Abstract
This note introduces a method of representing and reasoning about the actions of a class of proof procedures. A graph-like structure, called a proof diagram, is introduced in which conclusions of inferences can be shared. A version of Kruskal's Tree Theorem is developed for these structures and from there a notion of minimal proof is introduced. The notion of minimal proof allows us to make a link between standard treatments of proof and proofs generated by mechanical theorem provers. We discuss various uses for the proof diagrams including the development of tactics by graph reduction and propose a graph based meta-language for theorem provers.
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