YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(a(b(x0))) → c(b(a(a(a(x0)))))
a(c(x0)) → b(a(x0))

Proof

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
b(a(a(x0))) → a(a(a(b(c(x0)))))
c(a(x0)) → a(b(x0))

1.1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
b#(a(a(x0))) → c#(x0)
b#(a(a(x0))) → b#(c(x0))
c#(a(x0)) → b#(x0)

1.1.1 Reduction Pair Processor

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[c#(x1)] =
0 1 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[a(x1)] =
0 0 1
1 0 0
0 1 0
· x1 +
0 0 0
0 0 0
1 0 0
[b#(x1)] =
1 0 0
0 0 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
[b(x1)] =
1 0 0
1 0 0
1 0 0
· x1 +
0 0 0
1 0 0
0 0 0
[c(x1)] =
0 1 0
0 1 0
0 1 1
· x1 +
0 0 0
0 0 0
1 0 0
the pair
c#(a(x0)) → b#(x0)
remains.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.