Certification Problem

Input (TPDB TRS_Standard/AG01/#3.24)

The rewrite relation of the following TRS is considered.

f(0) s(0) (1)
f(s(0)) s(0) (2)
f(s(s(x))) f(f(s(x))) (3)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Constant to Unary

Every constant is turned into a unary function symbol to obtain the TRS
f(0'(x)) s(0'(x)) (4)
f(s(0'(x))) s(0'(x)) (5)
f(s(s(x))) f(f(s(x))) (3)

1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
0'(f(x)) 0'(s(x)) (6)
0'(s(f(x))) 0'(s(x)) (7)
s(s(f(x))) s(f(f(x))) (8)

1.1.1 Bounds

The given TRS is match-(raise)-bounded by 3. This is shown by the following automaton.