Certification Problem

Input (COPS 425)

We consider the TRS containing the following rules:

f(a,f(a,f(a,f(a,y)))) → f(a,f(a,f(a,g(y,f(a,y))))) (1)
f(x,y) → g(y,f(x,y)) (2)

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by csi @ CoCo 2021)

1 Redundant Rules Transformation

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

f(x,y) → g(y,f(x,y)) (2)

All redundant rules that were added or removed can be simulated in 1 steps .

1.1 Redundant Rules Transformation

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

f(x,y) → g(y,f(x,y)) (2)
f(x,y) → g(y,g(y,f(x,y))) (3)

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

1.1.1 Parallel Closed

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