Certification Problem

Input (COPS 571)

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)

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(x),+(s(x20),x21))
→ +(s(x),s(+(x20,x21)))
→ s(+(+(x20,x21),s(x)))
→ s(s(+(x,+(x20,x21))))
= t3

t0 = +(s(x),+(s(x20),x21))
→ s(+(x,+(s(x20),x21)))
→ s(+(x,s(+(x20,x21))))
→ s(s(+(+(x20,x21),x)))
= t3

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