Certification Problem

Input (TPDB TRS_Standard/HirokawaMiddeldorp_04/n008)

The rewrite relation of the following TRS is considered.

f(a) → f(a) (1)
a → b (2)

Property / Task

Prove or disprove termination.

Answer / Result

No.

Proof (by AProVE @ termCOMP 2023)

1 Constant to Unary

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

1.1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a'(f(x)) → a'(f(x)) (5)
a'(x) → b'(x) (4)

1.1.1 Rule Removal

The following rules have been removed.
a'(x) → b'(x) (4)

1.1.1.1 Innermost Lhss Increase

We add the following left hand sides to the innermost strategy.
a'(f(x0))

1.1.1.1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a'#(f(x)) → a'#(f(x)) (6)
It remains to prove infiniteness of the resulting DP problem.

1.1.1.1.1.1 Pair and Rule Removal

Some pairs and rules have been removed and it remains to prove infiniteness of the remaing problem. The following pairs have been deleted. and the following rules have been deleted.
a'(f(x)) → a'(f(x)) (5)

1.1.1.1.1.1.1 Innermost Lhss Removal Processor

We restrict the innermost strategy to the following left hand sides.

There are no lhss.

1.1.1.1.1.1.1.1 Loop

The following loop proves infiniteness of the DP problem.

t0 = a'#(f(x))
→P a'#(f(x))
= t1
where t1 = t0σ and σ = {x/x}