Certification Problem

Input (TPDB TRS_Standard/Waldmann_06/jwteparla1)

The rewrite relation of the following TRS is considered.

f(x,f(a,f(f(a,a),a))) → f(f(a,x),x) (1)

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.
f#(x,f(a,f(f(a,a),a))) → f#(f(a,x),x) (2)
f#(x,f(a,f(f(a,a),a))) → f#(a,x) (3)

1.1.1 Reduction Pair Processor

Using the
prec(f#) = 0 stat(f#) = mul
prec(f) = 2 stat(f) = mul
prec(a) = 1 stat(a) = mul

π(f#) = [1,2]
π(f) = []
π(a) = []

the pair
f#(x,f(a,f(f(a,a),a))) → f#(a,x) (3)
could be deleted.

1.1.1.1 Narrowing Processor

We consider all narrowings of the pair below position ε to get the following set of pairs
f#(f(a,f(f(a,a),a)),f(a,f(f(a,a),a))) → f#(f(f(a,a),a),f(a,f(f(a,a),a))) (4)

1.1.1.1.1 Usable Rules Processor

We restrict the rewrite rules to the following usable rules of the DP problem.

There are no rules.

1.1.1.1.1.1 Instantiation Processor

We instantiate the pair to the following set of pairs

There are no rules.

1.1.1.1.1.1.1 P is empty

There are no pairs anymore.