Certification Problem

Input (COPS 238)

We consider the TRS containing the following rules:

a(x) g(g(b(x))) (1)
a(x) g(c(x)) (2)
b(x) g(b(x)) (3)

The underlying signature is as follows:

{a/1, g/1, b/1, c/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2023)

1 Non-Joinable Fork

The system is not confluent due to the following forking derivations.

t0 = a(a(x10))
a(g(g(b(x10))))
a(g(g(g(b(x10)))))
g(c(g(g(g(b(x10))))))
= t3

t0 = a(a(x10))
g(g(b(a(x10))))
g(g(g(b(a(x10)))))
g(g(g(g(b(a(x10))))))
= t3

The two resulting terms cannot be joined for the following reason: