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Antimatroids were discovered by Dilworth in the context of lattices [4] andintroduced by Edelman and Jamison as convex geometries in[5]. The author of thecurrent paper independently discovered (possibly infinite) antimatroids in thecontext of proof systems in mathematical logic [1]. Carlson, a logician, makesimplicit use of this view of proof systems as possibly infinite antimatroids in[2]. Though antimatroids are in a sense dual to matroids, far fewer antimatroidforbidden minor theorems are known. Some results of this form are proved in[6], [7], [8], and [9]. This paper proves two forbidden induced minor theoremsfor these objects, which we think of as proof systems.  Our first main theorem gives a new proof of the forbidden induced minorcharacterization of partial orders as proof systems, proved in [8] in thefinite case and stated in [10] for what we call strong aut descendable proofsystems. It essentially states that, pathologies aside, there is a certainunique simplest nonposet. Our second main theorem states the new result that,pathologies aside, there is a certain unique simplest proof system containingpoints $x$ and $y$ such that $x$ needs $y$ in one context, yet $y$ needs $x$ inanother.

Two Forbidden Induced Minor Theorems for Antimatroids