Certification Problem

Input (COPS 961)

We consider the TRS containing the following rules:

h(1(1(x))) 1(h(x)) (1)
1(1(h(b(x)))) 1(1(s(b(x)))) (2)
1(s(x)) s(1(x)) (3)
b(s(x)) b(h(x)) (4)
h(1(b(x))) t(1(1(b(x)))) (5)
1(t(x)) t(1(1(1(x)))) (6)
b(t(x)) b(h(x)) (7)

The underlying signature is as follows:

{h/1, 1/1, b/1, s/1, t/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2020)

1 Non-Joinable Fork

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

t0 = h(1(1(s(c_1))))
h(1(s(1(c_1))))
= t1

t0 = h(1(1(s(c_1))))
1(h(s(c_1)))
= t1

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