Unless you have proof of your claim that you sent the proof of correction and payment -- such as a receipt for certified mail, or a canceled check or money order -- the court is very unlikely to dismiss the civil assessment for failure to appear.
Important documents should ALWAYS be sent via certified mail, Federal Express, or some other shipping method that proves they were delivered... or better yet, hand delivered. Payments should also be made by check or money order. If you have to pay in cash, DO NOT send it in the mail; go to the court clerk's office and make sure to save your receipt.
When you file something with the court, you should make a copy and have the clerk stamp it to show it was filed. If you are mailing the document to the court, make sure to include a return envelope, addressed to yourself with sufficient postage attached, so the clerk can send you the stamped copy.
My colleague's answer is exactly right. However, if you now find yourself in a situation where you do not have any proof of having paid the ticket, then your best bet is to go before a judge and request leniency. There's no harm in trying to ask for a reduction of the fines.