Certification Problem

Input (COPS 15)

We consider the TRS containing the following rules:

f(x,f(y,z)) f(f(x,y),f(x,z)) (1)
f(f(x,y),z) f(f(x,z),f(y,z)) (2)
f(f(x,y),f(y,z)) y (3)

The underlying signature is as follows:

{f/2}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2022)

1 Non-Joinable Fork

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

t0 = f(f(c_1,f(c_1,f(c_1,c_1))),f(c_1,f(c_1,c_1)))
f(f(f(c_1,f(c_1,f(c_1,c_1))),c_1),f(f(c_1,f(c_1,f(c_1,c_1))),f(c_1,c_1)))
f(f(f(f(c_1,f(c_1,f(c_1,c_1))),c_1),f(c_1,f(c_1,f(c_1,c_1)))),f(f(f(c_1,f(c_1,f(c_1,c_1))),c_1),f(c_1,c_1)))
f(c_1,f(f(f(c_1,f(c_1,f(c_1,c_1))),c_1),f(c_1,c_1)))
f(c_1,c_1)
= t4

t0 = f(f(c_1,f(c_1,f(c_1,c_1))),f(c_1,f(c_1,c_1)))
f(f(c_1,f(f(c_1,c_1),f(c_1,c_1))),f(c_1,f(c_1,c_1)))
f(f(c_1,c_1),f(c_1,f(c_1,c_1)))
c_1
= t3

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