Certification Problem

Input (COPS 63)

We consider the TRS containing the following rules:

-(0,0) 0 (1)
-(s(x),0) s(x) (2)
-(x,s(y)) -(d(x),y) (3)
d(s(x)) x (4)
-(s(x),s(y)) -(x,y) (5)
-(d(x),y) -(x,s(y)) (6)

The underlying signature is as follows:

{-/2, 0/0, s/1, d/1}

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:

-(d(x),y) -(x,s(y)) (6)
-(s(x),s(y)) -(x,y) (5)
d(s(x)) x (4)
-(x,s(y)) -(d(x),y) (3)
-(s(x),0) s(x) (2)
-(0,0) 0 (1)
-(d(x),y) -(d(x),y) (7)
-(x,s(y)) -(x,s(y)) (8)

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).