Certification Problem

Input (COPS 574)

We consider the TRS containing the following rules:

+(0,y) y (1)
+(s(0),y) s(y) (2)
+(+(x,y),z) +(x,+(y,z)) (3)
+(x,y) +(y,x) (4)
s(s(x)) s(x) (5)
s(x) s(s(x)) (6)

The underlying signature is as follows:

{+/2, 0/0, s/1}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2020)

1 Non-Joinable Fork

The system is not confluent due to the following forking derivations.

t0 = +(+(s(0),f99),f98)
+(s(f99),f98)
= t1

t0 = +(+(s(0),f99),f98)
+(s(0),+(f99,f98))
s(+(f99,f98))
= t2

The two resulting terms cannot be joined for the following reason: