Certification Problem

Input (COPS 795)

We consider the TRS containing the following rules:

f(x,f(y,z)) f(f(x,y),f(x,z)) (1)
f(f(x,y),z) f(f(x,z),f(y,z)) (2)
f(f(x,y),f(y,z)) y (3)

The underlying signature is as follows:

{f/2}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by csi @ CoCo 2022)

1 Redundant Rules Transformation

To prove that the TRS is (non-)confluent, we show (non-)confluence of the following modified system:

f(x,f(y,z)) f(f(x,y),f(x,z)) (1)
f(f(x,y),z) f(f(x,z),f(y,z)) (2)
f(f(x,y),f(y,z)) y (3)
f(y,f(y,z)) y (4)
f(f(x,z),z) z (5)

All redundant rules that were added or removed can be simulated in 3 steps .

1.1 Non-Joinable Fork

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

t0 = f(y,f(y,f(y,x378)))
f(y,y)
= t1

t0 = f(y,f(y,f(y,x378)))
y
= t1

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