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

Proof

1 String Reversal

Since only unary symbols occur, one can reverse all terms and obtains the TRS
a(x0) → x0
a(x0) → c(b(b(x0)))
a(c(c(x0))) → c(c(a(a(x0))))

1.1 Dependency Pair Transformation

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

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
1 -∞
· x1 +
0 -∞
3 -∞
[a#(x1)] =
-∞ 1
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
0 -∞
1 0
· x1 +
2 -∞
3 -∞
[b(x1)] =
0 -∞
0 -∞
· x1 +
0 -∞
2 -∞
together with the usable rules
a(x0) → x0
a(x0) → c(b(b(x0)))
a(c(c(x0))) → c(c(a(a(x0))))
(w.r.t. the implicit argument filter of the reduction pair), the pair
a#(c(c(x0))) → a#(a(x0))
remains.

1.1.1.1 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)] =
-∞ 1
0 0
· x1 +
2 -∞
0 -∞
[a#(x1)] =
0 1
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
0 1
-∞ 0
· x1 +
2 -∞
0 -∞
[b(x1)] =
-∞ 0
-∞ 0
· x1 +
0 -∞
0 -∞
together with the usable rules
a(x0) → x0
a(x0) → c(b(b(x0)))
a(c(c(x0))) → c(c(a(a(x0))))
(w.r.t. the implicit argument filter of the reduction pair), all pairs could be removed.

1.1.1.1.1 P is empty

There are no pairs anymore.