YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

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

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
[c#(x1)] =
0 2
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[c(x1)] =
0 -∞
1 0
· x1 +
-∞ -∞
-∞ -∞
[a#(x1)] =
1 2
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[b#(x1)] =
0 2
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 0
2 1
· x1 +
-∞ -∞
-∞ -∞
[b(x1)] =
0 0
2 1
· x1 +
-∞ -∞
-∞ -∞
together with the usable rules
a(a(x0)) → b(a(c(x0)))
b(b(x0)) → a(a(x0))
c(b(x0)) → a(x0)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
a#(a(x0)) → b#(a(c(x0)))
b#(b(x0)) → a#(a(x0))
remain.

1.1.1 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 -∞
0 0 -∞
0 0 -∞
· x1 +
-∞ -∞ -∞
0 -∞ -∞
-∞ -∞ -∞
[a#(x1)] =
0 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1 +
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[b#(x1)] =
1 0 0
-∞ -∞ -∞
-∞ -∞ -∞
· x1 +
0 -∞ -∞
-∞ -∞ -∞
-∞ -∞ -∞
[a(x1)] =
0 -∞ 0
0 -∞ 0
1 1 0
· x1 +
-∞ -∞ -∞
-∞ -∞ -∞
1 -∞ -∞
[b(x1)] =
1 1 0
0 1 0
0 1 -∞
· x1 +
1 -∞ -∞
1 -∞ -∞
0 -∞ -∞
together with the usable rules
a(a(x0)) → b(a(c(x0)))
b(b(x0)) → a(a(x0))
c(b(x0)) → a(x0)
(w.r.t. the implicit argument filter of the reduction pair), the pair
a#(a(x0)) → b#(a(c(x0)))
remains.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.