2 \usepackage[sc,slantedGreek]{mathpazo}
5 % Comment these out if you don't want a slide with just the
6 % part/section/subsection/subsubsection title:
7 \AtBeginPart{\frame{\partpage}}
8 \AtBeginSection{\frame{\sectionpage}}
9 \AtBeginSubsection{\frame{\subsectionpage}}
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11 \beamertemplatenavigationsymbolsempty
13 \usepackage{centernot}
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17 \setlength{\parskip}{6pt plus 2pt minus 1pt}
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23 \renewcommand{\AgdaKeywordFontStyle}[1]{\ensuremath{\mathrm{\underline{#1}}}}
24 \renewcommand{\AgdaFunction}[1]{\textbf{\textcolor{AgdaFunction}{#1}}}
25 \renewcommand{\AgdaField}{\AgdaFunction}
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34 \DeclareUnicodeCharacter{9655}{\ensuremath{\rhd}}
36 \newcommand{\mysmall}{}
37 \newcommand{\mysyn}{\AgdaKeyword}
38 \newcommand{\mytyc}[1]{\textup{\AgdaDatatype{#1}}}
39 \newcommand{\mydc}[1]{\textup{\AgdaInductiveConstructor{#1}}}
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42 \newcommand{\myb}[1]{\AgdaBound{$#1$}}
43 \newcommand{\myfield}{\AgdaField}
44 \newcommand{\myind}{\AgdaIndent}
45 \newcommand{\mykant}{{\rmfamily\scshape Bertus}}
46 \newcommand{\mysynel}[1]{#1}
47 \newcommand{\myse}{\mysynel}
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55 \newcommand{\mycumul}{\preceq}
57 \newcommand{\mydesc}[3]{
63 \hfill \textup{\phantom{ygp}\textbf{#1}} $#2$
64 \framebox[\textwidth]{
79 \newcommand{\mytmt}{\mysynel{t}}
80 \newcommand{\mytmm}{\mysynel{m}}
81 \newcommand{\mytmn}{\mysynel{n}}
82 \newcommand{\myred}{\leadsto}
83 \newcommand{\myredd}{\stackrel{*}{\leadsto}}
84 \newcommand{\myreddd}{\stackrel{*}{\reflectbox{$\leadsto$}}}
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96 \newcommand{\myjudd}[3]{#1 \vdash #2 : #3}
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100 \newcommand{\myunit}{\mytyc{Unit}}
101 \newcommand{\mypair}[2]{\mathopen{\mydc{$\langle$}}#1\mathpunct{\mydc{,}} #2\mathclose{\mydc{$\rangle$}}}
102 \newcommand{\myfst}{\myfld{fst}}
103 \newcommand{\mysnd}{\myfld{snd}}
104 \newcommand{\myconst}{\myse{c}}
105 \newcommand{\myemptyctx}{\varepsilon}
106 \newcommand{\myhole}{\AgdaHole}
107 \newcommand{\myfix}[3]{\mysyn{fix} \myappsp #1 {:} #2 \mapsto #3}
108 \newcommand{\mysum}{\mathbin{\textcolor{AgdaDatatype}{+}}}
109 \newcommand{\myleft}[1]{\mydc{left}_{#1}}
110 \newcommand{\myright}[1]{\mydc{right}_{#1}}
111 \newcommand{\myempty}{\mytyc{Empty}}
112 \newcommand{\mycase}[2]{\mathopen{\myfun{[}}#1\mathpunct{\myfun{,}} #2 \mathclose{\myfun{]}}}
113 \newcommand{\myabsurd}[1]{\myfun{absurd}_{#1}}
114 \newcommand{\myarg}{\_}
115 \newcommand{\myderivsp}{}
116 \newcommand{\myderivspp}{\vspace{0.3cm}}
117 \newcommand{\mytyp}{\mytyc{Type}}
118 \newcommand{\myneg}{\myfun{$\neg$}}
119 \newcommand{\myar}{\,}
120 \newcommand{\mybool}{\mytyc{Bool}}
121 \newcommand{\mytrue}{\mydc{true}}
122 \newcommand{\myfalse}{\mydc{false}}
123 \newcommand{\myitee}[5]{\myfun{if}\,#1 / {#2.#3}\,\myfun{then}\,#4\,\myfun{else}\,#5}
124 \newcommand{\mynat}{\mytyc{$\mathbb{N}$}}
125 \newcommand{\myrat}{\mytyc{$\mathbb{R}$}}
126 \newcommand{\myite}[3]{\myfun{if}\,#1\,\myfun{then}\,#2\,\myfun{else}\,#3}
127 \newcommand{\myfora}[3]{(#1 {:} #2) \myarr #3}
128 \newcommand{\myexi}[3]{(#1 {:} #2) \myprod #3}
129 \newcommand{\mypairr}[4]{\mathopen{\mydc{$\langle$}}#1\mathpunct{\mydc{,}} #4\mathclose{\mydc{$\rangle$}}_{#2{.}#3}}
130 \newcommand{\mynil}{\mydc{[]}}
131 \newcommand{\mycons}{\mathbin{\mydc{∷}}}
132 \newcommand{\myfoldr}{\myfun{foldr}}
133 \newcommand{\myw}[3]{\myapp{\myapp{\mytyc{W}}{(#1 {:} #2)}}{#3}}
134 \newcommand{\mynodee}{\mathbin{\mydc{$\lhd$}}}
135 \newcommand{\mynode}[2]{\mynodee_{#1.#2}}
136 \newcommand{\myrec}[4]{\myfun{rec}\,#1 / {#2.#3}\,\myfun{with}\,#4}
137 \newcommand{\mylub}{\sqcup}
138 \newcommand{\mydefeq}{\cong}
139 \newcommand{\myrefl}{\mydc{refl}}
140 \newcommand{\mypeq}{\mytyc{=}}
141 \newcommand{\myjeqq}{\myfun{$=$-elim}}
142 \newcommand{\myjeq}[3]{\myapp{\myapp{\myapp{\myjeqq}{#1}}{#2}}{#3}}
143 \newcommand{\mysubst}{\myfun{subst}}
144 \newcommand{\myprsyn}{\myse{prop}}
145 \newcommand{\myprdec}[1]{\mathopen{\mytyc{$\llbracket$}} #1 \mathclose{\mytyc{$\rrbracket$}}}
146 \newcommand{\myand}{\mathrel{\mytyc{$\wedge$}}}
147 \newcommand{\mybigand}{\mathrel{\mytyc{$\bigwedge$}}}
148 \newcommand{\myprfora}[3]{\forall #1 {:} #2.\, #3}
149 \newcommand{\myimpl}{\mathrel{\mytyc{$\Rightarrow$}}}
150 \newcommand{\mybot}{\mytyc{$\bot$}}
151 \newcommand{\mytop}{\mytyc{$\top$}}
152 \newcommand{\mycoe}{\myfun{coe}}
153 \newcommand{\mycoee}[4]{\myapp{\myapp{\myapp{\myapp{\mycoe}{#1}}{#2}}{#3}}{#4}}
154 \newcommand{\mycoh}{\myfun{coh}}
155 \newcommand{\mycohh}[4]{\myapp{\myapp{\myapp{\myapp{\mycoh}{#1}}{#2}}{#3}}{#4}}
156 \newcommand{\myjm}[4]{(#1 {:} #2) \mathrel{\mytyc{=}} (#3 {:} #4)}
157 \newcommand{\myeq}{\mathrel{\mytyc{=}}}
158 \newcommand{\myprop}{\mytyc{Prop}}
159 \newcommand{\mytmup}{\mytmsyn\uparrow}
160 \newcommand{\mydefs}{\Delta}
161 \newcommand{\mynf}{\Downarrow}
162 \newcommand{\myinff}[3]{#1 \vdash #2 \Uparrow #3}
163 \newcommand{\myinf}[2]{\myinff{\myctx}{#1}{#2}}
164 \newcommand{\mychkk}[3]{#1 \vdash #2 \Downarrow #3}
165 \newcommand{\mychk}[2]{\mychkk{\myctx}{#1}{#2}}
166 \newcommand{\myann}[2]{#1 : #2}
167 \newcommand{\mydeclsyn}{\myse{decl}}
168 \newcommand{\myval}[3]{#1 : #2 \mapsto #3}
169 \newcommand{\mypost}[2]{\mysyn{abstract}\ #1 : #2}
170 \newcommand{\myadt}[4]{\mysyn{data}\ #1 #2\ \mysyn{where}\ #3\{ #4 \}}
171 \newcommand{\myreco}[4]{\mysyn{record}\ #1 #2\ \mysyn{where}\ #3\{ #4 \}}
172 \newcommand{\myelabt}{\vdash}
173 \newcommand{\myelabf}{\rhd}
174 \newcommand{\myelab}[2]{\myctx \myelabt #1 \myelabf #2}
175 \newcommand{\mytele}{\Delta}
176 \newcommand{\mytelee}{\delta}
177 \newcommand{\mydcctx}{\Gamma}
178 \newcommand{\mynamesyn}{\myse{name}}
179 \newcommand{\myvec}{\overrightarrow}
180 \newcommand{\mymeta}{\textsc}
181 \newcommand{\myhyps}{\mymeta{hyps}}
182 \newcommand{\mycc}{;}
183 \newcommand{\myemptytele}{\varepsilon}
184 \newcommand{\mymetagoes}{\Longrightarrow}
185 % \newcommand{\mytesctx}{\
186 \newcommand{\mytelesyn}{\myse{telescope}}
187 \newcommand{\myrecs}{\mymeta{recs}}
188 \newcommand{\myle}{\mathrel{\lcfun{$\le$}}}
189 \newcommand{\mylet}{\mysyn{let}}
190 \newcommand{\myhead}{\mymeta{head}}
191 \newcommand{\mytake}{\mymeta{take}}
192 \newcommand{\myix}{\mymeta{ix}}
193 \newcommand{\myapply}{\mymeta{apply}}
194 \newcommand{\mydataty}{\mymeta{datatype}}
195 \newcommand{\myisreco}{\mymeta{record}}
196 \newcommand{\mydcsep}{\ |\ }
197 \newcommand{\mytree}{\mytyc{Tree}}
198 \newcommand{\myproj}[1]{\myfun{$\pi_{#1}$}}
199 \newcommand{\mysigma}{\mytyc{$\Sigma$}}
200 \newcommand{\mynegder}{\vspace{-0.3cm}}
201 \newcommand{\myquot}{\Uparrow}
202 \newcommand{\mynquot}{\, \Downarrow}
203 \newcommand{\mycanquot}{\ensuremath{\textsc{quote}{\Downarrow}}}
204 \newcommand{\myneuquot}{\ensuremath{\textsc{quote}{\Uparrow}}}
205 \newcommand{\mymetaguard}{\ |\ }
206 \newcommand{\mybox}{\Box}
207 \newcommand{\mytermi}[1]{\text{\texttt{#1}}}
208 \newcommand{\mysee}[1]{\langle\myse{#1}\rangle}
209 \newcommand{\mycomp}{\mathbin{\myfun{$\circ$}}}
210 \newcommand{\mylist}[1]{\mytyc{List}\myappsp #1}
211 \newcommand{\mylistt}[1]{\mathopen{\mydc{$[$}} #1 \mathclose{\mydc{$]$}}}
212 \newcommand{\myplus}{\mathbin{\myfun{$+$}}}
213 \newcommand{\mytimes}{\mathbin{\myfun{$*$}}}
214 \newcommand{\mysuc}{\mydc{suc}}
215 \newcommand{\myzero}{\mydc{zero}}
217 \renewcommand{\[}{\begin{equation*}}
218 \renewcommand{\]}{\end{equation*}}
219 \newcommand{\mymacol}[2]{\text{\textcolor{#1}{$#2$}}}
221 \title{\mykant: Implementing Observational Equality}
222 \author{Francesco Mazzoli \texttt{<fm2209@ic.ac.uk>}}
229 \frametitle{\mykant?}
231 \mykant\ is an \emph{interactive theorem prover}, implemented in
234 It is similar in scope to Agda or Coq, but with a more powerful notion
237 We figured out its theory, but do not have a complete
238 implementation---although most of the work is done.
242 \frametitle{Theorem provers are short-sighted}
244 Two functions dear to the Haskell practitioner:
247 \myfun{map} : (\myb{a} \myarr \myb{b}) \myarr \mylist{\myb{a}} \myarr \mylist{\myb{b}} \\
248 \begin{array}{@{}l@{\myappsp}c@{\myappsp}c@{\ }c@{\ }l}
249 \myfun{map} & \myb{f} & \mynil & = & \mynil \\
250 \myfun{map} & \myb{f} & (\myb{x} \mycons \myb{xs}) & = & \myapp{\myb{f}}{\myb{x}} \mycons \myfun{map} \myappsp \myb{f} \myappsp \myb{xs} \\
254 (\myfun{${\circ}$}) : (\myb{b} \myarr \myb{c}) \myarr (\myb{a} \myarr \myb{b}) \myarr (\myb{a} \myarr \myb{c}) \\
255 (\myb{f} \mathbin{\myfun{$\circ$}} \myb{g}) \myappsp \myb{x} = \myapp{\myb{g}}{(\myapp{\myb{f}}{\myb{x}})}
261 \frametitle{Theorem provers are short-sighted}
262 $\myfun{map}$'s composition law states that:
264 \forall \myb{f} {:} (\myb{b} \myarr \myb{c}), \myb{g} {:} (\myb{a} \myarr \myb{b}). \myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g} \myeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g})
266 We can convince Coq or Agda that
268 \forall \myb{f} {:} (\myb{b} \myarr \myb{c}), \myb{g} {:} (\myb{a} \myarr \myb{b}), \myb{l} {:} \mylist{\myb{a}}. (\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g}) \myappsp \myb{l} \myeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) \myappsp \myb{l}
270 But we cannot get rid of the $\myb{l}$. Why?
274 \frametitle{\mykant\ and observational equality}
276 \emph{Observational} equality solves this and other annoyances.
278 \mykant\ is a system aiming at making observational equality more
283 \frametitle{Theorem provers, dependent types}
286 types $\leftrightarrow$ propositions
288 programs $\leftrightarrow$ proofs
293 \frametitle{Theorem provers, dependent types} First class types: we
294 can return them, have them as arguments, etc.
296 \begin{array}{@{}l@{\ }l@{\ \ \ }l}
297 \mysyn{data}\ \myempty & & \text{No members.} \\
298 \mysyn{data}\ \myunit & = \mytt & \text{One member.} \\
299 \mysyn{data}\ \mynat & = \mydc{zero} \mydcsep \mydc{suc}\myappsp\mynat & \text{Natural numbers.}
302 $\myempty : \mytyp$, $\myunit : \mytyp$, $\mynat : \mytyp$.
304 $\myunit$ is trivially inhabitable: it corresponds to $\top$ in
307 $\myempty$ is \emph{not} inhabitable: it corresponds to $\bot$.
309 \myfun{absurd} : \myempty \myarr \myb{A}
314 \frametitle{Theorem provers, dependent types}
315 % TODO examples first
317 We can express relations:
320 (\myfun{${>}$}) : \mynat \myarr \mynat \myarr \mytyp \\
321 \begin{array}{@{}c@{\,}c@{\,}c@{\ }l}
322 \mydc{zero} & \mathrel{\myfun{$>$}} & \myb{m} & = \myempty \\
323 \myb{n} & \mathrel{\myfun{$>$}} & \mydc{zero} & = \myunit \\
324 (\mydc{suc} \myappsp \myb{n}) & \mathrel{\myfun{$>$}} & (\mydc{suc} \myappsp \myb{m}) & = \myb{n} \mathrel{\myfun{$>$}} \myb{m}
329 A term of type $\myb{m} \mathrel{\myfun{$>$}} \myb{n}$ represents a
330 \emph{proof} that $\myb{m}$ is indeed greater than $\myb{n}$.
333 3 \mathrel{\myfun{$>$}} 1 \myred \myunit \\
334 2 \mathrel{\myfun{$>$}} 2 \myred \myempty \\
335 0 \mathrel{\myfun{$>$}} \myb{m} \myred \myempty
341 \frametitle{Example: safe $\myfun{head}$ function}
345 \mysyn{data}\ \mylist{\myb{A}} = \mynil \mydcsep \myb{A} \mycons \mylist{\myb{A}} \\
347 \myfun{length} : \mylistt{\myb{A}} \myarr \mynat \\
348 \begin{array}{@{}l@{\myappsp}c@{\ }c@{\ }l}
349 \myfun{length} & \mynil & \mapsto & \mydc{zero} \\
350 \myfun{length} & (\myb{x} \mycons \myb{xs}) & \mapsto & \mydc{suc} \myappsp (\myfun{length} \myappsp \myb{xs})
353 \myfun{head} : \myfora{\myb{l}}{\mytyc{List}\myappsp\myb{A}}{ \myfun{length}\myappsp\myb{l} \mathrel{\myfun{$>$}} 0 \myarr \myb{A}} \\
354 \begin{array}{@{}l@{\myappsp}c@{\myappsp}c@{\ }c@{\ }l}
355 \myfun{head} & \mynil & \myb{p} & \mapsto & \only<1,2>{\myhole{?}}\only<3>{\myabsurd\myappsp\myb{p}} \\
356 \myfun{head} & (\myb{x} \mycons \myb{xs}) & \myb{p} & \mapsto & \myb{x}
362 The logic equivalent would be
364 \forall \myb{l} {:} \mylist{\myb{A}}.\ \myfun{length}\myappsp\myb{l} \mathrel{\myfun{$>$}} 0 \myarr \myb{A}
366 `For all non-empty lists of type $\myb{A}$, we can get an element of $\myb{A}$.'
369 The type of $\myb{p}$ in the $\myhole{?}$ is $\myempty$, since
370 \[\myfun{length} \myappsp \mynil \mathrel{\myfun{$>$}} 0 \myred 0 \mathrel{\myfun{$>$}} 0 \myred \myempty \]}
373 \[ \myfun{absurd} : \myempty \myarr \myb{A} \]
378 \frametitle{How do we type check this thing?}
380 \myfun{head} \myappsp \mylistt{3} : \myfun{length} \myappsp \mylistt{3} \mathrel{\myfun{$>$}} 0 \myarr \mynat
383 \[ \mytt : \myfun{length} \myappsp \mylistt{3} \mathrel{\myfun{$>$}} 0 \]
385 \[ \myfun{head} \myappsp \mylistt{3} : \myunit \myarr \mynat \]
387 Yes: to typecheck, we reduce terms fully before comparing:
389 \begin{array}{@{}r@{\ }c@{\ }c@{\ }c@{\ }c@{\ }c@{\ }l}
390 \myunit & \myredd & \myunit & \mydefeq & \myunit & \myreddd & \myfun{length} \myappsp \mylistt{3} \mathrel{\myfun{$>$}} 0 \\
391 \myfun{length} \myappsp \mynil \mathrel{\myfun{$>$}} 0 & \myredd & \myempty & \mydefeq & \myempty & \myreddd & \myempty \\
392 (\myabs{\myb{x}\, \myb{y}}{\myb{y}}) \myappsp \myunit \myappsp \myappsp \mynat & \myredd & \mynat & \mydefeq & \mynat & \myreddd & (\myabs{\myb{x}\, \myb{y}}{\myb{x}}) \myappsp \mynat \myappsp \myunit \\
397 \mydefeq\ \text{takes the name of \textbf{definitional equality}.}
402 \frametitle{Propositional equality} Using definitional equality, we
403 can give the user a type-level notion of term equality.
405 (\myeq) : \myb{A} \myarr \myb{A} \myarr \mytyp\ \ \ \text{internalises equality \textbf{as a type}}
407 We introduce members of $\myeq$ by reflexivity, for example
409 \myrefl\myappsp5 : 5 \myeq 5
414 \myrefl\myappsp 5 : (3 + 2) \myeq (1 + 4)\text{, since}\\
415 (3 + 2) \myeq (1 + 4) \myredd 5 \myeq 5
418 Then we can use a substitution law to derive other
419 laws---transitivity, congruence, etc.
420 \[ \myeq\ \text{takes the name of \textbf{propositional equality}} \]
424 \frametitle{The problem with prop. equality}
425 Going back to $\myfun{map}$, we can prove that
426 \[ \forall \myb{f} {:} (\myb{b} \myarr \myb{c}), \myb{g} {:} (\myb{a} \myarr \myb{b}), \myb{l} {:} \mylist{\myb{a}}.\ (\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g}) \myappsp \myb{l} \myeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) \myappsp \myb{l} \]
429 (\myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g})\myappsp \myb{l} \mydefeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g}) \myappsp \myb{l}
431 By induction on $\myb{l}$.
433 Without the $\myb{l}$ we cannot compute, so we are stuck with
435 \myfun{map}\myappsp \myb{f} \mycomp \myfun{map}\myappsp \myb{g} \not\mydefeq \myfun{map}\myappsp (\myb{f} \mycomp \myb{g})
440 \frametitle{Observational equality}
442 Instead of basing its equality on definitional equality, look at the
443 structure of the type to decide.
445 For functions this will mean that proving
447 \myfun{map}\myappsp \myse{f} \mycomp \myfun{map}\myappsp \myse{g} \myeq \myfun{map}\myappsp (\myse{f} \mycomp \myse{g})
449 Will reduce to proving that
451 (\myb{l} : \mylist{\myb{A}}) \myarr
452 (\myfun{map}\myappsp \myse{f} \mycomp \myfun{map}\myappsp \myse{g})\myappsp\myb{l} \myeq \myfun{map}\myappsp (\myse{f} \mycomp \myse{g})\myappsp \myb{l}
454 Which is what we want. The interesting part is how to make the system
457 This extends to other structures (tuples, inductive types, \dots).
458 Moreover, if we can deem two \emph{types} equal, we can \emph{coerce}
459 values from one to the other.
464 {\huge \mykant' features}
469 \frametitle{Inductive data}
471 \mysyn{data}\ \mytyc{List}\myappsp (\myb{A} : \mytyp) \mapsto \mynil \mydcsep \myb{A} \mycons \mytyc{List}\myappsp\myb{A}
473 Each with an induction principle:
475 \begin{array}{@{}l@{\ }l}
476 \mytyc{List}.\myfun{elim} : & \AgdaComment{-{}- The property that we want to prove:} \\
477 & (\myb{P} : \mytyc{List}\myappsp\myb{A} \myarr \mytyp) \myarr \\
478 & \AgdaComment{-{}- If it holds for the base case:} \\
479 & \myb{P} \myappsp \mynil \myarr \\
480 & \AgdaComment{-{}- And for the inductive step:} \\
481 & ((\myb{x} : \myb{A}) \myarr (\myb{l} : \mytyc{List}\myappsp \myb{A}) \myarr \myb{P} \myappsp \myb{l} \myarr \myb{P} \myappsp (\myb{x} \mycons \myb{l})) \myarr \\
482 & \AgdaComment{-{}- Then it holds for every list:} \\
483 & (\myb{l} : \mytyc{List}\myappsp\myb{A}) \myarr \myb{P} \myappsp \myb{l}
486 Induction is also computation, via structural recursion:
488 \begin{array}{@{}l@{\ }l}
489 \mytyc{List}.\myfun{elim} \myappsp \myse{P} \myappsp \myse{pn} \myappsp \myse{pc} \myappsp \mynil & \myred \myse{pn} \\
490 \mytyc{List}.\myfun{elim} \myappsp \myse{P} \myappsp \myse{pn} \myappsp \myse{pc} \myappsp (\mytmm \mycons \mytmn) & \myred \myse{pc} \myappsp \mytmm \myappsp \mytmn \myappsp (\mytyc{List}.\myfun{elim} \myappsp \myse{P} \myappsp \myse{pn} \myappsp \myse{ps} \myappsp \mytmt )
496 \frametitle{Dependent defined types} \emph{Unlike} Haskell, we can
497 have fields of a data constructor to depend on earlier fields:
500 \mysyn{record}\ \mytyc{Tuple}\myappsp(\myb{A} : \mytyp)\myappsp\myhole{$(\myb{B} : \myb{A} \myarr \mytyp)$} \mapsto \\
501 \myind{2}\mydc{tuple}\ \{ \myfun{fst} : \myb{A}, \myfun{snd} : \myb{B}\myappsp\myb{fst} \}
505 The \emph{type} of the second element depends on the \emph{value} of
508 \begin{array}{@{}l@{\ }l}
509 \myfun{fst} & : \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B} \myarr \myb{A} \\
510 \myfun{snd} & : (\myb{x} : \mytyc{Tuple}\myappsp\myb{A}\myappsp\myb{B}) \myarr \myb{B} \myappsp (\myfun{fst} \myappsp \myb{x})
513 Where the projection's reduction rules are predictably
515 \begin{array}{@{}l@{\ }l}
516 \myfun{fst}\myappsp&(\mydc{tuple}\myappsp\mytmm\myappsp\mytmn) \myred \mytmm \\
517 \myfun{snd}\myappsp&(\mydc{tuple}\myappsp\mytmm\myappsp\mytmn) \myred \mytmn \\
523 \frametitle{Example: the type of even numbers}
524 For example we can define the type of even numbers:
528 \myfun{even} : \mynat \myarr \mytyp \\
529 \begin{array}{@{}l@{\myappsp}c@{\ }l}
530 \myfun{even} & \myzero & \mapsto \myunit \\
531 \myfun{even} & (\mysuc \myappsp \myzero) & \mapsto \myempty \\
532 \myfun{even} & (\mysuc \myappsp (\mysuc \myappsp \myb{n})) & \mapsto \myfun{even} \myappsp \myb{n}
535 \mytyc{Even} : \mytyp \\
536 \mytyc{Even} \mapsto \mytyc{Tuple}\ \mynat\ \myfun{even}
539 In logic this would be
540 \[ \exists x \in \mathbb{N}.\ even(x) \]
544 \frametitle{Type hierarchy}
545 \[\{\mynat, \mybool, \mytyc{List}\myappsp\mynat, \cdots\} : \mytyp\]
546 What is the type of $\mytyp$?
548 \cancel{\mytyp : \mytyp}\ \ \ \text{\textbf{inconsistent}}
550 Much like na{\"i}ve set theory is (Girard's paradox).
552 Instead, we have a hierarchy:
554 \{\mynat, \mybool, \mytyc{List}\myappsp\mynat, \cdots\} : \mytyp_0 : \mytyp_1 : \cdots
556 We talk of types in $\mytyp_0$ as `smaller' than types in $\mytyp_1$.
560 \frametitle{Cumulativity and typical ambiguity}
562 \[ \mytyp_0 : \mytyp_1 \ \ \ \text{but}\ \ \ \mytyp_0 \centernot{:} \mytyp_2\]
563 We have a cumulative hierarchy, so that
564 \[ \mytyp_n : \mytyp_m \ \ \ \text{iff}\ \ \ n < m \]
566 \[ \mynat : \mytyp_0\ \ \ \text{and}\ \ \ \mynat : \mytyp_1\ \ \ \text{and}\ \ \ \mynat : \mytyp_{50} \]
568 But in \mykant, you can forget about levels: the consistency is
569 checked automatically---a mechanised version of what Russell called
570 \emph{typical ambiguity}.
574 \frametitle{OTT + user defined types}
576 For each type, we need to:
578 \item Describe when two types formed by the defined type constructors
580 \[ \mylist{\mytya_1} \myeq \mylist{\mytya_2} \]
581 \item Describe when two values of the defined type are equal;
583 \begin{array}{@{}c@{\ \ \ \ \ \ }c}
584 \mynil \myeq \mynil & \mynil \myeq \mytmm \mycons \mytmn \\
585 \mytmm \mycons \mytmn \myeq \mynil & \mytmm_1 \mycons \mytmn_1 \myeq \mytmm_2 \mycons \mytmn_2
588 \item Describe how to transport values of the defined type.
593 \frametitle{OTT + hierarchy}
595 Since equalities reduce to functions abstracting over various things,
596 we need to make sure that the hierarchy is respected.
598 For example we have that
601 ((\myb{x_1} {:} \mytya_1) \myarr \mytyb_1 : \mytyp) \myeq ((\myb{x_2} {:} \mytya_2) \myarr \mytyb_2 : \mytyp) \myred \\
602 \myind{2} (\mytya_1 : \mytyp) \myeq (\mytya_2 : \mytyp) \myand \\
603 \myind{2} (\myb{x_1} : \mytya_1) \myarr (\myb{x_2} : \mytya_2) \myarr (\mytyb_1 : \mytyp) \myeq (\mytyb_2 : \mytyp)
607 Subtle point---I discovered a problem in the theory after
608 submitting... but I have a fix.
612 \frametitle{Bidirectional type checking}
614 \mysyn{data}\ \mytyc{List}\myappsp (\myb{A} : \mytyp) \mapsto \mydc{nil} \mydcsep \mydc{cons} \myappsp \myb{A}\myappsp (\mytyc{List}\myappsp\myb{A})
617 With no type inference, we have explicit types for the constructors:
619 \begin{array}{@{}l@{\ }l}
620 \mydc{nil} & : (\myb{A} : \mytyp) \myarr \mytyc{List}\myappsp\myb{A} \\
621 \mydc{cons} &: (\myb{A} : \mytyp) \myarr \myb{A} \myarr \mytyc{List}\myappsp\myb{A} \myarr \mytyc{List}\myappsp\myb{A}\\
626 \mydc{cons}\myappsp \mynat\myappsp 1 \myappsp (\mydc{cons}\myappsp \mynat \myappsp 2 \myappsp (\mydc{cons}\myappsp \mynat \myappsp 3 \myappsp (\mydc{nil}\myappsp \mynat)))
628 Instead, annotate terms and propagate the type:
630 \mydc{cons}\myappsp 1 \myappsp (\mydc{cons}\myappsp 2 \myappsp (\mydc{cons} \myappsp 3 \myappsp \mydc{nil})) : \mytyc{List}\myappsp\mynat
632 Conversely, when we use eliminators the type can be inferred.
636 \frametitle{Bidirectional type checking}
638 This technique is known as \emph{bidirectional} type checking---some
639 terms get \emph{checked}, some terms \emph{infer} types.
641 Usually used for pre-defined types or core calculi, \mykant\ extends
642 to user-defined types.