Proof of cancellation of nodes

Proposition (Cancellation of nodes). For any:

we can find h such that:

Proof. First, show a result about bisimulation on graphs; if:

then either:

or we can find h such that:

We now proceed by induction on f, which (by normalization) we can assume is in normal form. We have three cases:

If f is: then we have a contradiction, since f is bisimilar to a graph with a node connected to an outgoing edge.
If f is: then: then find permutation q such that: = so we have: By induction, we can find h1 such that: =
so: = = = = = and:
If f is: then by the above result about bisimulation either:
- we have:
  and we are finished,
- or we can find g1 such that:
  so by induction we can find h1 such that: =
  Then we have two cases depending on the relative positions of c and d, but the proofs are identical: = = = and:

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