I think core of the problem in property-based testing that the property/specification needs to be quite simple compared to the implementation.

I did some property-based testing in Haskell and in some cases the implementation was the specification verbatum. So what properties should I test? It was clearer where my function should be symmetric in the arguments or that there is a neutral element, etc..

If the property is basically your specification which (as the language is very expressive) is your implementation then you're just going in circles.

Yeah, reimplementing the solution just to have something to check against is a bad idea.

I find that most tutorials talk about "properties of function `foo`", whereas I prefer to think about "how is function `foo` related to other functions". Those relationships can be expressed as code, by plugging outputs of one function into arguments of another, or by sequencing calls in a particular order, etc. and ultimately making assertions. However, there will usually be gaps; filling in those gaps is what a property's inputs are for.

Another good source of properties is trying to think of ways to change an expression/block which are irrelevant. For example, when we perform a deletion, any edits made beforehand should be irrelevant; boom, that's a property. If something would filter out negative values, then it's a property that sprinkling negative values all over the place has no effect. And so on.