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

Proof

1 Dependency Pair Transformation

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

1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the arctic semiring over the integers
[c#(x1)] =
0 0 1
-∞ -∞ -∞
-∞ -∞ -∞
· x1 +
1 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[b(x1)] =
-∞ -∞ -∞
-∞ -∞ 0
0 -∞ -∞
· x1 +
0 -∞ -∞
0 -∞ -∞
0 -∞ -∞
[a#(x1)] =
0 0 -∞
-∞ -∞ -∞
-∞ -∞ -∞
· x1 +
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[a(x1)] =
1 1 0
0 0 0
0 0 0
· x1 +
1 -∞ -∞
0 -∞ -∞
-∞ -∞ -∞
[c(x1)] =
0 0 1
0 0 0
0 -∞ 0
· x1 +
1 -∞ -∞
1 -∞ -∞
0 -∞ -∞
together with the usable rules
a(x0) → x0
a(b(b(a(x0)))) → a(c(c(b(x0))))
c(x0) → a(b(x0))
(w.r.t. the implicit argument filter of the reduction pair), the pairs
a#(b(b(a(x0)))) → c#(b(x0))
a#(b(b(a(x0)))) → c#(c(b(x0)))
a#(b(b(a(x0)))) → a#(c(c(b(x0))))
remain.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.