Certification Problem

Input (COPS 631)

We consider the TRS containing the following rules:

inv(0) 0 (1)
inv(s(x)) p(inv(x)) (2)
inv(p(x)) s(inv(x)) (3)
minus(x,0) x (4)
minus(x,p(y)) s(minus(x,y)) (5)
minus(x,s(y)) p(minus(x,y)) (6)
minus(0,x) inv(x) (7)
minus(s(x),s(y)) minus(x,y) (8)
minus(p(x),p(y)) minus(x,y) (9)
inv(x) minus(0,x) (10)
s(p(x)) x (11)
p(s(x)) x (12)

The underlying signature is as follows:

{inv/1, 0/0, s/1, p/1, minus/2}

Property / Task

Prove or disprove confluence.

Answer / Result

No.

Proof (by ACP @ CoCo 2020)

1 Non-Joinable Fork

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

t0 = minus(p(c_1),p(s(c_2)))
minus(c_1,s(c_2))
p(minus(c_1,c_2))
= t2

t0 = minus(p(c_1),p(s(c_2)))
minus(p(c_1),c_2)
= t1

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