Emergent sums and differences in graphic lambda calculus

See the page Graphic lambda calculus for background.

Here I want to discuss the treatment of one identity concerning approximate sums and differences in emergent algebras. The identity is the following: $\Delta^{x}_{\varepsilon}(u, \Sigma^{x}_{\varepsilon}(u,v)) = v$

The approximate sum (maybe emergent sum would be a better name) $\Sigma^{x}_{\varepsilon}(u,w)$ has the following associated graph in $GRAPH$: The letters in red “ $x, u, w, \Sigma$” are there only for the convenience of the reader.

Likewise, the graph in $GRAPH$ which corresponds to the approximate difference (or emergent difference) $\Delta^{x}_{\varepsilon}(u,w)$ is the following: The graph which corresponds to $\Delta^{x}_{\varepsilon}(u, \Sigma^{x}_{\varepsilon}(u,v))$ is this one: By a succession of CO-ASSOC moves we arrive to this graph: We are ready to apply an R2 move to get: We use now an ext2 move at the node marked by “1” followed by local pruning Here comes the funny part! We cannot continue unless we work with a graph where at the edges marked by the red letters “ $x, u$” we put two disjoint (not connected by edges) graphs in $GRAPH$, say $X, U$: Let us suppose that from the beginning we had $X, U$ connected at the edges marked by the red letters $x, u$, and proceed further. My claim is that by three  GLOBAL FAN-OUT  moves we can make the following move We use this move and we obtain: As previously, we use an R2 move and another ext2 move to finally obtain this: which is the answer we were looking for. We could use GLOBAL PRUNING to get rid of the part of the graph which ends by a termination gate.