Definitions and proofs in extended abstract(mathematics)
Before you start writing, try to create a 5-minute elevator pitch talk about your paper. Imagine you're describing the results to your friend (or actually invite a friend) and talk through the main idea of the paper. Practice it several times and note which things you keep in, and which you decide to leave out. This will give you an idea of what to include in your extended abstract.

It is simply not possible to squeeze everything from 20 pages into a 4-pages without losing something, so stop trying. Inevitably you will have to omit some results, or simply name them without going into details. You will have to drop some technicalities (which can contain some useful ideas as well) and so on. The aim of the extended abstract is to communicate the core ideas, not fully elaborated proofs and  methods. The elevator pitch technique above helps you to tell what's central and what's not.