YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 Dependency Pair Transformation

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

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
[c#(x1)] =
0 0
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[c(x1)] =
0 -∞
0 -∞
· x1 +
-∞ -∞
-∞ -∞
[a#(x1)] =
1 3
-∞ -∞
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
1 3
1 0
· x1 +
-∞ -∞
-∞ -∞
[b(x1)] =
0 0
1 0
· x1 +
-∞ -∞
-∞ -∞
together with the usable rules
a(x0) → b(b(c(x0)))
c(a(b(x0))) → a(a(c(x0)))
c(b(x0)) → x0
(w.r.t. the implicit argument filter of the reduction pair), the pair
c#(a(b(x0))) → a#(a(c(x0)))
remains.

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.