YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

a(b(x0)) → b(r(x0))
r(a(x0)) → d(r(x0))
r(x0) → d(x0)
d(a(x0)) → a(a(d(x0)))
d(x0) → a(x0)

Proof

1 Dependency Pair Transformation

The following set of initial dependency pairs has been identified.
a#(b(x0)) → r#(x0)
r#(a(x0)) → r#(x0)
r#(a(x0)) → d#(r(x0))
r#(x0) → d#(x0)
d#(a(x0)) → d#(x0)
d#(a(x0)) → a#(d(x0))
d#(a(x0)) → a#(a(d(x0)))
d#(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
[a#(x1)] =
0 -∞
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 -∞
0 0
· x1 +
-∞ -∞
-∞ -∞
[d(x1)] =
0 -∞
0 0
· x1 +
-∞ -∞
2 -∞
[r#(x1)] =
0 -∞
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[b(x1)] =
1 -∞
2 -∞
· x1 +
0 -∞
-∞ -∞
[d#(x1)] =
0 -∞
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[r(x1)] =
0 -∞
1 1
· x1 +
-∞ -∞
3 -∞
together with the usable rules
a(b(x0)) → b(r(x0))
r(a(x0)) → d(r(x0))
r(x0) → d(x0)
d(a(x0)) → a(a(d(x0)))
d(x0) → a(x0)
(w.r.t. the implicit argument filter of the reduction pair), the pairs
r#(a(x0)) → r#(x0)
r#(a(x0)) → d#(r(x0))
r#(x0) → d#(x0)
d#(a(x0)) → d#(x0)
d#(a(x0)) → a#(d(x0))
d#(a(x0)) → a#(a(d(x0)))
d#(x0) → a#(x0)
remain.

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.