Certification Problem

Input (TPDB TRS_Innermost/Mixed_innermost/innermost3)

The rewrite relation of the following TRS is considered.

f(a(x),y) → g(x,y) (1)
g(x,y) → h(x,y) (2)
h(b,y) → f(y,y) (3)
a(b) → c (4)
The evaluation strategy is innermost.

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by AProVE @ termCOMP 2023)

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
f#(a(x),y) → g#(x,y) (5)
g#(x,y) → h#(x,y) (6)
h#(b,y) → f#(y,y) (7)

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 Innermost Lhss Removal Processor

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

a(b)

1.1.1.1 Instantiation Processor

We instantiate the pair to the following set of pairs
f#(a(x0),a(x0)) → g#(x0,a(x0)) (8)

1.1.1.1.1 Instantiation Processor

We instantiate the pair to the following set of pairs
g#(z0,a(z0)) → h#(z0,a(z0)) (9)

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.