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

Proof

1 String Reversal

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

1.1 Dependency Pair Transformation

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

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

1.1.1.1.1 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
[b#(x1)] =
1 1
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[b(x1)] =
0 -∞
0 -∞
· x1 +
0 -∞
0 -∞
[a#(x1)] =
1 0
-∞ -∞
· x1 +
0 -∞
-∞ -∞
[a(x1)] =
1 0
0 0
· x1 +
3 -∞
0 -∞
[c(x1)] =
-∞ 0
1 -∞
· x1 +
0 -∞
3 -∞
together with the usable rules
a(x0) → x0
b(a(x0)) → a(c(b(x0)))
a(c(c(x0))) → a(b(a(x0)))
(w.r.t. the implicit argument filter of the reduction pair), the pairs
a#(c(c(x0))) → b#(a(x0))
a#(c(c(x0))) → a#(b(a(x0)))
remain.

1.1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.