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

Proof

1 Rule Removal

Using the linear polynomial interpretation over (2 x 2)-matrices with strict dimension 1 over the arctic semiring over the integers
[a(x1)] =
0 0
0 2
· x1 +
-∞ -∞
-∞ -∞
[b(x1)] =
0 -∞
2 1
· x1 +
-∞ -∞
-∞ -∞
[c(x1)] =
0 -∞
0 -∞
· x1 +
-∞ -∞
-∞ -∞
the rules
b(c(x0)) → c(b(b(x0)))
b(a(x0)) → a(c(b(x0)))
remain.

1.1 String Reversal

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

1.1.1 Dependency Pair Transformation

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

1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 2 components.