Certification Problem

Input (COPS 976)

We consider the TRS containing the following rules:

0(2(1(x))) 0(4(0(1(4(5(1(5(0(1(x)))))))))) (1)
1(1(3(x))) 1(2(1(2(2(2(5(1(4(2(x)))))))))) (2)
0(0(0(3(x)))) 4(2(4(0(2(5(3(3(4(5(x)))))))))) (3)
1(0(2(3(x)))) 1(1(2(5(4(1(2(4(3(2(x)))))))))) (4)
1(3(5(3(x)))) 3(5(4(5(2(4(3(2(5(4(x)))))))))) (5)
0(2(3(1(3(x))))) 1(0(1(2(1(3(1(3(1(2(x)))))))))) (6)
0(3(4(5(3(x))))) 2(0(2(5(1(2(4(4(5(5(x)))))))))) (7)
0(5(2(1(3(x))))) 2(4(2(5(2(4(3(0(2(4(x)))))))))) (8)
2(1(3(1(0(x))))) 0(1(4(5(1(5(5(2(3(0(x)))))))))) (9)
0(5(2(2(2(0(x)))))) 2(5(4(3(0(2(5(1(2(1(x)))))))))) (10)
2(0(0(5(2(0(x)))))) 4(0(4(2(1(4(4(4(0(1(x)))))))))) (11)
2(0(5(3(0(2(x)))))) 2(5(3(5(1(4(5(0(0(2(x)))))))))) (12)
2(1(0(2(1(5(x)))))) 2(5(4(1(3(2(2(5(4(5(x)))))))))) (13)
5(1(5(1(0(2(x)))))) 4(5(0(0(4(3(1(1(0(4(x)))))))))) (14)
0(5(2(2(2(1(0(x))))))) 0(3(2(3(1(4(1(0(1(0(x)))))))))) (15)
0(5(3(5(3(1(5(x))))))) 0(1(3(4(0(1(4(5(1(5(x)))))))))) (16)
1(1(5(1(4(4(3(x))))))) 1(0(3(4(4(1(0(2(5(5(x)))))))))) (17)
1(3(2(3(0(5(3(x))))))) 1(4(0(1(5(4(0(3(2(5(x)))))))))) (18)
1(5(2(4(2(1(1(x))))))) 4(4(1(4(1(4(3(1(0(3(x)))))))))) (19)
2(0(2(0(2(1(0(x))))))) 2(0(2(3(4(2(4(4(4(0(x)))))))))) (20)
2(4(5(5(1(3(5(x))))))) 2(1(2(1(4(4(4(3(4(4(x)))))))))) (21)
3(0(0(5(5(2(1(x))))))) 0(2(4(3(2(3(2(1(0(3(x)))))))))) (22)
3(1(5(2(3(0(5(x))))))) 5(3(4(0(4(5(2(0(0(4(x)))))))))) (23)

The underlying signature is as follows:

{0/1, 2/1, 1/1, 4/1, 5/1, 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 = 2(0(5(3(0(2(1(x3017)))))))
2(0(5(3(0(4(0(1(4(5(1(5(0(1(x3017))))))))))))))
= t1

t0 = 2(0(5(3(0(2(1(x3017)))))))
2(5(3(5(1(4(5(0(0(2(1(x3017)))))))))))
= t1

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