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<p class="MsoNormal"><span style="font-size:11.0pt">I can switch off positivity checking locally by saying<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">{-# NO_POSITIVITY_CHECK #-}<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 I can’t say<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">{-# TYPE-IN-TYPE #-}<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">data Prop : Set<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">T : Prop → Set<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">data Prop where<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> pi : (A : Set)(b : A → Prop) → Prop<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">T (pi A b) = (x : A) → T (b x)<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">Or is there a difference syntax? This would be useful for a simple-minded encoding of the calculus of constructions.<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">Thorsten<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">P.S. I know I can avoid this by adding another level:<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">data Prop : Set<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">T : Prop → Set<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">data U : Set<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">El : U → Set<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">{-# NO_POSITIVITY_CHECK #-}<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">data U where<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> prop : U<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> pi : (a : U)(b : El a → U) → U<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> emb : Prop → U<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">El prop = Prop<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">El (pi a b) = (x : El a) → El (b x)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">El (emb p) = T p<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">data Prop where<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"> pi : (a : U)(b : El a → Prop) → Prop<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">T (pi a b) = (x : El a) → T (b x)<o:p></o:p></span></p>
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