Mathematicians at every level use diagrams to prove theorems. Mathematical Reasoning with Diagrams investigates the possibilities of mechanizing this sort of diagrammatic reasoning in a formal computer proof system, even offering a semi-automatic formal proof system—called Diamond—which allows users to prove arithmetical theorems using diagrams.

Abstract: "Theorems in automated theorem proving are usually proved by logical formal proofs. However, there is a subset of problems which can also be proved in a more informal way by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is more clearly perceived in these than in the corresponding logical proofs: they capture an intuitive notion of truthfulness that humans find easy to see and understand. The proposed research project is to identify and ultimately automate this diagrammatic reasoning on mathematical theorems. The system that we are in the process of implementing will be given a theorem and will (initially) interactively prove it by the use of geometric manipulations on the diagram that the user chooses to be the appropriate ones. These operations will be the inference steps of the proof. The constructive w-rule will be used as a tool to capture the generality of diagrammatic proofs. In this way, we hope to verify and to show that the diagrammatic proof of a given theorem can be formalised."

This thesis is on the automation of diagrammatic proofs, a novel approach to mechanised mathematical reasoning. Theorems in automated theorem proving are usually proved by formal logical proofs. However, there are some conjectures which humans can prove by the use of geometric operations on diagrams that somehow represent these conjectures, so called diagrammatic proofs. Insight is often more clearly perceived in these diagrammatic proofs than in the algebraic proofs. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete rather than general diagrams are used to prove ground instances of a universally quantified theorem. The diagrammatic proof in constructed by applying geometric operations to the diagram. These operations are in the inference steps of the proof. A general schematic proof is extracted from the ground instances of a proof. it is represented as a recursive program that consists of a general number of applications of geometric operations. When gien a particular diagram, a schematic proof generates a proof for that diagram. To verify that the schematic proof produces a correct proof of the conjecture for each ground instance we check its correctness in a theory of diagrams. We use the constructive omega-rule and schematic proofs to make a translation from concrete instances to a general argument about the diagrammatic proof. The realisation of our ideas is a diagrammatic reasoning system DIAMOND. DIAMOND allows a user to interactively construct instances of a diagrammatic proof. It then automatically abstracts these into a general schematic proof and checks the correctness of this proof using an inductive theorem prover.

Abstract: "Theorems in automated theorem proving are usually proved by logical formal proofs. However, there is a subset of problems which humans can prove in a different way by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is more clearly perceived in these than in the corresponding algebraic proofs: they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are identifying and automating this diagrammatic reasoning on mathematical theorems. The user gives the system, called DIAMOND, a theorem and then interactively proves it by the use of geometric manipulations on the diagram. These operations are the 'inference steps' of the proof. DIAMOND then automatically derives from these example proofs a generalised proof. The constructive [omega]-rule is used as a mathematical basis to capture the generality of inductive diagrammatic proofs. In this way, we explore the relation between diagrammatic and algebraic proofs."