Certification Problem

Input (COPS 975)

We consider the TRS containing the following rules:

5(5(x)) 0(5(4(0(2(5(4(5(2(1(x)))))))))) (1)
5(5(x)) 3(4(1(1(1(1(4(4(0(4(x)))))))))) (2)
2(5(5(x))) 4(2(5(4(4(0(0(1(1(2(x)))))))))) (3)
5(2(4(x))) 0(5(0(2(3(3(4(2(4(2(x)))))))))) (4)
5(5(2(x))) 0(1(3(2(3(0(3(2(5(3(x)))))))))) (5)
5(5(3(x))) 0(3(5(4(4(1(0(1(5(0(x)))))))))) (6)
5(5(5(x))) 5(3(4(1(0(1(4(5(0(0(x)))))))))) (7)
2(5(0(4(x)))) 4(4(3(2(4(4(5(1(0(0(x)))))))))) (8)
4(5(2(4(x)))) 4(1(5(5(2(0(3(1(3(3(x)))))))))) (9)
4(5(5(5(x)))) 1(5(1(2(0(3(2(1(0(5(x)))))))))) (10)
0(2(5(3(4(x))))) 3(2(4(3(1(5(1(1(3(4(x)))))))))) (11)
2(5(5(3(4(x))))) 4(5(4(3(1(4(0(2(4(4(x)))))))))) (12)
5(5(5(1(4(x))))) 3(3(0(5(0(4(3(4(4(0(x)))))))))) (13)
0(4(4(5(5(5(x)))))) 0(4(4(4(3(3(4(1(3(1(x)))))))))) (14)
1(2(4(5(2(4(x)))))) 3(3(5(3(0(4(0(3(1(3(x)))))))))) (15)
4(1(5(5(0(4(x)))))) 1(0(3(0(4(2(4(4(3(4(x)))))))))) (16)
4(2(5(5(1(5(x)))))) 2(3(4(2(1(1(3(4(2(5(x)))))))))) (17)
5(2(5(5(0(4(x)))))) 0(4(2(3(3(5(2(1(4(4(x)))))))))) (18)
5(5(2(4(5(0(x)))))) 2(1(1(4(2(4(0(4(2(0(x)))))))))) (19)
0(1(5(5(5(3(5(x))))))) 5(3(2(5(1(0(1(2(0(5(x)))))))))) (20)
4(4(5(2(4(2(2(x))))))) 4(0(5(5(4(5(1(2(2(1(x)))))))))) (21)

The underlying signature is as follows:

{5/1, 0/1, 4/1, 2/1, 1/1, 3/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2022)

1 Non-Joinable Fork

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

t0 = 5(5(5(x2017)))
5(0(5(4(0(2(5(4(5(2(1(x2017)))))))))))
= t1

t0 = 5(5(5(x2017)))
0(5(4(0(2(5(4(5(2(1(5(x2017)))))))))))
= t1

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