How can I delete duplicates with Abs?

Question

I have this list with some duplicate elements and I want to delete duplicate ones.

mylist = {ConditionalExpression[1, -1 <= x <= 0], 
   ConditionalExpression[-x, x <= -1], 
   ConditionalExpression[-x, x <= -1], 
   ConditionalExpression[Abs[x], x <= -1]};

Now I tried to delete duplicate elements with this.

DeleteDuplicates[mylist]

This doesn't work. I expected that two of these three elements {ConditionalExpression[-x, x <= -1], ConditionalExpression[-x, x <= -1], ConditionalExpression[Abs[x], x <= -1]} should be deleted.

Then I thought probably that was due to Mathematica thinking that x is complex, so I added the assumption that x is real, but this didn't work either.

Assuming[x \[Element] Reals, DeleteDuplicates[mylist]]

How can I delete duplicate elements so the three elements above are considered duplicate?

Answer 1

DeleteDuplicates[Refine[mylist, Assumptions -> x ∈ Reals]]

gives what you want

Answer 2

Try this:

DeleteDuplicates[Refine[Map[LogicalExpand, mylist], Assumptions -> x \[Element] Reals]]

Answer 3

Assuming you do want to keep one of the equivalent ConditionalExpressions

DeleteDuplicatesBy[mylist, Refine[#, #[[2]]] &]

{ConditionalExpression[1, -1 <= x <= 0], ConditionalExpression[-x, x <= -1]}

Answer 4

list =
  {ConditionalExpression[1, -1 <= x <= 0],
   ConditionalExpression[-x, x <= -1],
   ConditionalExpression[-x, x <= -1],
   ConditionalExpression[Abs[x], x <= -1]};

In this case we only need

DeleteDuplicates @ Refine @ list

From the documentation:

"Refine[expr] uses default assumptions ..."