YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

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

Proof

1 String Reversal

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

1.1 Rule Removal

Using the linear polynomial interpretation over (3 x 3)-matrices with strict dimension 1 over the naturals
[b(x1)] =
1 0 0
0 0 0
0 1 1
· x1 +
0 0 0
0 0 0
0 0 0
[a(x1)] =
1 0 1
0 0 0
0 0 1
· x1 +
0 0 0
1 0 0
0 0 0
[c(x1)] =
1 0 0
0 1 0
0 0 0
· x1 +
0 0 0
0 0 0
0 0 0
the rules
c(a(x0)) → a(c(c(x0)))
c(c(c(x0))) → b(c(b(x0)))
remain.

1.1.1 String Reversal

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

1.1.1.1 Bounds

The given TRS is match-bounded by 1. This is shown by the following automaton.