Certification Problem

Input (COPS 591)

We consider the TRS containing the following rules:

+(0,y) → y (1)
+(s(x),y) → s(+(x,y)) (2)
*(0,y) → 0 (3)
*(s(x),y) → +(*(x,y),y) (4)
*(x,s(y)) → +(x,*(x,y)) (5)

The underlying signature is as follows:

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

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(x),+(s(x84),x85))
→ *(s(x),s(+(x84,x85)))
→ +(s(x),*(s(x),+(x84,x85)))
→ s(+(x,*(s(x),+(x84,x85))))
= t3

t0 = *(s(x),+(s(x84),x85))
→ +(*(x,+(s(x84),x85)),+(s(x84),x85))
→ +(*(x,s(+(x84,x85))),+(s(x84),x85))
→ +(+(x,*(x,+(x84,x85))),+(s(x84),x85))
= t3

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