Certification Problem

Input (COPS 213)

We consider the TRS containing the following rules:

a → b (1)
a → f(a) (2)
b → f(f(b)) (3)
f(f(f(b))) → b (4)

The underlying signature is as follows:

{a/0, b/0, f/1}

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by csi @ CoCo 2023)

1 Redundant Rules Transformation

To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:

a → b (1)
b → f(f(b)) (3)
f(f(f(b))) → b (4)

All redundant rules that were added or removed can be simulated in 4 steps .

1.1 Strongly closed

Confluence is proven since the TRS is strongly closed. The joins can be performed using 7 step(s).