[Agda] simple question

[Andreas Abel]

Besides the point that induction is not needed and a simple

   eq-succ (refl _) = refl _

suffices, the error you get stems from the fact that you confused proof 
and proposition in the induction step.  You write the proposition

   ((succ i) + m) == (plus (succ i) m)

where you should have written a /proof/ of this proposition.

Cheers,
Andreas

On 14.10.2014 04:13, Andrew Harris wrote:
> Hello,
>
>     I'm trying to follow along in the very good paper "Dependent Types
> At Work" written by Ana Bove and Peter Dybjer.  I'm stuck at trying to
> prove "eq-succ", which is one of the exercises in Section 4.4.  The
> closest I can make it is the following:
>
> {- proof of eq-succ -}
> eq-succ : {n m : Nat} → n == m → succ n == succ m
> eq-succ (refl m) = natrec {(\k → (k + m) == (plus k m))} (refl m) (\i h
> → ((succ i) + m) == (plus (succ i) m)) m
>
> But this does not typecheck, I get the error:
>
> Set !=< (succ i + m) == plus (succ i) m of type Set₁
> when checking that the expression (succ i + m) == plus (succ i) m
> has type (succ i + m) == plus (succ i) m
>
> I'm stuck -- I don't quite understand what I'm supposed to create an
> element of Set_1 that has a signature of (succ i + m) == plus (succ i)
> m, (if that's what I'm supposed to do).  Any hints would be appreciated!
>
> -andrew
>
>
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-- 
Andreas Abel  <><      Du bist der geliebte Mensch.

Department of Computer Science and Engineering
Chalmers and Gothenburg University, Sweden

andreas.abel at gu.se
http://www2.tcs.ifi.lmu.de/~abel/