Proof of completeness for cyclic graphs

Proposition (Completeness). If [[f]] [[g]] then f = g.

Proof. Let R be the bisimulation between [[f]] and [[g]]. We can regard this as a graph G:

Nodes are pairs (N,N) R.
Edges are pairs (E,E) R.
Incoming edges, outgoing edges, and labellings inherited from [[f]] and [[g]] (since R is a graph relation, this is consistent).

By expressivity, let h be the term such that [[h]]

G. Find normal forms for f, g and h:

Let E₁...E_m be the outgoing edges of [[f1]] which form the cycles in [[f]], and F₁...F_n be the outgoing edges of [[h1]] which form the cycles in [[h]]. Construct a shuffle s with:

Edges E₁...E_m.
Incoming edges E₁...E_m.
Outgoing edges E₁...E_n, where E_i = E_j whenever F_i = (E_j,_).
Labelling inherited from [[f1]].

We can then show that:

and since we have completeness for acyclic graphs, this means:

= Then since trace is uniform wrt shuffles:

= and so

f = h. We can show

g = h similarly.

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