Certification Problem

Input (COPS 590)

We consider the TRS containing the following rules:

+(0,0) → 0 (1)
+(s(x),y) → s(+(x,y)) (2)
+(x,s(y)) → s(+(y,x)) (3)
s(s(x)) → x (4)

The underlying signature is as follows:

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

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2022)

1 Non-Joinable Fork

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

t0 = +(+(s(x585),x586),s(y))
→ +(s(+(x585,x586)),s(y))
→ s(+(+(x585,x586),s(y)))
→ s(s(+(y,+(x585,x586))))
→ +(y,+(x585,x586))
= t4

t0 = +(+(s(x585),x586),s(y))
→ s(+(y,+(s(x585),x586)))
→ s(+(y,s(+(x585,x586))))
→ s(s(+(+(x585,x586),y)))
→ +(+(x585,x586),y)
= t4

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