[Agda] what is happening here?

[Thorsten Altenkirch]

Ok, but shouldn’t it always work if there is a trivial solution, i.e. the goal matches given exactly (i.e. upto alpha conversion)?

I mean one could always try “exact” – or is this a bad idea?

Thorsteen

From: Ulf Norell <ulf.norell at gmail.com>
Date: Sunday, 8 March 2020 at 18:47
To: Thorsten Altenkirch <psztxa at exmail.nottingham.ac.uk>
Cc: agda-list <agda at lists.chalmers.se>
Subject: Re: [Agda] what is happening here?

Hi Thorsten,

The constraints you get are

  ?l + (?m + ?n) = l + (m + n) : ℕ
  ?l + ?m + ?n = l + m + n : ℕ

You are right that ?l := l, ?m := m, ?n := n is the only solution, but this
requires some non-trivial reasoning. For instance, the first constraint
can be satisfied by

  ?l := 0, ?m := l, ?n := (m + n)

so Agda would need to look at both constraints together to rule out this solution.

/ Ulf

On Sun, Mar 8, 2020 at 7:26 PM Thorsten Altenkirch <Thorsten.Altenkirch at nottingham.ac.uk<mailto:Thorsten.Altenkirch at nottingham.ac.uk>> wrote:
I define

record isMonoid {A : Set}(e : A)(_•_ : A → A → A) : Prp where
  field
    lneutr : {x : A} → e • x ≡ x
    rneutr : {x : A} → x • e ≡ x
    assoc : {x y z : A} → (x • y) • z ≡ x • (y • z)

and I have shown all 3 properties, in particular:

assoc+ : {l m n : ℕ} → (l + m) + n ≡ l + (m + n)

but when I try

isMonoid : isMonoid zero _+_
+isMonoid = record {
  lneutr = lneutr+ ;
  rneutr = rneutr+ ;
  assoc = assoc+ }

I get some unsolved constraints, even though my proof and the goal are alpha-equivalent! Ok I can fix this by eta expanding:

  assoc = λ {x}{y}{z} → assoc+ {x}{y}{z}

But this shouldn’t be necessary.




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