Certification Problem

Input (TPDB TRS_Standard/SK90/4.18)

The rewrite relation of the following TRS is considered.

gcd(x,0) → x (1)
gcd(0,y) → y (2)
gcd(s(x),s(y)) → if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y))) (3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Switch to Innermost Termination

The TRS is overlay and locally confluent:

10

Hence, it suffices to show innermost termination in the following.

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
gcd#(s(x),s(y)) → gcd#(s(x),-(y,x)) (4)
gcd#(s(x),s(y)) → gcd#(-(x,y),s(y)) (5)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.