Certification Problem

Input (TPDB TRS_Standard/SK90/4.52)

The rewrite relation of the following TRS is considered.

s(a) a (1)
s(s(x)) x (2)
s(f(x,y)) f(s(y),s(x)) (3)
s(g(x,y)) g(s(x),s(y)) (4)
f(x,a) x (5)
f(a,y) y (6)
f(g(x,y),g(u,v)) g(f(x,u),f(y,v)) (7)
g(a,a) a (8)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(g) = 0 weight(g) = 0
prec(f) = 2 weight(f) = 0
prec(s) = 3 weight(s) = 0
prec(a) = 1 weight(a) = 1
all of the following rules can be deleted.
s(a) a (1)
s(s(x)) x (2)
s(f(x,y)) f(s(y),s(x)) (3)
s(g(x,y)) g(s(x),s(y)) (4)
f(x,a) x (5)
f(a,y) y (6)
f(g(x,y),g(u,v)) g(f(x,u),f(y,v)) (7)
g(a,a) a (8)

1.1 R is empty

There are no rules in the TRS. Hence, it is terminating.