elevó a definitivas sus conclusiones provisionales
submitted their opening argument as their summation (US)/ final submission (UK)

You may have to add the word “written” to argument, summation or final submission.

conclusión provisional: opening argument, opening statement.
conclusión definitiva: closing argument, closing statement.

Black’s Law Dictionary
closing argument. In a trial, a lawyer's final statement to the judge or jury before deliberation begins, in which the lawyer requests the judge or jury to consider the evidence and to apply the law in his or her client's favor. ● After the closing arguments in a jury trial, the judge ordinarily instructs the jury on the law that governs the case. — Also termed closing statement; final argument; jury summation; summing up; summation; closing argument; (in English law) final submission; (in English law) final speech.

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Note added at 4 hrs (2013-06-22 16:26:05 GMT)
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Perhaps it would be better to say opening statement than opening argument.

BLD
opening statement. At the outset of a trial, an advocate's statement giving the fact-finder a preview of the case and of the evidence to be presented. ● Although the opening statement is not supposed to be argumentative, lawyers — purposefully or not — often include some form of argument. The term is thus sometimes referred to as opening argument.