Monotone can be increasing or *decreasing* about proofs HOT 8 CLOSED

Your code only take care of strictly increasing monotone function.

It's non-strict.

It doesn't take into account the strictly decreasing one, IMHO

The theorem isn't true for functions which are strictly decreasing. The definition of continuous most commonly used in domain theory is that the function preserves suprema of directed subsets. That only implies monotone in the increasing direction, and this variant is commonly taken to be the definition of monotone in domain theory.

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I removed strict which wasn't true. and thanks for the detailed answer. Domain theory looks even more interesting now !

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What is the issue? There are a few different flavors of monotone functions. This one is not strict (because partial orders are reflexive), so it doesn't match the title of the issue.

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This is the standard definition for monotone in domain theory (the one that is implied by Scott continuity), so I will close this. But maybe I am missing something, so feel free to re-open if that is the case.

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Sorry I wrote it too fast
Your code only take care of strictly increasing monotone function. It doesn't take into account the strictly decreasing one, IMHO

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But am I correct that in the sample you provide, we could use different definition for ⊑_A and ⊑_B, hence in this case we could say ⊑_A ≝ \leq and ⊑_B ≝ \geq and hence it covers increasing and decreasing monotonic ?

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we could say ⊑_A ≝ \leq and ⊑_B ≝ \geq and hence it covers increasing and decreasing monotonic

You are asking whether we can define monotone as: x <= y implies f(y) <= f(x)?

No, because this has the same problem that I mentioned before: it is not implied by Scott-continuity.

Also, that would not be very useful; think about what a monotonically decreasing function would mean in terms of the information order. It makes no sense for computable functions, which are the kind studied in domain theory.

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Also, that would not be very useful; think about what a monotonically decreasing function would mean in terms of the information order. It makes no sense for computable functions, which are the kind studied in domain theory.

Very good point !!!

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