YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 String Reversal

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

1.1 Dependency Pair Transformation

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

1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[b#(x1)] =
0 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[c(x1)] =
1 0
0 -∞
· x1 +
0 -∞
0 -∞
[a#(x1)] =
0 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[c#(x1)] =
0 -∞
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
0 1
-∞ 0
· x1 +
1 -∞
0 -∞
[b(x1)] =
-∞ 0
0 1
· x1 +
0 -∞
1 -∞
together with the usable rules
a(x0) → c(b(x0))
b(a(b(x0))) → a(a(a(x0)))
c(c(x0)) → a(x0)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
a#(x0) → b#(x0)
a#(x0) → c#(b(x0))
c#(c(x0)) → a#(x0)
remain.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.