Certification Problem

Input (COPS 1655)

We consider the TRS containing the following rules:

f(x1,g(x2)) f(x1,g(x1)) (1)
f(g(y1),y2) f(g(y1),g(y1)) (2)
g(a) g(b) (3)
b a (4)

The underlying signature is as follows:

{f/2, g/1, a/0, b/0}

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:

b a (4)
g(a) g(b) (3)
f(g(y1),y2) f(g(y1),g(y1)) (2)
f(x1,g(x2)) f(x1,g(x1)) (1)
g(a) g(a) (5)
f(g(y1),y2) f(g(y1),g(g(y1))) (6)

All redundant rules that were added or removed can be simulated in 2 steps .

1.1 Development Closed

Confluence is proven since the TRS is development closed. The joins can be performed using 1 development step(s).