ProveDetails Command

ProveDetails( <Boolean Expression> )

Returns some details of the result of the automated proof.

Normally, GeoGebra decides whether a boolean expression is true or not by using numerical computations. However, the ProveDetails command uses symbolic methods to determine whether a statement is true or false in general. This command works like the Prove command, but also returns some details of the result as a list:

• An empty list {} if GeoGebra cannot determine the answer.
• A list with one element: {false}, if the statement is not true in general.
• A list with one element: {true}, if the statement is always true (in all cases when the diagram can be constructed).
• A list with more elements, containing the boolean value true and another list for some so-called non-degeneracy conditions, if the statement is true under certain conditions, e.g. {true, {"AreCollinear[A,B,C],AreEqual[C,D]"}}. This means that if none of the conditions are true (and the diagram can be constructed), then the statement will be true.
• A list {true,{"..."}}, if the statement is true under certain conditions, but these conditions cannot be translated to human readable form for some reasons.

Example:

Let us define a triangle with vertices A, B and C, and define D=MidPoint[B,C], E=MidPoint[A,C], p=Line[A,B], q=Line[D,E]. Now ProveDetails[p∥q] returns {true}, which means that if the diagram can be constructed, then the midline DE of the triangle is parallel to the side AB.

Example:

Let AB be the segment a, and define C=MidPoint[A,B], b=PerpendicularBisector[A,B], D=Intersect[a,b]. Now ProveDetails[C==D] returns {true,{"AreEqual[A,B]"}}: it means that if the points A and B differ, then the points C and D will coincide.

Example:

Let AB be the segment a, and define l=Line[A,B]. Let C be an arbitrary point on line l, moreover let b=Segment[B,C], c=Segment[A,C]. Now ProveDetails[a==b+c] returns {true,{"a+b==c", "b==a+c"}}: it means that if neither a+b=c, nor b=a+c, then a=b+c.

It is possible that the list of the non-degeneracy conditions is not the simplest possible set. For the above example, the simplest set would be the empty set.

Note: See also Prove command, Boolean values, GeoGebra Automated Reasoning Tools: A Tutorial and technical details of the algorithms.