Witness and Proof - Most court procedure is accomplished by motions and stipulations. Motions are requests for the court to do something with a formal paper notifying someone that this request will be considered by the court on a certain date and time and in a specified courthouse. The court generally can't take action without the other interested people having advance notice.
In motions, the evidence comes to the court in the form of affidavits which are sworn written statements. Agreements are usually handled in writing and should be memorialized by writing which has been signed by the parties and submitted to the court for court approval. The last resort is to take a matter to trial.
At trial, witnesses speak the sworn evidence into the record and the Judge makes a decision. Experts are brought to give opinions on specialized issues and large amounts f preparation are needed to get the information to the court in sworn testimony and in the amount of time the court has to take the testimony.