<div dir="ltr"><div><div>If you want to stick to your Ints datastructure then you need a stronger dependent type theory that allows you to reason about "double-promoted" types.<br><br></div>However in this case you don't really need that complexity:<br><br>data Ints<br> = Zero<br> | Minus Nat<br> | Plus Nat<br><br></div>is sufficient enough. Although this definition is probably not the one you want to use... take a look at <a href="http://www.cse.chalmers.se/~nad/repos/lib/src/Data/Integer.agda">http://www.cse.chalmers.se/~nad/repos/lib/src/Data/Integer.agda</a> for hints<br></div><div class="gmail_extra"><br><div class="gmail_quote">On 14 May 2015 at 18:36, Nicholls, Mark <span dir="ltr"><<a href="mailto:nicholls.mark@vimn.com" target="_blank">nicholls.mark@vimn.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
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<p class="MsoNormal">Hello,<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">I clearly don’t really know what I’m doing…but at least I know it….<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">Here we defined the Naturals…and then attempt to construct the Integers….<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">> {-# LANGUAGE DataKinds #-}<u></u><u></u></p>
<p class="MsoNormal">> {-# LANGUAGE ExplicitForAll #-}<u></u><u></u></p>
<p class="MsoNormal">> {-# LANGUAGE FlexibleContexts #-}<u></u><u></u></p>
<p class="MsoNormal">> {-# LANGUAGE FlexibleInstances #-}<u></u><u></u></p>
<p class="MsoNormal">> {-# LANGUAGE GADTs #-}<u></u><u></u></p>
<p class="MsoNormal">> {-# LANGUAGE MultiParamTypeClasses #-}<u></u><u></u></p>
<p class="MsoNormal">> {-# LANGUAGE PolyKinds #-}<u></u><u></u></p>
<p class="MsoNormal">> {-# LANGUAGE StandaloneDeriving #-}<u></u><u></u></p>
<p class="MsoNormal">> {-# LANGUAGE TypeFamilies #-}<u></u><u></u></p>
<p class="MsoNormal">> {-# LANGUAGE TypeOperators #-}<u></u><u></u></p>
<p class="MsoNormal">> {-# LANGUAGE UndecidableInstances #-}<u></u><u></u></p>
<p class="MsoNormal">> {-# LANGUAGE ScopedTypeVariables #-}<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">> import Prelude hiding (head, tail, (++), (+), replicate)<u></u><u></u></p>
<p class="MsoNormal">> import qualified Prelude as P<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">naturals<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">> data Nat where<u></u><u></u></p>
<p class="MsoNormal">> Z :: Nat<u></u><u></u></p>
<p class="MsoNormal">> S :: Nat -> Nat<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">I can now define + and * and prove things about them…1 * x == x etc….nice..but lets put that on one side.<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">Borrow bits and bobs from singletons<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">> data family Sing (a :: k)<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">Borrow bits and bobs from singletons..i.e. the isomorphic values…my proofs in nat now map to SNat…double nice.<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">> type SNat = (Sing :: Nat -> *)<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">Create the integers by following my nose……(the integers are the equivalence class of pairs of naturals….)<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">i.e. we have “positive” or “negative” or “zero”…<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">> data Ints (a :: Nat) (b :: Nat) where<u></u><u></u></p>
<p class="MsoNormal">> Minus :: SNat a -> Ints 'Z a<u></u><u></u></p>
<p class="MsoNormal">> Zero :: Ints 'Z 'Z<u></u><u></u></p>
<p class="MsoNormal">> Plus :: SNat a -> Ints a 'Z<u></u><u></u></p>
<p class="MsoNormal">> <u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">Ok….this works as a set of values….but….<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">I can’t prove anything about these because the data constructors for my integers aren’t “promotable”…..so I cant do the same trick I did with Nat.<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">“:k Zero” …..<u></u><u></u></p>
<p class="MsoNormal" style="text-autospace:none"><span style="font-family:Consolas;color:#af0000"> Data constructor ‘Zero’ comes from an un-promotable type ‘Ints’<u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-family:Consolas;color:#af0000"> In a type in a GHCi command: Zero</span><u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">I’ve tried rejigging this in various futile and ignorant manners….but the bottom line is its…”un-promotable”…..look at the docs, and it says something vague about GADT’s…but that isnt the problem directly.<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">Is there a nifty way around this log jam? <u></u><u></u></p>
<p class="MsoNormal">(I can start proving things about ‘(Nat,Nat)….but this will soon become a bit clumsy….)<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">Or do I just stop here and wrestle with getting agda installed and lose another few months in the agda-café (which appears to be a very nice place somewhere in finland….hmmmm…I think that’s something different).<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">(I’ve already allegedly descended into cabal hell following my nose with an agda install…<u></u><u></u></p>
<p class="MsoNormal">setup.exe: The program cpphs version >=1.18.6 && <1.19 is required but the<u></u><u></u></p>
<p class="MsoNormal">version found at C:\Users\nichom\AppData\Roaming\cabal\bin\cpphs.exe is<u></u><u></u></p>
<p class="MsoNormal">version 1.19).<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal">computer science would be good, if it wasn’t for the computers.<u></u><u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
<p class="MsoNormal"><u></u> <u></u></p>
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