Certification Problem

Input (COPS 974)

We consider the TRS containing the following rules:

0(1(2(3(4(5(1(x))))))) 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(x))))))))))))))))))) (1)
0(1(2(3(4(5(1(x))))))) 1(2(3(4(5(1(1(0(1(2(3(4(5(0(1(2(3(4(5(0(1(2(3(4(5(x))))))))))))))))))))))))) (2)

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2020)

1 Non-Joinable Fork

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

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

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

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