YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 String Reversal

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

1.1 Dependency Pair Transformation

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

1.1.1 Reduction Pair Processor with Usable Rules

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

1.1.1.1 Reduction Pair Processor with Usable Rules

Using the linear polynomial interpretation over (4 x 4)-matrices with strict dimension 1 over the arctic semiring over the integers
[b#(x1)] =
0 0 0 0
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
· x1 +
0 -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
[b(x1)] =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
· x1 +
0 -∞ -∞ -∞
0 -∞ -∞ -∞
-∞ -∞ -∞ -∞
0 -∞ -∞ -∞
[a#(x1)] =
-∞ 0 0 -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
· x1 +
0 -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
[c#(x1)] =
0 1 0 -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
· x1 +
0 -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
-∞ -∞ -∞ -∞
[a(x1)] =
0 1 0 1
-∞ 0 0 0
0 0 0 0
-∞ 0 0 0
· x1 +
0 -∞ -∞ -∞
-∞ -∞ -∞ -∞
0 -∞ -∞ -∞
0 -∞ -∞ -∞
[c(x1)] =
-∞ 1 0 0
-∞ 0 -∞ 0
-∞ 1 0 0
0 0 0 0
· x1 +
0 -∞ -∞ -∞
-∞ -∞ -∞ -∞
0 -∞ -∞ -∞
0 -∞ -∞ -∞
together with the usable rules
a(a(x0)) → b(b(x0))
b(a(x0)) → c(c(x0))
c(b(x0)) → a(a(x0))
(w.r.t. the implicit argument filter of the reduction pair), the pairs
a#(a(x0)) → b#(b(x0))
b#(a(x0)) → c#(c(x0))
remain.

1.1.1.1.1 Dependency Graph Processor

The dependency pairs are split into 0 components.