Certification Problem

Input (TPDB TRS_Standard/Transformed_CSR_04/Ex16_Luc06_FR)

The rewrite relation of the following TRS is considered.

f(X,X) → f(a,n__b) (1)
b → a (2)
b → n__b (3)
activate(n__b) → b (4)
activate(X) → X (5)

Property / Task

Prove or disprove termination.

Answer / Result

Yes.

Proof (by ttt2 @ termCOMP 2023)

1 Rule Removal

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the naturals
[activate(x1)] =
1 0 0 1
0 1 0 0
0 0 1 0
0 0 0 1
· x1 +
1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
[a] =
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
[b] =
1 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
[f(x1, x2)] =
1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
· x1 +
1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
· x2 +
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
[n__b] =
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
all of the following rules can be deleted.
b → a (2)
b → n__b (3)
activate(n__b) → b (4)
activate(X) → X (5)

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
f#(X,X) → f#(a,n__b) (6)

1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.