If one steps on a cockroach on Shabbos and the cockroach is then mutilated to the point that it will surely die, is one then allowed to kill it?
Is it allowed to kill the cockroach because essentially it is now a living corpse?
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R Dovid Ribiat answers your question in his book The 39 melochos (vol 3, p. 888)
It is questionable wether killing a mortally ill or wounded animal is the melocho m'deoraisa of Schochait. In any case, doing so is not actually permitted. "Mercy killing" situations are common examples of this [question].
Example: One who notices a badly wounded bird or similar creature and perceives that it is suffering great pain, may not kill it on Shabbos to put it out of its misery. However he may apprise a non-Jew of the problem and allow the non-Jew to kill the animal.