YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

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

1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the arctic semiring over the integers
[b#(x1)] =
0 0 0 0
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
· x1 +
0 -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
[b(x1)] =
-∞ 0 -∞ 0
-∞ 0 -∞ -∞
0 1 0 0
-∞ 0 -∞ -∞
· x1 +
0 -∞ -∞ -∞
0 -∞ -∞ -∞
1 -∞ -∞ -∞
0 -∞ -∞ -∞
[a#(x1)] =
0 -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
· x1 +
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
[a(x1)] =
0 0 0 0
-∞ 0 -∞ -∞
0 0 0 0
0 0 -∞ 0
· x1 +
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
0 -∞ -∞ -∞
0 -∞ -∞ -∞
[c(x1)] =
1 0 0 0
0 0 -∞ 0
0 1 -∞ 0
0 0 0 0
· x1 +
0 -∞ -∞ -∞
0 -∞ -∞ -∞
0 -∞ -∞ -∞
0 -∞ -∞ -∞
together with the usable rules
a(x0) → x0
a(a(b(x0))) → c(b(b(a(a(x0)))))
b(c(x0)) → a(x0)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
a#(a(b(x0))) → a#(x0)
a#(a(b(x0))) → a#(a(x0))
a#(a(b(x0))) → b#(a(a(x0)))
a#(a(b(x0))) → b#(b(a(a(x0))))
remain.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.