Certification Problem

Input (COPS 163)

We consider the TRS containing the following rules:

+(+(x,y),z) → +(x,+(y,z)) (1)
+(x,+(y,z)) → +(+(x,y),z) (2)

The underlying signature is as follows:

{+/2}

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by csi @ CoCo 2022)

1 Redundant Rules Transformation

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

+(x,+(y,z)) → +(+(x,y),z) (2)
+(+(x,y),z) → +(x,+(y,z)) (1)
+(x,+(y,z)) → +(x,+(y,z)) (3)
+(+(x,y),z) → +(+(x,y),z) (4)

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

1.1 Strongly closed

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