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  • $\begingroup$ @Naïm Favier I don't think you can avoid the use of UIP here, but I would be delighted if you have an idea of how this proof can go around it. Also, I don't see how opacity got in your way, everything here is just fine? $\endgroup$
    – Meven Lennon-Bertrand ♦
    Commented Jul 5 at 10:44
  • 1
    $\begingroup$ No, I changed my mind after thinking it through; you do need UIP. The opacity problems I had were in proving that Vladimir's le_n' was equal to le_n, without using the general fact. $\endgroup$
    – Naïm Favier
    Commented Jul 5 at 10:46
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    $\begingroup$ We're clear on the fac tthat UIP for nat is provable, yes? This isn't some extra assumption we're talking about. $\endgroup$
    – Andrej Bauer
    Commented Jul 5 at 12:02
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    $\begingroup$ @NaïmFavier: I think a better way of stating what you're saying is: "This wouldn't hold if nat were replaced with some type in which non-trivial loops exist." (Saying "if nat had non-trivial loops" is like saying "if 6 were an odd number".) $\endgroup$
    – Andrej Bauer
    Commented Jul 5 at 13:49
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    $\begingroup$ In fact proving that it implies UIP is rather straightforward since a <= b /\ b <= a gives you a Hedberg function. $\endgroup$
    – JoJoModding
    Commented Jul 5 at 18:56