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<p class="MsoNormal"><span style="mso-fareast-language:EN-US">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)?<o:p></o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US">I mean one could always try “exact” – or is this a bad idea?<o:p></o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US">Thorsteen<o:p></o:p></span></p>
<p class="MsoNormal"><span style="mso-fareast-language:EN-US"><o:p> </o:p></span></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><b><span style="font-size:12.0pt;color:black">From:
</span></b><span style="font-size:12.0pt;color:black">Ulf Norell <ulf.norell@gmail.com><br>
<b>Date: </b>Sunday, 8 March 2020 at 18:47<br>
<b>To: </b>Thorsten Altenkirch <psztxa@exmail.nottingham.ac.uk><br>
<b>Cc: </b>agda-list <agda@lists.chalmers.se><br>
<b>Subject: </b>Re: [Agda] what is happening here?<o:p></o:p></span></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">Hi Thorsten,<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">The constraints you get are <o:p>
</o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"> ?l + (?m + ?n) = l + (m + n) : <span style="font-family:"Cambria Math",serif">
ℕ</span><br>
?l + ?m + ?n = l + m + n : <span style="font-family:"Cambria Math",serif">ℕ</span><o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">You are right that ?l := l, ?m := m, ?n := n is the only solution, but this<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">requires some non-trivial reasoning. For instance, the first constraint<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">can be satisfied by<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"> ?l := 0, ?m := l, ?n := (m + n)<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">so Agda would need to look at both constraints together to rule out this solution.<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">/ Ulf<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:36.0pt">On Sun, Mar 8, 2020 at 7:26 PM Thorsten Altenkirch <<a href="mailto:Thorsten.Altenkirch@nottingham.ac.uk">Thorsten.Altenkirch@nottingham.ac.uk</a>> wrote:<o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
I define<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
record isMonoid {A : Set}(e : A)(_•_ : A → A → A) : Prp where<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
field<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
lneutr : {x : A} → e • x ≡ x<o:p></o:p></p>
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<span lang="DE">rneutr : {x : A} → x • e ≡ x</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<span lang="DE"> </span><span lang="PL">assoc : {x y z : A} → (x • y) • z ≡ x • (y • z)</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<span lang="PL"> </span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
and I have shown all 3 properties, in particular:<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
assoc+ : {l m n : <span style="font-family:"Cambria Math",serif">ℕ</span>} → (l + m) + n ≡ l + (m + n)<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
but when I try<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
isMonoid : isMonoid zero _+_<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
+isMonoid = record {<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
lneutr = lneutr+ ;<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
rneutr = rneutr+ ;<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
assoc = assoc+ }<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
I get some unsolved constraints, even though my proof and the goal are alpha-equivalent! Ok I can fix this by eta expanding:<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
assoc = λ {x}{y}{z} → assoc+ {x}{y}{z}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
But this shouldn’t be necessary.<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:36.0pt">
<o:p></o:p></p>
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