Certification Problem

Input (COPS 1138)

The rewrite relation of the following conditional TRS is considered.

f(y,a) g(x,x) | h(z) ≈ x, yk(z), xb
f(y,z) k(x) | zh(a), yk(a), ax
k(x) h(x)
k(x) b
k(x) a
a b
g(x,x) a

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ConCon @ CoCo 2020)

1 Removal of Infeasible Rules

We may safely remove rules with infeasible conditions. They do not influence the rewrite relation in any way.

1.1 Rules with Infeasible Conditions

1.2 Inlining of Conditions

Inlining of conditions results in the following transformed CTRS having the same multistep rewrite relation.
f(y,z) k(a) | zh(a), yk(a)
k(x) h(x)
k(x) b
k(x) a
a b
g(x,x) a

1.2.1 Non-Joinable Fork

The system is not confluent due to the following forking derivations.
k(z) * h(z)
k(z) * b
The two resulting terms cannot be joined for the following reason: