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<p class="MsoNormal"><span style="font-size:11.0pt">I define<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">record isMonoid {A : Set}(e : A)(_•_ : A → A → A) : Prp where<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> field<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> lneutr : {x : A} → e • x ≡ x<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> </span><span lang="DE" style="font-size:11.0pt">rneutr : {x : A} → x • e ≡ x<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="DE" style="font-size:11.0pt"> </span><span lang="PL" style="font-size:11.0pt">assoc : {x y z : A} → (x • y) • z ≡ x • (y • z)<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="PL" style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">and I have shown all 3 properties, in particular:<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">assoc+ : {l m n : </span><span style="font-size:11.0pt;font-family:"Cambria Math",serif">ℕ</span><span style="font-size:11.0pt">} → (l + m) + n ≡ l + (m + n)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">but when I try<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">isMonoid : isMonoid zero _+_<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">+isMonoid = record {<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> lneutr = lneutr+ ;<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> rneutr = rneutr+ ;<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> assoc = assoc+ }<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">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></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> assoc = λ {x}{y}{z} → assoc+ {x}{y}{z}<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">But this shouldn’t be necessary.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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