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<p class="MsoNormal">Hello,<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">I clearly don’t really know what I’m doing…but at least I know it….<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Here we defined the Naturals…and then attempt to construct the Integers….<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> {-# LANGUAGE DataKinds #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE ExplicitForAll #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE FlexibleContexts #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE FlexibleInstances #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE GADTs #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE MultiParamTypeClasses #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE PolyKinds #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE StandaloneDeriving #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE TypeFamilies #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE TypeOperators #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE UndecidableInstances #-}<o:p></o:p></p>
<p class="MsoNormal">> {-# LANGUAGE ScopedTypeVariables #-}<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> import Prelude hiding (head, tail, (++), (+), replicate)<o:p></o:p></p>
<p class="MsoNormal">> import qualified Prelude as P<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">naturals<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> data Nat where<o:p></o:p></p>
<p class="MsoNormal">> Z :: Nat<o:p></o:p></p>
<p class="MsoNormal">> S :: Nat -> Nat<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></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.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Borrow bits and bobs from singletons<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> data family Sing (a :: k)<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></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.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> type SNat = (Sing :: Nat -> *)<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Create the integers by following my nose……(the integers are the equivalence class of pairs of naturals….)<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">i.e. we have “positive” or “negative” or “zero”…<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">> data Ints (a :: Nat) (b :: Nat) where<o:p></o:p></p>
<p class="MsoNormal">> Minus :: SNat a -> Ints 'Z a<o:p></o:p></p>
<p class="MsoNormal">> Zero :: Ints 'Z 'Z<o:p></o:p></p>
<p class="MsoNormal">> Plus :: SNat a -> Ints a 'Z<o:p></o:p></p>
<p class="MsoNormal">> <o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Ok….this works as a set of values….but….<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></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.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">“:k Zero” …..<o:p></o:p></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’<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:Consolas;color:#AF0000"> In a type in a GHCi command: Zero</span><o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></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.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Is there a nifty way around this log jam? <o:p></o:p></p>
<p class="MsoNormal">(I can start proving things about ‘(Nat,Nat)….but this will soon become a bit clumsy….)<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></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).<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">(I’ve already allegedly descended into cabal hell following my nose with an agda install…<o:p></o:p></p>
<p class="MsoNormal">setup.exe: The program cpphs version >=1.18.6 && <1.19 is required but the<o:p></o:p></p>
<p class="MsoNormal">version found at C:\Users\nichom\AppData\Roaming\cabal\bin\cpphs.exe is<o:p></o:p></p>
<p class="MsoNormal">version 1.19).<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">computer science would be good, if it wasn’t for the computers.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
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