Definition:Tableau Proof (Natural Deduction)

Definition
A tableau proof by natural deduction is a technique for presenting a logical argument in a straightforward, standard form.

A tableau proof is a sequence of lines specifying the order of premises, assumptions, inferences and conclusion in support of an argument.

Each line of a tableau proof has a particular format. It consists of the following five parts:


 * Line: The line number of the proof. This is a simple numbering from 1 upwards.
 * Pool: The list of all the lines containing the pool of assumptions for the formula introduced on this line.
 * Formula: The formula or statement form introduced on this line.
 * Rule: The justification for introducing this line. This should be the appropriate abbreviation of the proof rule being used to derive this line, for example:
 * 1) $\mathrm A$ (for the Rule of Assumption);
 * 2) $\implies \mathcal I$ (for the Rule of Implication)
 * Depends on: The lines (if any) upon which this line directly depends. For assumptions, this field will be empty.

Premises
At the end of a tableau proof, the lines upon which the proof depends contain the premises under which the formula in the last line is guaranteed to hold, as demonstrated by the tableau proof under consideration.

To improve the readability of a tableau proof, it is advisable to use the letter $\textrm P$ for introducing premises, and the letter $\textrm A$ for assumptions that will be discharged later on.