Certification Problem

Input (COPS 504)

We consider the TRS containing the following rules:

+(x,y) +(y,x) (1)
*(+(x,y),z) +(*(x,z),*(y,z)) (2)
*(+(y,x),z) +(*(x,z),*(y,z)) (3)

The underlying signature is as follows:

{+/2, */2}

Property / Task

Prove or disprove confluence.

Answer / Result

Yes.

Proof (by csi @ CoCo 2020)

1 Redundant Rules Transformation

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

*(+(y,x),z) +(*(x,z),*(y,z)) (3)
*(+(x,y),z) +(*(x,z),*(y,z)) (2)
+(x,y) +(y,x) (1)
*(+(y,x),z) +(*(y,z),*(x,z)) (4)
*(+(x,y),z) +(*(y,z),*(x,z)) (5)
+(x,y) +(x,y) (6)

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

1.1 Parallel Closed

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