Veriication of Diagrammatic Proofs

Mateja Jamnik, Alan Bundy, Ian Green Department of Arti cial Intelligence, 80 South Bridge Edinburgh, EH1 1HN, UK [email protected], [email protected], [email protected] Abstract Human mathematicians often \prove" theorems by the use of diagrams and manipulations on them. We call these diagrammatic proofs. In (Jamnik, Bundy, & Green 1997) we presented how \informal" reasoning with diagrams can be automated. Three stages of proof extraction procedure were identi ed. First, concrete rather than general diagrams are used to prove particular instances of the universally quanti ed theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. Second, an abstracted schematic proof of the universally quantied theorem is automatically induced from these proof instances. Third, the nal step is to con rm that the abstraction of the schematic proof from the proof instances is sound. Our main focus in this paper is on the third stage, the veri cation of schematic proofs. We de ne a theory of diagrams where we prove the correctness of a schematic proof. We give an example of an extraction of a schematic proof for a theorem about the sum of odd naturals, and prove its correctness in the theory of diagrams. Introduction