + &*& $\KindWellFormed$ & by (?) \\ + &(12)& $\WellFormedAtKind{\typ[2]}$ & by (10) \\ + &*& $\KindWellFormed[{\SKind[\knd][\typ[2]]}]$ & by (43) \\ + \end{tabular*} \end{proof} \vphantom{.} \begin{lemma} From cc97e96c0430adcabe63711324a379a0a90a32ee Mon Sep 17 00:00:00 2001 From: Johnson He Date: Wed, 21 Jul 2021 01:01:26 +0000 Subject: [PATCH 041/181] slow progess --- .../latex/11-type-constructors.pdf | Bin 223332 -> 223482 bytes .../latex/11-type-constructors.tex | 14 ++++++++++---- 11-type-constructors/latex/hejohns-hazel.sty | 6 +++--- 3 files changed, 13 insertions(+), 7 deletions(-) diff --git a/11-type-constructors/latex/11-type-constructors.pdf b/11-type-constructors/latex/11-type-constructors.pdf index b4de959566d47711e3730ad39ff7884286b2dd4e..7e5ef6711f520ccaedc43058ecddd989457a31b7 100644 GIT binary patch delta 41416 zcmV(zK<2;X&<*<04Ui_6@ooYskx#CF!sq^jIYy4)FR!ih$*FW17>1yvmzN{AObU{y zkV&*FiMO9FmnYgB&weX}e0g^C-y@*wc?Ox%OgoV&B^g+gz5j6Z<{i(@>Hl73oC#;M zUyEOU$fUDOgQU-k?E9ndN*&R5fH|d#?SK(XAW0MS9m4AFuP-joSBoe8fm4NlxEor3 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246d505..464747b 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -108,6 +108,8 @@ \subsection*{} \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\KindEquiv{\SKind[\SKind[\knd][\typ[1]]][\typ]}{\SKind[\knd][\typ[1]]}} \and \inferrule{\WellFormedAtKind{\typ}[\DepFunKind]}{\KindEquiv{\SKind[\DepFunKind][\typ]}{\DepFunKind[][\knd[1]][\SKind[\knd[2]][\TypeAp[\typ][\typvar]]]}} + \and + \inferrule{\KindEquiv{\knd[1]}{\knd[2]} \\ \KindEquiv[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\knd[3]}{\knd[4]}}{\KindEquiv{\DepFunKind[][\knd[1]][\knd[2]]}{\DepFunKind[][\knd[3]][\knd[4]]}} \end{mathpar} \end{minipage} \subsection*{} @@ -122,8 +124,14 @@ \subsection*{} \and \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\Type] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\Type]}{\TypeEquivAtKind{\binop}{\binop[\typ[3]][\typ[4]]}[\Type]} \and - \inferrule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]] \\ \WellFormedAtKind{\typ[2]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\SKind[\knd][\typ[2]]]} + \inferrule{\KindEquiv{}{} \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ[1]}{\typ[2]}}{\TypeEquivAtKind{\TypeFun[][\knd[1]][\typ[1]]}{\TypeFun[][\knd[2]][\typ[2]]}[\DepFunKind[][\knd[1]][\knd]]} \and + \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[1]}{\typvar}{\knd[2]}]} + \and + \inferrule{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} + \\ + \inferrule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]] \\ \WellFormedAtKind{\typ[2]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\SKind[\knd][\typ[2]]]} + \\ \mprset{myfraction=\inferruledotfrac} \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \end{mathpar} @@ -147,6 +155,23 @@ \subsection*{} \judgbox{\PrincipalKind{\typ}}{$\typ$ has principal kind $\knd$} \begin{mathpar} \inferrule{ }{\PrincipalKind{\bse}[\SKind[\Type][\bse]]} + \and + \inferrule{\typvar\KndAssump\knd\in\TypVarCtx[]}{\PrincipalKind{\typvar}[\SKind[\knd][\typvar]]} \end{mathpar} \end{minipage} +\subsection*{} + \begin{minipage}{\textwidth} + \judgbox{\MatchedPiKind{\knd}}{$\knd$ has matched $\Pi$-kind $\DepFunKind$} + \begin{mathpar} + \inferrule{ }{\MatchedPiKind{\KHole}[\DepFunKind[][\KHole][\KHole]]} + \and + \inferrule{ }{\MatchedPiKind{\DepFunKind}} + \and + \inferrule{\KindEquiv{\knd}{\DepFunKind}}{\MatchedPiKind{\SKind[\knd][\typ]}[\DepFunKind[][\knd[1]][\SKind[\knd[2]][\TypeAp[\typ][\typvar]]]]} + \end{mathpar} + \end{minipage} +\subsection*{} + \begin{minipage}{\textwidth} + \judgbox{\WellFormedAtKind{\typ}}{$\typ$ is well formed at kind $\knd$} + \end{minipage} \end{document} From 6d376f85af6477512aae567ce7ad0fb668d0f5f6 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Tue, 13 Jul 2021 05:57:15 +0000 Subject: [PATCH 025/181] forgot some stuff --- .../latex/kind-judgements.pdf | Bin 170625 -> 170705 bytes .../latex/kind-judgements.tex | 4 +++- 2 files changed, 3 insertions(+), 1 deletion(-) diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index 2184e17abaeacbd12dbf0d975b20f8c04958c552..0c39224fcd9fe86120fae9944fe5868e255e6fdd 100644 GIT binary patch delta 5959 zcmV-N7r5wwwF=R-3XmlNH#n0a?kJH@7JtM|o%S?swoh(yqS#8SRM~Y_avJy7cR)%M z0g@myqnGOV>w@}SBuSZGyCS|6@LizlylEhmX|AYIYuYm${x3~8B?4_^;(*&3lj@}%oA%Y`C z2}FRYC~iL7Zf8ngFMchFe13EF^A*GUX~7Dqh_bwpQV?cz@#d$i*Wb|ME&lISK?%2} z_-}dTPld3CC?@dxy7>C)Z#x+wJAaH=DNA;kW`yA-lH+f{s?}etZvXmjw|j~rWL6@L zd&j`x&%Zlfq-+us$O+1XR#g5;8TCXEt_^}vWA~RofBxx+B|#FY4=YmeRxDnZTKO@F 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zs-XsIp$_Vy0UDtRnxO?+WsyqX+a{YH@`}oi=e^ox7o}buvfDPVPTA|4SC{O2(5qW2 zIrr+3`jcM0QsT#}PfEtU`lXvsuL0@l!)s7_fA$)ZzV5w-6;e;UMik0lyhatGIbLH5 zu_Lb@afQ=IuL*_AF0V=0g#$Q*BZaH8|KGTKvp+AAChV74ngSmRF*Y*_B_%~qMhXhu BOKJcB diff --git a/9-type-aliases-redux/latex/kind-judgements.tex b/9-type-aliases-redux/latex/kind-judgements.tex index e505ea7..cd05986 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -73,8 +73,8 @@ \section{syntax} & \Int \vt \Float \vt \Bool \\ \textrm{User Types} & \utyp & ::= & \utypvar \vt \bse \vt \ubinop \vt \ETypeHole \vt \NETypeHole[\utyp] \vt \TypeFun[][\Type][\utyp] \vt \TypeAp[\utyp[1]][\utyp[2]] \\ - \textrm{Internal Types} & \utyp & ::= - & \utypvar \vt \bse \vt \binop \vt \ETypeHole \vt \NETypeHole \vt \TypeFun \vt \TypeAp \\ + \textrm{Internal Types} & \typ & ::= + & \typvar \vt \bse \vt \binop \vt \ETypeHole \vt \NETypeHole \vt \TypeFun \vt \TypeAp \\ \textrm{Type Pattern} & \\ \textrm{User Expression} & \\ \textrm{Internal Expression} & \\ @@ -87,7 +87,7 @@ \subsection*{} \inferrule{ }{\ConsistentSubKind{\KHole}{\knd}} \and \inferrule{ }{\ConsistentSubKind{\knd}{\KHole}} - \and + \\ \inferrule{\KindEquiv{\knd[1]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} \and \inferrule{\WellFormedAtKind{\typ}}{\ConsistentSubKind{\SKind}{\knd}} @@ -167,9 +167,7 @@ \subsection*{} \begin{mathpar} \inferrule{ }{\MatchedPiKind{\KHole}[\DepFunKind[][\KHole][\KHole]]} \and - \inferrule{ }{\MatchedPiKind{\DepFunKind}} - \and - \inferrule{\KindEquiv{\knd}{\DepFunKind}}{\MatchedPiKind{\SKind[\knd][\typ]}[\DepFunKind[][\knd[1]][\SKind[\knd[2]][\TypeAp[\typ][\typvar]]]]} + \inferrule{\KindEquiv{\knd}{\DepFunKind}}{\MatchedPiKind{\knd}} \end{mathpar} \end{minipage} \subsection*{} From 7bccc52ad4cdc3968ee112d44e2508e41f29515a Mon Sep 17 00:00:00 2001 From: Johnson He Date: Wed, 14 Jul 2021 02:09:45 +0000 Subject: [PATCH 027/181] circularity mostly worked out? except the rules are obnoxiously opaque and 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zRvI+_-qmLn_rLm*Vbr-*NY}_t3?zMIHV)D}GS-Lz362~vLk34S7$VV(qPZ5M=prwR gAJ!o^d*h4dTyU^+JK_C%+JQfxOO+~!_0oaeNdH?_b diff --git a/9-type-aliases-redux/latex/kind-judgements.tex b/9-type-aliases-redux/latex/kind-judgements.tex index cd05986..03254b8 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -118,6 +118,9 @@ \subsection*{} \begin{minipage}{\textwidth} \judgbox{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{$\typ[1]$ is equivalent to $\typ[2]$ at kind $\knd$} \begin{mathpar} + \inferrule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]] \\ \WellFormedAtKind{\typ[2]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd]} + \\ + \mprset{myfraction=\inferruledotfrac} \inferrule{\WellFormedAtKind{\typ}}{\TypeEquivAtKind{\typ}{\typ}} \and \inferrule{\TypeEquivAtKind{\typ[2]}{\typ[1]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} @@ -131,10 +134,7 @@ \subsection*{} \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[3]}[\DepFunKind] \\ \TypeEquivAtKind{\typ[2]}{\typ[4]}[\knd[1]]}{\TypeEquivAtKind{\TypeAp}{\TypeAp[\typ[3]][\typ[4]]}[\subst{\typ[1]}{\typvar}{\knd[2]}]} \and \inferrule{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} - \\ - \inferrule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]] \\ \WellFormedAtKind{\typ[2]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\SKind[\knd][\typ[2]]]} - \\ - \mprset{myfraction=\inferruledotfrac} + \and \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} \end{mathpar} \end{minipage} @@ -159,6 +159,18 @@ \subsection*{} \inferrule{ }{\PrincipalKind{\bse}[\SKind[\Type][\bse]]} \and \inferrule{\typvar\KndAssump\knd\in\TypVarCtx[]}{\PrincipalKind{\typvar}[\SKind[\knd][\typvar]]} + \and + \inferrule{\WellFormedAtKind{\typ[1]}[\Type] \\ \WellFormedAtKind{\typ[2]}[\Type]}{\PrincipalKind{\binop}[\SKind[\Type][\binop]]} + \and + \inferrule{\hole\KndAssump\knd\in\HoleCtx[]}{\PrincipalKind{\ETypeHole}} + \and + \inferrule{\hole\KndAssump\knd\in\HoleCtx[] \\ \WellFormedAtKind{\typ}[\knd[1]]}{\PrincipalKind{\NETypeHole}} + \and + \inferrule{\hole\KndAssump\knd\in\HoleCtx[] \\ \typvar\notin\dom{\TypVarCtx[]}}{\PrincipalKind{\UnboundTypeVar}} + \and + \inferrule{\PrincipalKind[\HoleCtx\TypVarCtx[],\typvar\KndAssump\knd[1]]{\typ}[\knd[2]]}{\PrincipalKind{\TypeFun[][\knd[1]]}[\SKind[\DepFunKind][\TypeFun[][\knd[1]]]]} + \and + \inferrule{\PrincipalKind{\typ[1]} \\ \MatchedPiKind{\knd} \\ \WellFormedAtKind{\typ[2]}[\knd[1]]}{\PrincipalKind{\TypeAp}[\subst{\typ[2]}{\typvar}{\knd[2]}]} \end{mathpar} \end{minipage} \subsection*{} @@ -173,5 +185,10 @@ \subsection*{} \subsection*{} \begin{minipage}{\textwidth} \judgbox{\WellFormedAtKind{\typ}}{$\typ$ is well formed at kind $\knd$} + \begin{mathpar} + \inferrule{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} + \and + \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ}} + \end{mathpar} \end{minipage} \end{document} From d9c0c83aaf41b2e0567112ab1182c61c20b45987 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Wed, 14 Jul 2021 03:04:24 +0000 Subject: [PATCH 028/181] progress. I think --- 9-type-aliases-redux/latex/hejohns-hazel.sty | 3 +++ .../latex/kind-judgements.pdf | Bin 179546 -> 178871 bytes .../latex/kind-judgements.tex | 22 ++++-------------- 3 files changed, 8 insertions(+), 17 deletions(-) diff --git a/9-type-aliases-redux/latex/hejohns-hazel.sty b/9-type-aliases-redux/latex/hejohns-hazel.sty index 55c16a4..7bacef8 100644 --- a/9-type-aliases-redux/latex/hejohns-hazel.sty +++ b/9-type-aliases-redux/latex/hejohns-hazel.sty @@ -167,6 +167,8 @@ \newcommand*{\SigKndAsc}{\black{::_{\sig}}} \newcommand*{\SigKndAssump}{\SigKndAsc} + \NewDocumentCommand{\WellFormed}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}m}{\mathrm{#1 \vdash #2~\textsf{OK}}} + % kinds \newcommand*{\ConsistentSubKindOp}{\lesssim} \NewDocumentCommand{\ConsistentSubKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mm}{\mathrm{#1 \vdash #2 \ConsistentSubKindOp #3}} @@ -186,6 +188,7 @@ \NewDocumentCommand{\PrincipalKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\knd}}{\mathrm{#1 \vdash 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\inferrule{\KindWellFormed}{\KindEquiv{\knd}{\knd}} \and \inferrule{\KindEquiv{\knd[2]}{\knd[1]}}{\KindEquiv{\knd[1]}{\knd[2]}} \and @@ -136,20 +136,6 @@ \subsection*{} \inferrule{\WellFormedAtKind{\typ[1]}[\DepFunKind[][\knd[1]][\knd[3]]] \\ \WellFormedAtKind{\typ[2]}[\DepFunKind[][\knd[1]][\knd[4]]] \\ \TypeEquivAtKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[1]]{\TypeAp[\typ[1]][\typvar]}{\TypeAp[\typ[2]][\typvar]}[\knd[2]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\DepFunKind]} \and \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]] \\ \KindEquiv{\knd[1]}{\knd}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}} - \end{mathpar} - \end{minipage} -\subsection*{} - \begin{minipage}{\textwidth} - \judgbox{\TypeEquiv{\typ[1]}{\typ[2]}}{$\typ[1]$ is equivalent to $\typ[2]$ at ``top" kind} - \begin{mathpar} - \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{\TypeEquiv{\typ[1]}{\typ[2]}} - \and - \mprset{myfraction=\inferruledotfrac} - \inferrule{\WellFormedAtKind{\typ}}{\TypeEquiv{\typ}{\typ}} - \and - \inferrule{\TypeEquiv{\typ[2]}{\typ[1]}}{\TypeEquiv{\typ[1]}{\typ[2]}} - \and - \inferrule{\TypeEquiv{\typ[1]}{\typ[3]} \\ \TypeEquiv{\typ[3]}{\typ[2]}}{\TypeEquiv{\typ[1]}{\typ[2]}} \end{mathpar} \end{minipage} \subsection*{} @@ -186,9 +172,11 @@ \subsection*{} \begin{minipage}{\textwidth} \judgbox{\WellFormedAtKind{\typ}}{$\typ$ is well formed at kind $\knd$} \begin{mathpar} - \inferrule{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} + \inferrule{\PrincipalKind{\typ}[\SKind[\knd][\typ]]}{\WellFormedAtKind{\typ}} \and - \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ}} + \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]] \\ \WellFormedAtKind{\typ[1]}}{\WellFormedAtKind{\typ}} + \and + \inferrule{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} \end{mathpar} \end{minipage} \end{document} From cd20c0759f788e0f3287ed01539832794515194a Mon Sep 17 00:00:00 2001 From: Johnson He Date: Thu, 15 Jul 2021 01:47:51 +0000 Subject: [PATCH 029/181] ::>, :: better, but equivalence at superkinds still broken --- 9-type-aliases-redux/latex/hejohns-hazel.sty | 2 +- .../latex/kind-judgements.pdf | Bin 178871 -> 179813 bytes .../latex/kind-judgements.tex | 18 +++++++++++++----- 3 files changed, 14 insertions(+), 6 deletions(-) diff --git a/9-type-aliases-redux/latex/hejohns-hazel.sty b/9-type-aliases-redux/latex/hejohns-hazel.sty index 7bacef8..ac0815f 100644 --- a/9-type-aliases-redux/latex/hejohns-hazel.sty +++ b/9-type-aliases-redux/latex/hejohns-hazel.sty @@ -178,6 +178,7 @@ \newcommand*{\MatchedPiKindOp}{\stackrel{\blacktriangleright}{{}_{\Pi}}} \NewDocumentCommand{\MatchedPiKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\DepFunKind}}{\mathrm{#1 \vdash #2 \MatchedPiKindOp #3}} + \newcommand*{\KindWellFormed}[1][\knd]{\WellFormed{#1}} % types \newcommand*{\TypeEquivOp}{\equiv} \newcommand*{\TypeEquivAtKindOp}[1][\knd]{\stackrel{#1}{\TypeEquivOp}} @@ -188,7 +189,6 @@ \NewDocumentCommand{\PrincipalKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\knd}}{\mathrm{#1 \vdash #2 \PrincipalKindOp #3}} \NewDocumentCommand{\WellFormedAtKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mO{\knd}}{\mathrm{#1 \vdash #2 \KndAsc #3}} - \newcommand*{\KindWellFormed}[1][\knd]{\WellFormed{#1}} % signature kinds \newcommand{\ConsistentSubSigKindOp}{\lesssim_{\sigknd}} \newcommand{\ConsistentSubSigKind}[3]{\mathrm{#1 \vdash #2 \ConsistentSubSigKindOp #3}} diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index 4bf002d78d46c09fc5830b7b566a41ad630f7bc2..91465c2c4a8c09a0129752a2937c75fcfb2e7e3e 100644 GIT binary patch delta 27716 zcmZUZLv)}G6QyI@wr#W1VW;Dyla6hhZ*1GPZ6_Vuwryj+nSVK(I?twRS#{5?+wur^ z_yHG74#CRH#g&kkOaokM{&HOBLg{}0Q_F4nm$a8Dd-NgRe1prJ+I&BI&Z(S2rb4Ao z`Iq_r*Bg^m7#RtwS7?0_ZqCp79kBP_7Gf9a?9aH{i*@v^oA;ZKgDu=U1fF?36_>AM zra0Q@G_G*_!HbXIPkRp?qrm*DStNm6&L?k&Fy60f`As)oFmJ%)Q=g+a=hTfY=Qq~; 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a/9-type-aliases-redux/latex/kind-judgements.tex b/9-type-aliases-redux/latex/kind-judgements.tex index e0f504c..949aeda 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -84,14 +84,18 @@ \subsection*{} \begin{minipage}{\textwidth} \judgbox{\ConsistentSubKind{\knd[1]}{\knd[2]}}{$\knd[1]$ is a consistent subkind of $\knd[2]$} \begin{mathpar} - \inferrule{ }{\ConsistentSubKind{\KHole}{\knd}} + \inferrule{\KindWellFormed}{\ConsistentSubKind{\KHole}{\knd}} \and - \inferrule{ }{\ConsistentSubKind{\knd}{\KHole}} + \inferrule{\KindWellFormed}{\ConsistentSubKind{\knd}{\KHole}} \\ \inferrule{\KindEquiv{\knd[1]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} \and + \inferrule{\KindEquiv{\knd[1]}{\knd[3]} \\ \ConsistentSubKind{\knd[3]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} + \and \inferrule{\WellFormedAtKind{\typ}}{\ConsistentSubKind{\SKind}{\knd}} \and + \inferrule{\ConsistentSubKind{\knd[1]}{\knd[2]} \\ \TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd[1]]}{\ConsistentSubKind{\SKind[\knd[1]][\typ[1]]}{\SKind[\knd[2]][\typ[2]]}} + \and \inferrule{\ConsistentSubKind{\knd[3]}{\knd[1]} \\ \ConsistentSubKind[\HoleCtx\TypVarCtx,\typvar\KndAssump\knd[3]]{\knd[2]}{\knd[4]}}{\ConsistentSubKind{\DepFunKind}{\DepFunKind[][\knd[3]][\knd[4]]}} \end{mathpar} \end{minipage} @@ -118,7 +122,7 @@ \subsection*{} \begin{minipage}{\textwidth} \judgbox{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{$\typ[1]$ is equivalent to $\typ[2]$ at kind $\knd$} \begin{mathpar} - \inferrule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]] \\ \WellFormedAtKind{\typ[2]}}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd]} + \inferrule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]}{\TypeEquivAtKind{\typ[1]}{\typ[2]}[\knd]} \\ \mprset{myfraction=\inferruledotfrac} \inferrule{\WellFormedAtKind{\typ}}{\TypeEquivAtKind{\typ}{\typ}} @@ -172,11 +176,15 @@ \subsection*{} \begin{minipage}{\textwidth} \judgbox{\WellFormedAtKind{\typ}}{$\typ$ is well formed at kind $\knd$} \begin{mathpar} - \inferrule{\PrincipalKind{\typ}[\SKind[\knd][\typ]]}{\WellFormedAtKind{\typ}} + \inferrule{\PrincipalKind{\typ}}{\WellFormedAtKind{\typ}} \and - \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]] \\ \WellFormedAtKind{\typ[1]}}{\WellFormedAtKind{\typ}} + \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ}} \and \inferrule{\PrincipalKind{\typ}[\knd[1]] \\ \ConsistentSubKind{\knd[1]}{\knd}}{\WellFormedAtKind{\typ}} + \\ + \inferrule{\WellFormedAtKind{\typ[2]}[\SKind[\knd][\typ[1]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} + \and + \inferrule{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[3]]] \\ \WellFormedAtKind{\typ[3]}[\SKind[\knd][\typ[2]]]}{\WellFormedAtKind{\typ[1]}[\SKind[\knd][\typ[2]]]} \end{mathpar} \end{minipage} \end{document} From ebc1cfde23cc88349ae4b113a473dbbb093d8186 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Thu, 15 Jul 2021 02:41:59 +0000 Subject: [PATCH 030/181] is this 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032/181] change ordering --- .../latex/kind-judgements.pdf | Bin 179843 -> 179856 bytes .../latex/kind-judgements.tex | 110 +++++++++--------- 2 files changed, 55 insertions(+), 55 deletions(-) diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index caaba1787dba9fa821c5998f7e76c3d63b71fdca..bcc528b7917326b4c854e659e019f8d52a874dc5 100644 GIT binary patch delta 11744 zcmaiaQ*b2!)8)juaVEBH+qP}nnE1xFZA@(2_T)wrb7C9&eY^GV(^hSDKb-37x320s zeNKmb!X^GE$sk$T*jO`)s=(=h8F=l80Xs%4{wa!l>|on@2mCMWj{D>lo%RL#C*qAO zx^+F>TWNTF{e8Q!jACfy_J2u}{?kwf-Wtsy@B{;UIp?M3OQ#2z zAt`!$@H;J36UD>iiy0`k{d(4BTC*4g(M9{Xh2F1m7T)4$Cc&V9VkZA$$VbmqKUS+_ zR*&p_gN|U+gybYl8n{~Kp9-l{xAes5RNLA(1jumhx3cdOw5AmaiY?$sF(?tG7{h!w z-`}REJ)XTpeR8=y2YLgEA{?R$+jIAni2)ElQz6(C$2OP9sQ%KA*0AUhOWm6Ep=ZP1 z8$D5yj?GG}M8MTdKdd50&^ME2 zoH2aa7-klrxy8=u@*4?xv7OZfVaSZdSVU0bW^Ntqv71tz6*VmQj`a)e@Jr7z{lv*& z|AfNGmVY`)+!Rk3Kp))mK<$&3=jhe40Lg<9W?)*T~{BI_$+0tGPQ$#v|CG(gtJi 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macros -\def\OPTIONConf{1}% -\usepackage{joshuadunfield} - - -\newcommand{\elabAna}[5]{#1 \vdash #2 \Leftarrow #3 \leadsto #4 \dashv #5} -\newcommand{\elabSyn}[5]{#1 \vdash #2 \Rightarrow #3 \leadsto #4 \dashv #5} -\newcommand{\patMatch}[5]{#1 \vdash #2 : #3 \vartriangleright #4 \dashv #5} -\newcommand{\kconsubkind}[3]{#1 \vdash #2 \lesssim #3} -\newcommand{\kindAssign}[3]{#1 \vdash #2 : #3} -\newcommand{\kindSyn}[3]{#1 \vdash #2 \Rightarrow #3} -\newcommand{\kindAna}[3]{#1 \vdash #2 \Leftarrow #3} -\newcommand{\kformation}[2]{#1 \vdash #2 \textt{ kind} } -\newcommand{\kequiv}[3]{#1 \vdash #2 \equiv #3} -\newcommand{\kcequiv}[3]{#1 \vdash #2 \equiv #3} -\newcommand{\ksubkind}[3]{#1 \vdash #2 <:: #3} -\newcommand{\tyvarValid}[1]{#1 \textt{ valid} } -\newcommand{\tyvarNotValid}[1]{\lnot(#1 \textt{ }\textt{valid}) } -\newcommand{\dexpTypeAssign}[3]{#1 \vdash #2 : #3} -\newcommand{\kindLeqLattice}[3]{#1 \vdash #2 \leq #3} - -\newcommand{\hPhi}{\Phi} -\newcommand{\hGamma}{\Gamma} -\newcommand{\EmptyhPhi}{\cdot} -\newcommand{\EmptyDelta}{\cdot} - -\newcommand{\hexp}{e} -\newcommand{\dexp}{d} -\newcommand{\htau}{\hat{\tau}} -\newcommand{\hkappa}{\kappa} -\newcommand{\dtau}{\tau} -\newcommand{\hrho}{\rho} - -\newcommand{\Ty}{\textt{Ty}} -\newcommand{\KHole}{\textt{KHole}} -\newcommand{\KSing}[1]{\textt{S(}#1{\textt{)}}} -\newcommand{\hlist}[1]{\textt{list(}#1\textt{)}} - -\newcommand{\llparenthesiscolor}{\textcolor{violet}{\llparenthesis}} -\newcommand{\rrparenthesiscolor}{\textcolor{violet}{\rrparenthesis}} -\newcommand{\llparenthesiscolorc}{\textcolor{cyan}{\llparenthesis}} -\newcommand{\rrparenthesiscolorc}{\textcolor{cyan}{\rrparenthesis}} -\newcommand{\hthole}[1]{\llparenthesiscolor\rrparenthesiscolor^{#1}} -\newcommand{\hhole}[2]{\llparenthesiscolor#1\rrparenthesiscolor^{#2}} -\newcommand{\patehole}{\llparenthesiscolorc\rrparenthesiscolorc} -\newcommand{\pathole}[1]{\llparenthesiscolorc#1\rrparenthesiscolorc} - -\newcommand{\Dbinding}[2]{#1 :: #2} -\newcommand{\domOf}[1]{\mathsf{dom}(#1)} - -\newcommand{\htdefine}[3]{\textt{type }#1 \textt{ = } #2\textt{ in }#3} -\newcommand{\dtdefine}[4]{\textt{type }#1 \textt{ = } #2 : #3\textt{ in }#4} - +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% standard header + \documentclass[12pt]{article} + \usepackage[margin=2cm]{geometry} + \usepackage[utf8x]{inputenc} % for utf8 + \usepackage{titlesec} % see below + \usepackage{xcolor} % \color \textcolor + \usepackage{hyperref} % \href + \usepackage[normalem]{ulem} % normalem retains \emph as italic + % \uline \uuline \uwave \sout + \usepackage{enumitem} + % \begin{enumerate}[label=\Alpha*.] + % \begin{enumerate}[label=\roman*.] + % \begin{itemize} + \usepackage{graphicx} + % \includegraphicswidth=0.5\textwidth,trim=[0cm 0cm 0cm 0cm,clip]{file.png} + \usepackage{pdfpages} % \includepdf[pages=1]{file.pdf} + % for prose + %\usepackage[doublespacing]{setspace} + \usepackage{csquotes} % \blockquote + \usepackage{xparse} % \NewDocumentCommand{\name}{O{#1 default}mO{#3 default}}{def} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \renewcommand{\rmdefault}{cmr} + \renewcommand{\sfdefault}{cmss} + \renewcommand{\ttdefault}{cmtt} + \renewcommand{\familydefault}{\rmdefault} + \setlength{\titlewidth}{\textwidth} + + \titlespacing{\section}{0pt}{0pt}{0pt} + \titlespacing{\subsection}{0pt}{0pt}{0pt} + \titlespacing{\subsubsection}{0pt}{0pt}{0pt} + \titlespacing{\paragraph}{0pt}{0pt}{0pt} + % https://www.overleaf.com/learn/latex/How_to_write_a_LaTeX_class_file_and_design_your_own_CV_(Part_1) + % \titleformat{command}[shape]{format}{label}{sep}{before-code}[after-code] + \titleformat{\section} % Customise the \section command + [hang] + {\Large\scshape\raggedright} % Make the \section headers large (\Large), + % small capitals (\scshape) and left aligned (\raggedright) + {}{0em} % Can be used to give a prefix to all sections, like 'Section ...' + {} % Can be used to insert code before the heading + [\titlerule] % Inserts a horizontal line after the heading + \titleformat{\subsection} + [hang] + {\large\scshape\raggedright} + {}{0em} + {} + [] +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% packages + \usepackage{hejohns-hazel} +% commands + \newcommand*{\vt}{~\vert~} +% get rid of 10-modules stuff + \renewcommand*{\TypVarCtx}[1][]{\Phi} + \renewcommand*{\ModVarCtx}[1][]{} + \renewcommand*{\SigVarCtx}[1][]{} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\pagenumbering{gobble} +\nonfrenchspacing +%\frenchspacing % when monospaced \begin{document} - -\begin{figure}[t] - $\arraycolsep=4pt\begin{array}{rllllll} - \mathsf{BinOp} & \oplus & ::= & - \textt{Product} ~\vert~ \textt{Sum} ~\vert~ \textt{Arrow} \\ - \mathsf{Kind} & \kappa & ::= & - \Ty ~\vert~ \KHole ~\vert~ \KSing{\tau} \\ - \mathsf{Constant Types} & c & ::= & - \textt{Int} ~\vert~ \textt{Float} ~\vert~ \textt{Bool} \\ - \mathsf{User HTyp} & \htau & ::= & - c ~\vert~ \htau_1 \oplus \htau_2 ~\vert~ \hlist{\htau} ~\vert~ \hthole{u} ~\vert~ \hhole{\htau}{u} \\ - \mathsf{Internal HTyp} & \dtau & ::= & - c ~\vert~ \dtau_1 \oplus \dtau_2 ~\vert~ \hlist{\dtau} ~\vert~ \hthole{u} ~\vert~ \hhole{\dtau}{u} \\ - \mathsf{Type Vars} & t & & \\ - \mathsf{Type Pattern} & \hrho & ::= & - t ~\vert~ \patehole ~\vert~ \pathole{t} \\ - \mathsf{User Expression} & \hexp & ::= & - \htdefine{\hrho}{\htau}{\hexp} ~\vert~ elided \\ - \mathsf{Internal Expression} & \dtau & ::= & - \dtdefine{\hrho}{\dtau}{\hkappa}{\dexp} ~\vert~ elided \\ - \end{array}$ -\end{figure} - -\begin{minipage}{\linewidth} - \judgbox - {\kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}} - {$\hkappa_1$ is a consistent subkind of $\hkappa_2$} - \begin{mathpar} - \inferrule[KCHoleL]{ }{ - \kconsubkind{\Delta;\hPhi}{\KHole}{\hkappa} - } - \and - \inferrule[KCHoleR]{ }{ - \kconsubkind{\Delta;\hPhi}{\hkappa}{\KHole} - } - \and - \inferrule[KCRespectEquiv]{ - \kequiv{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} \\ - } { - \kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} - } - \and - \inferrule[KCSubsumption]{ - \kindAna{\Delta;\hPhi}{\dtau}{\Ty} - } { - \kconsubkind{\Delta;\hPhi}{\KSing{\dtau}}{\Ty} - } - \end{mathpar} -\end{minipage} -\\ -\\ - - -\begin{minipage}{\linewidth} - \judgbox - {\tyvarValid{t}} - {$t$ is a valid type variable} \; \\ - $t$ is valid if it is not a builtin-type or keyword, begins with an alpha char or underscore, and only contains alphanumeric characters, underscores, and primes. -\end{minipage} -\\ -\\ - - -\begin{minipage}{\linewidth} - \judgbox - {\kformation{\Delta;\hPhi}{\hkappa}} - {$\hkappa$ forms a kind} \;\\ - \begin{mathpar} - \inferrule[KFTy] { }{ \kformation{\Delta;\hPhi}{\Ty} } - \and - \inferrule[KFHole] { }{ \kformation{\Delta;\hPhi}{\KHole} } - \and - \inferrule[KFSing]{ - \kindAna{\Delta;\hPhi}{\dtau}{\Ty} - } { - \kformation{\Delta;\hPhi}{\KSing{\dtau}} - } - \end{mathpar} -\end{minipage} -\\ -\\ - -\begin{minipage}{\linewidth} - \judgbox - {\kequiv{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}} - {$\hkappa_1$ is equivalent to $\hkappa_2$} \;\\ - \begin{mathpar} - \inferrule[KERefl]{ } { - \kequiv{\Delta;\hPhi}{\hkappa}{\hkappa} - } - \and - \inferrule[KESymm]{ - \kequiv{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} - } { - \kequiv{\Delta;\hPhi}{\hkappa_2}{\hkappa_1} - } - \and - \inferrule[KETrans]{ - \kequiv{\Delta;\hPhi}{\hkappa_1}{\hkappa_2} \\ - \kequiv{\Delta;\hPhi}{\hkappa_2}{\hkappa_3} - } { - \kequiv{\Delta;\hPhi}{\hkappa_1}{\hkappa_3} - } - \and - \inferrule[KESingEquiv]{ - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} - } { - \kequiv{\Delta;\hPhi}{\KSing{\dtau_1}}{\KSing{\dtau_2}} - } - \end{mathpar} -\end{minipage} -\\ -\\ - -\begin{minipage}{\linewidth} - \judgbox - {\kindSyn{\Delta;\hPhi}{\dtau}{\hkappa}} - {$\dtau$ synthesizes kind $\hkappa$} - \begin{mathpar} - \inferrule[KSConst]{ }{ - \kindSyn{\Delta;\hPhi}{c}{\KSing{c}} - } - \and - \inferrule[KSVar]{ - t : \hkappa \in \hPhi - } { - \kindSyn{\Delta;\hPhi}{t}{\hkappa} - } - \and - \inferrule[KSUVar]{ - t \not \in \domOf{\hPhi} - } { - \kindSyn{\Delta;\hPhi}{t}{\KHole} - } - \and - \inferrule[KSBinOp]{ - \kindAna{\Delta;\hPhi}{\dtau_1}{\KSing{\dtau_1}} \\ - \kindAna{\Delta;\hPhi}{\dtau_2}{\KSing{\dtau_2}} - } { - \kindSyn{\Delta;\hPhi}{\dtau_1 \oplus \dtau_2}{\KSing{\dtau_1 \oplus \dtau_2}} - } - \and - \inferrule[KSList]{ - \kindAna{\Delta;\hPhi}{\dtau}{\KSing{\dtau}} - } { - \kindSyn{\Delta;\hPhi}{\hlist{\dtau}}{\KSing{\hlist{\dtau}}} - } - \and - \inferrule[KSEHole]{ - \Dbinding{u}{\hkappa} \in \Delta - } { - \kindSyn{\Delta;\hPhi}{\hthole{u}}{\hkappa} - } - \and - \inferrule[KSNEHole]{ - \Dbinding{u}{\hkappa} \in \Delta \\ - \kindSyn{\Delta;\hPhi}{\dtau}{\hkappa'} - } { - \kindSyn{\Delta;\hPhi}{\hhole{\dtau}{u}}{\hkappa} - } - - - \end{mathpar} -\end{minipage} -\\ -\\ - - - -\begin{minipage}{\linewidth} - \judgbox - {\kindAna{\Delta;\hPhi}{\dtau}{\hkappa}} - {$\dtau$ analyzes against kind $\hkappa$} - \begin{mathpar} - \inferrule[KAASubsume]{ - \kindSyn{\hPhi}{\dtau}{\hkappa'} \\ - \kconsubkind{\Delta;\hPhi}{\hkappa'}{\hkappa} - }{ - \kindAna{\Delta;\hPhi}{\dtau}{\hkappa} - } - - - \end{mathpar} -\end{minipage} -\\ -\\ - - - -\begin{minipage}{\linewidth} - \judgbox - {\kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}} - {$\dtau_1$ is equivalent to $\dtau_2$} \;\\ - \begin{mathpar} - \inferrule[KCESymm]{ - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} - } { - \kcequiv{\Delta;\hPhi}{\dtau_2}{\dtau_1} - } - \and - \inferrule[KCETrans]{ - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} \\ - \kcequiv{\Delta;\hPhi}{\dtau_2}{\dtau_3} - } { - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_3} - } - \and - \inferrule[KCESingEquiv]{ - \kindAna{\Delta;\hPhi}{\dtau_1}{\KSing{\dtau_2}} - } { - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} - } - \and - \inferrule[KCEConst]{ }{ - \kcequiv{\Delta;\hPhi}{c}{c} - } - \and - \inferrule[KCEVar]{ - t : \hkappa \in \hPhi - } { - \kcequiv{\Delta;\hPhi}{t}{t} - } - \and - \inferrule[KCEBinOp]{ - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} \\ - \kcequiv{\Delta;\hPhi}{\dtau_3}{\dtau_4} - } { - \kcequiv{\Delta;\hPhi}{\dtau_1 \oplus \dtau_3}{\dtau_2 \oplus \dtau_4} - } - \and - \inferrule[KCEList]{ - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} - } { - \kcequiv{\Delta;\hPhi}{\hlist{\dtau_1}}{\hlist{\dtau_2}} - } - \and - \inferrule[KCEEHole]{ - \Dbinding{u}{\hkappa} \in \Delta - } { - \kcequiv{\Delta;\hPhi}{\hthole{u}}{\hthole{u}} - } - \and - \inferrule[KCENEHole]{ - \Dbinding{u}{\hkappa} \in \Delta \\ - \kindAna{\Delta;\hPhi}{\dtau}{\hkappa'} - } { - \kcequiv{\Delta;\hPhi}{\hhole{\dtau}{u}}{\hhole{\dtau}{u}} - } - \end{mathpar} -\end{minipage} -\\ -\\ - -\begin{minipage}{\linewidth} - \judgbox - {\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}} - {$\htau$ synthesizes kind $\hkappa$ and elaborates to $\dtau$} - \begin{mathpar} - \inferrule[TElabSConst]{ }{ - \elabSyn{\hPhi}{c}{\KSing{c}}{c}{\EmptyDelta} - } - \and - \inferrule[TElabSBinOp]{ - \elabAna{\hPhi}{\htau_1}{\Ty}{\dtau_1}{\Delta_1} \\ - \elabAna{\hPhi}{\htau_2}{\Ty}{\dtau_2}{\Delta_2} - }{ - \elabSyn{\hPhi}{\htau_1 \oplus \htau_2}{\KSing{\dtau_1 \oplus \dtau_2}}{\dtau_1 \oplus \dtau_2}{\Delta_1 \cup \Delta_2} - } - \and - \inferrule[TElabSList]{ - \elabAna{\hPhi}{\htau}{\Ty}{\dtau}{\Delta} - } { - \elabSyn{\hPhi}{\hlist{\htau}}{\KSing{\hlist{\dtau}}}{\hlist{\dtau}}{\Delta} - } - \and - \inferrule[TElabSVar]{ - t : \hkappa \in \hPhi - } { - \elabSyn{\hPhi}{t}{\KSing{t}}{t}{\EmptyDelta} - } - \and - \inferrule[TElabSUVar]{ - t \not\in \domOf{\hPhi} - } { - \elabSyn{\hPhi}{t}{\KHole}{\hhole{t}{u}}{\Dbinding{u}{\KHole} } - } - \and - \inferrule[TElabSHole]{ } { - \elabSyn{\hPhi}{\hthole{u}}{\KHole}{\hthole{u}}{\Dbinding{u}{\KHole}} - } - \and - \inferrule[TElabSNEHole]{ - \elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta} - } { - \elabSyn{\hPhi}{\hhole{\htau}{u}}{\KHole}{\hhole{\dtau}{u}}{\Delta,\Dbinding{u}{\KHole}} - } - \end{mathpar} -\end{minipage} -\\ -\\ - -\begin{minipage}{\linewidth} - \judgbox - {\elabAna{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}} - {$\htau$ analyzes against kind $\hkappa_1$ and elaborates to $\dtau$} - \begin{mathpar} - \inferrule[TElabASubsume]{ - \htau \neq \hthole{u} \\ - \htau \neq \hhole{\htau'}{u} \\ - \elabSyn{\hPhi}{\htau}{\hkappa'}{\dtau}{\Delta} \\ - \kconsubkind{\Delta;\hPhi}{\hkappa'}{\hkappa} - }{ - \elabAna{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta} - } - \and - \inferrule[TElabAEHole]{ } { - \elabAna{\hPhi}{\hthole{u}}{\hkappa}{\hthole{u}}{\Dbinding{u}{\kappa}} - } - \and - \inferrule[TElabANEHole]{ - \elabSyn{\hPhi}{\htau}{\hkappa'}{\dtau}{\Delta} - } { - \elabAna{\hPhi}{\hhole{\htau}{u}}{\hkappa}{\hhole{\dtau}{u}}{\Delta,\Dbinding{u}{\kappa}} - } - - \end{mathpar} -\end{minipage} -\\ -\\ - -\begin{minipage}{\linewidth} - \judgbox - {\patMatch{\hPhi_1}{\dtau}{\hkappa}{\hrho}{\hPhi_2}} - {$\hrho$ matches against $\dtau : \hkappa$ extending $\hPhi$ if necessary} - \begin{mathpar} - \inferrule[RESVar]{ - \tyvarValid{t} - }{ - \patMatch{\hPhi}{\dtau}{\hkappa}{t}{\hPhi,t :: \hkappa} - } - \and - \inferrule[RESEHole]{ }{ - \patMatch{\hPhi}{\dtau}{\hkappa}{\patehole}{\hPhi} - } - \and - \inferrule[RESVarHole]{ - \tyvarNotValid{t} - }{ - \patMatch{\hPhi}{\dtau}{\hkappa}{\pathole{t}}{\hPhi} - } - \end{mathpar} -\end{minipage} -\\ -\\ - -\begin{minipage}{\linewidth} - \judgbox - {\elabSyn{\hGamma;\hPhi}{\hexp}{\dtau}{\dexp}{\Delta}} - {$\hexp$ synthesizes type $\dtau$ and elaborates to $\dexp$} - \begin{mathpar} - \inferrule[ESDefine]{ - \elabSyn{\hPhi_1}{\htau}{\hkappa}{\dtau}{\Delta_1} \\ - \patMatch{\hPhi_1}{\dtau}{\hkappa}{\hrho}{\hPhi_2} \\ - \elabSyn{\hGamma;\hPhi_2}{\hexp}{\dtau_1}{\dexp}{\Delta_2} - }{ - \elabSyn{\hGamma;\hPhi_1}{\htdefine{\hrho}{\htau}{\hexp}}{\dtau_1}{\dtdefine{\hrho}{\dtau}{\hkappa}{\dexp}}{\Delta_1 \cup \Delta_2} - } - - - \end{mathpar} -\end{minipage} -\\ -\\ - -\begin{minipage}{\linewidth} - \judgbox - {\dexpTypeAssign{\Delta;\hGamma;\hPhi}{\dexp}{\dtau}} - {$\dexp$ is assigned type $\dtau$} - \begin{mathpar} - \inferrule[DEDefine]{ - \patMatch{\hPhi_1}{\dtau_1}{\hkappa}{\hrho}{\hPhi_2} \\ - \dexpTypeAssign{\Delta;\hGamma;\hPhi_2}{\dexp}{\dtau_2} - }{ - \dexpTypeAssign{\Delta;\hGamma;\hPhi_1}{\dtdefine{\hrho}{\dtau_1}{\hkappa}{\dexp}}{\dtau_2} - } - - - \end{mathpar} -\end{minipage} -\\ -\\ - -\newtheorem{thm}{Theorem} -\begin{thm}[Well-Kinded Elaboration]\ \\ - (1) If $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ then $\kindSyn{\Delta;\hPhi}{\dtau}{\hkappa}$ \\ - (2) If $\elabAna{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ then $\kindAna{\Delta;\hPhi}{\dtau}{\hkappa}$ -\end{thm} -\noindent -This is like the Typed Elaboration theorem in the POPL19 paper. - -\begin{thm}[Elaborability]\ \\ - (1) $\exists \Delta$ s.t. if $\kindSyn{\Delta;\hPhi}{\dtau}{\hkappa}$ then $\exists \htau$ such that $\elabSyn{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ \\ - (2) $\exists \Delta$ s.t. if $\kindAna{\Delta;\hPhi}{\dtau}{\hkappa}$ then $\exists \htau$ such that $\elabAna{\hPhi}{\htau}{\hkappa}{\dtau}{\Delta}$ -\end{thm} -\noindent -This is similar but a little different from Elaborability theorem in the POPL19 paper. Choose the $\Delta$ that is emitted from elaboration and then there's an $\htau$ that elaborates to any of the $\dtau$ forms. Elaborability and Well-Kinded Elaboration implies we can just rely on the elaboration forms for the premises of any rules that demand kind synthesis/analysis. - -\begin{thm}[Type Elaboration Unicity]\ \\ - (1) If $\elabSyn{\hPhi}{\htau}{\hkappa_1}{\dtau_1}{\Delta_1}$ and $\elabSyn{\hPhi}{\htau}{\hkappa_2}{\dtau_2}{\Delta_2}$ then $\hkappa_1 = \hkappa_2$, $\dtau_1 = \dtau_2$, $\Delta_1 = \Delta_2$ \\ - (2) If $\elabAna{\hPhi}{\htau}{\hkappa}{\dtau_1}{\Delta_1}$ and $\elabAna{\hPhi}{\htau}{\hkappa}{\dtau_2}{\Delta_2}$ then $\dtau_1 = \dtau_2$, $\Delta_1 = \Delta_2$ -\end{thm} -\noindent -This is like the Elaboration Unicity theorem in the POPL19 paper. - - -\begin{thm}[Kind Synthesis Precision]\ \\ - If $\kindSyn{\Delta;\hPhi}{\dtau}{\hkappa_1}$ and $\kindAna{\Delta;\hPhi}{\dtau}{\hkappa_2}$ then $\kconsubkind{\Delta;\hPhi}{\hkappa_1}{\hkappa_2}$ -\end{thm} - -\noindent -Kind Synthesis Precision says that synthesis finds the most precise kappa possible for a given input type. This is somewhat trivial, but interesting to note because it means we can expect singletons wherever possible. - - - - +\title{Hazel Phi: 9-type-aliases} +\author{} +\date{\today} +\maketitle +\section{syntax} + \begin{longtable}{RCRL} + \textrm{BinOp} & \binop[][] & ::= + & \ProdType \vt \SumType \vt \FunType \\ + \textrm{Kind} & \knd & ::= + & \Type \vt \KHole \vt \SKind \vt \DepFunKind \\ + \textrm{Base Types} & \bse & ::= + & \Int \vt \Float \vt \Bool \\ + \textrm{User Types} & \utyp & ::= + & \utypvar \vt \bse \vt \ubinop \vt \ETypeHole \vt \NETypeHole[\utyp] \vt \TypeFun[][\Type][\utyp] \vt \TypeAp[\utyp[1]][\utyp[2]] \\ + \textrm{Internal Types} & \utyp & ::= + & \utypvar \vt \bse \vt \binop \vt \ETypeHole \vt \NETypeHole \vt \TypeFun \vt \TypeAp \\ + \textrm{Type Pattern} & \\ + \textrm{User Expression} & \\ + \textrm{Internal Expression} & \\ + \end{longtable} +\section{Declaratives} +\subsection*{} + \begin{minipage}{\textwidth} + \judgbox{\ConsistentSubKind{\knd[1]}{\knd[2]}}{$\knd[1]$ is a consistent subkind of $\knd[2]$} + \begin{mathpar} + \inferrule{ }{\ConsistentSubKind{\KHole}{\knd}} + \and + \inferrule{ }{\ConsistentSubKind{\knd}{\KHole}} + \and + \inferrule{\KindEquiv{\knd[1]}{\knd[2]}}{\ConsistentSubKind{\knd[1]}{\knd[2]}} + \and + \inferrule{\WellFormedAtKind{\typ}}{\ConsistentSubKind{\SKind}{\knd}} + \end{mathpar} + \end{minipage} +\subsection*{} + \begin{minipage}{\textwidth} + \judgbox{\KindEquiv{\knd[1]}{\knd[2]}}{$\knd[1]$ is equivalent to $\knd[2]$} + \begin{mathpar} + \inferrule{ }{\KindEquiv{\knd}{\knd}} + \and + \inferrule{\KindEquiv{\knd[2]}{\knd[1]}}{\KindEquiv{\knd[1]}{\knd[2]}} + \and + \inferrule{\KindEquiv{\knd[1]}{\knd[3]} \\ \KindEquiv{\knd[3]}{\knd[2]}}{\KindEquiv{\knd[1]}{\knd[2]}} + \and + \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{\KindEquiv{\SKind[\knd][\typ[1]]}{\SKind[\knd][\typ[2]]}} + \and + \inferrule{\WellFormedAtKind{\typ}[\SKind[\knd][\typ[1]]]}{\KindEquiv{\SKind[\SKind[\knd][\typ[1]]][\typ]}{\SKind[\knd][\typ[1]]}} + \and + \inferrule{\WellFormedAtKind{\typ}[\DepFunKind]}{\KindEquiv{\SKind[\DepFunKind][\typ]}{\DepFunKind[][\knd[1]][\SKind[\knd[2]][\TypeAp[\typ][\typvar]]]}} + \end{mathpar} + \begin{mathpar} + \mprset{myfraction=\inferruledotfrac} + \end{mathpar} + \end{minipage} +\subsection*{} + \begin{minipage}{\textwidth} + \judgbox{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{$\typ[1]$ is equivalent to $\typ[2]$ at kind $\knd$} + \inferrule{\WellFormedAtKind{\typ}}{\TypeEquivAtKind{\typ}{\typ}} + \end{minipage} +\subsection*{} + \begin{minipage}{\textwidth} + \judgbox{\TypeEquiv{\typ[1]}{\typ[2]}}{$\typ[1]$ is equivalent to $\typ[2]$ at ``top" kind} + \inferrule{\TypeEquivAtKind{\typ[1]}{\typ[2]}}{\TypeEquiv{\typ[1]}{\typ[2]}} + \end{minipage} \end{document} From 247cd7984471575dbad697b48dac5ea26b589191 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Tue, 13 Jul 2021 03:23:17 +0000 Subject: [PATCH 023/181] ditto --- 9-type-aliases-redux/latex/hejohns-hazel.sty | 4 +- .../latex/kind-judgements.pdf | Bin 148553 -> 151308 bytes .../latex/kind-judgements.tex | 36 +++++++++++++++--- 3 files changed, 33 insertions(+), 7 deletions(-) diff --git a/9-type-aliases-redux/latex/hejohns-hazel.sty b/9-type-aliases-redux/latex/hejohns-hazel.sty index 7db2283..8da65f2 100644 --- a/9-type-aliases-redux/latex/hejohns-hazel.sty +++ b/9-type-aliases-redux/latex/hejohns-hazel.sty @@ -178,11 +178,11 @@ % types \newcommand*{\TypeEquivOp}{\equiv} \newcommand*{\TypeEquivAtKindOp}[1][\knd]{\stackrel{#1}{\TypeEquivOp}} - \NewDocumentCommand{\TypeEquivAtKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mmO{\knd}}{\mathrm{#1 \vdash #2 \TypeEquivAtKindOp[#4] #3}} + \NewDocumentCommand{\TypeEquivAtKind}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mmO{\knd}}{\mathrm{#1 \vdash #2 \TypeEquivAtKindOp[{#4}] #3}} \newcommand*{\TypeEquiv}[2]{\TypeEquivAtKind{#1}{#2}[]} 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.../latex/kind-judgements.tex | 27 +++++++++++++++++- 3 files changed, 28 insertions(+), 2 deletions(-) diff --git a/9-type-aliases-redux/latex/hejohns-hazel.sty b/9-type-aliases-redux/latex/hejohns-hazel.sty index 8da65f2..55c16a4 100644 --- a/9-type-aliases-redux/latex/hejohns-hazel.sty +++ b/9-type-aliases-redux/latex/hejohns-hazel.sty @@ -135,6 +135,7 @@ \newcommand*{\type}[2]{\mathsf{type(}#1, #2\mathsf{)}} \newcommand*{\submodule}[1]{\mathsf{submodule(}#1\mathsf{)}} \renewcommand*{\hole}[1][]{\blackrm{u#1}} + \newcommand*{\subst}[3]{\mathsf{[#1/#2]#3}} % https://tex.stackexchange.com/questions/128991/frac-or-inference-rule-with-dotted-line % for mathpartir's inferrule @@ -173,7 +174,7 @@ \newcommand*{\KindEquivOp}{\equiv} \NewDocumentCommand{\KindEquiv}{O{\HoleCtx\TypVarCtx\ModVarCtx\SigVarCtx[]}mm}{\mathrm{#1 \vdash #2 \KindEquivOp #3}} - \newcommand*{\MatchedPiKindOp}{\stackrel{\blacktriangleright}{\Pi}} + \newcommand*{\MatchedPiKindOp}{\stackrel{\blacktriangleright}{{}_{\Pi}}} 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ztaE^bO}jimC>$U!zWD*RSy0Ge)9_0R$`319Guu#FPZhQy4ew#c1#5{|t3#T~!o#$fBjoIOx&vS12FVPhu?+5#?9J>!PmuFh#4jCMY}BXmwi-E^4V zLwE)?Adj1!KFCiu09eBo27?7#J1N*$I(Q8!1b`2E{*6n}@&lfQY(~C;!ESUJBpAf#3z(4SS$_Gy=OpJfGy8M1 zD2TxW=6{9#iyN;cq+mj?g* Zv42fSEFPD3;{qQFIWjN`B_%~qMhbQdegFUf diff --git a/9-type-aliases-redux/latex/kind-judgements.tex b/9-type-aliases-redux/latex/kind-judgements.tex index ac09ef2..842ebb1 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -20,7 +20,7 @@ \newcommand{\kindAna}[3]{#1 \vdash #2 \Leftarrow #3} \newcommand{\kformation}[2]{#1 \vdash #2 \textt{ kind} } \newcommand{\kequiv}[3]{#1 \vdash #2 \equiv #3} -\newcommand{\kcequiv}[4]{#1 \vdash #2 \equiv #3 : #4} +\newcommand{\kcequiv}[3]{#1 \vdash #2 \equiv #3} \newcommand{\ksubkind}[3]{#1 \vdash #2 <:: #3} \newcommand{\tyvarValid}[1]{#1 \textt{ valid} } \newcommand{\tyvarNotValid}[1]{\lnot(#1 \textt{ }\textt{valid}) } @@ -165,7 +165,7 @@ } \and \inferrule[KESingEquiv]{ - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\Ty} + \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} } { \kequiv{\Delta;\hPhi}{\KSing{\dtau_1}}{\KSing{\dtau_2}} } @@ -251,62 +251,62 @@ \begin{minipage}{\linewidth} \judgbox - {\kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\hkappa}} - {$\dtau_1$ is equivalent to $\dtau_2$ and has kind $\hkappa_2$} \;\\ + {\kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}} + {$\dtau_1$ is equivalent to $\dtau_2$} \;\\ \begin{mathpar} \inferrule[KCESymm]{ - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\hkappa} + \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} } { - \kcequiv{\Delta;\hPhi}{\dtau_2}{\dtau_1}{\hkappa} + \kcequiv{\Delta;\hPhi}{\dtau_2}{\dtau_1} } \and \inferrule[KCETrans]{ - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\hkappa} \\ - \kcequiv{\Delta;\hPhi}{\dtau_2}{\dtau_3}{\hkappa} + \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} \\ + \kcequiv{\Delta;\hPhi}{\dtau_2}{\dtau_3} } { - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_3}{\hkappa} + \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_3} } \and \inferrule[KCESingEquiv]{ \kindAna{\Delta;\hPhi}{\dtau_1}{\KSing{\dtau_2}} } { - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\Ty} + \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} } \and \inferrule[KCEConst]{ }{ - \kcequiv{\Delta;\hPhi}{c}{c}{\Ty} + \kcequiv{\Delta;\hPhi}{c}{c} } \and \inferrule[KCEVar]{ t : \hkappa \in \hPhi } { - \kcequiv{\Delta;\hPhi}{t}{t}{\hkappa} + \kcequiv{\Delta;\hPhi}{t}{t} } \and \inferrule[KCEBinOp]{ - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\Ty} \\ - \kcequiv{\Delta;\hPhi}{\dtau_3}{\dtau_4}{\Ty} + \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} \\ + \kcequiv{\Delta;\hPhi}{\dtau_3}{\dtau_4} } { - \kcequiv{\Delta;\hPhi}{\dtau_1 \oplus \dtau_3}{\dtau_2 \oplus \dtau_4}{\Ty} + \kcequiv{\Delta;\hPhi}{\dtau_1 \oplus \dtau_3}{\dtau_2 \oplus \dtau_4} } \and \inferrule[KCEList]{ - \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2}{\Ty} + \kcequiv{\Delta;\hPhi}{\dtau_1}{\dtau_2} } { - \kcequiv{\Delta;\hPhi}{\hlist{\dtau_1}}{\hlist{\dtau_2}}{\Ty} + \kcequiv{\Delta;\hPhi}{\hlist{\dtau_1}}{\hlist{\dtau_2}} } \and \inferrule[KCEEHole]{ \Dbinding{u}{\hkappa} \in \Delta } { - \kcequiv{\Delta;\hPhi}{\hthole{u}}{\hthole{u}}{\hkappa} + \kcequiv{\Delta;\hPhi}{\hthole{u}}{\hthole{u}} } \and \inferrule[KCENEHole]{ \Dbinding{u}{\hkappa} \in \Delta \\ \kindAna{\Delta;\hPhi}{\dtau}{\hkappa'} } { - \kcequiv{\Delta;\hPhi}{\hhole{\dtau}{u}}{\hhole{\dtau}{u}}{\hkappa} + \kcequiv{\Delta;\hPhi}{\hhole{\dtau}{u}}{\hhole{\dtau}{u}} } \end{mathpar} \end{minipage} From 4410cd565ce717707e580e44f64868d3175fe2a6 Mon Sep 17 00:00:00 2001 From: Brandon Kase Date: Fri, 21 May 2021 00:36:52 +0200 Subject: [PATCH 019/181] Synthesize self for tvars --- .../latex/kind-judgements.pdf | Bin 189343 -> 189360 bytes .../latex/kind-judgements.tex | 4 ++-- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/9-type-aliases-redux/latex/kind-judgements.pdf b/9-type-aliases-redux/latex/kind-judgements.pdf index 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zksIq~@dG*x4s@(wsZ!N?3~C$&BUc_$w8TCM&@6SttKX~kr>D88YJjA;V9I?ySL@&~ z!AZbNU_c0_+!d~JHle6*-D1wbNs%rNN5}bHPG{DQe2^(669*d=)G`gFS^b9M>N=;Q zJ?d`^h)weVRNUsQY5h+z-0Nx@h+*TcyD=Ove2XqSK!~mh@bBmmfE|S)OGs8&@0r%; zFgir=`40#60hxpJ9EbEA0k`xV0#L;QG$4~5wZ?{iW~g>*!}@$ VAubk|hvNbu2{1DXB_%~qMhaaK8cqNJ diff --git a/9-type-aliases-redux/latex/kind-judgements.tex b/9-type-aliases-redux/latex/kind-judgements.tex index 842ebb1..cbe5478 100644 --- a/9-type-aliases-redux/latex/kind-judgements.tex +++ b/9-type-aliases-redux/latex/kind-judgements.tex @@ -186,7 +186,7 @@ \inferrule[KSVar]{ t : \hkappa \in \hPhi } { - \kindSyn{\Delta;\hPhi}{t}{\hkappa} + \kindSyn{\Delta;\hPhi}{t}{\KSing{t}} } \and \inferrule[KSUVar]{ @@ -338,7 +338,7 @@ \inferrule[TElabSVar]{ t : \hkappa \in \hPhi } { - \elabSyn{\hPhi}{t}{\hkappa}{t}{\EmptyDelta} + \elabSyn{\hPhi}{t}{\KSing{t}}{t}{\EmptyDelta} } \and \inferrule[TElabSUVar]{ From 8f533393dd79958ba639d7253403f6e6604633d2 Mon Sep 17 00:00:00 2001 From: Johnson He Date: Tue, 15 Jun 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