YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

c(c(x0)) → a(b(x0))
b(x0) → a(a(x0))
b(b(b(x0))) → a(c(b(x0)))
a(c(a(x0))) → a(c(c(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
[b(x1)] =
0 1
0 2
· x1 +
-∞ -∞
-∞ -∞
[c(x1)] =
0 2
0 -∞
· x1 +
-∞ -∞
-∞ -∞
[a(x1)] =
0 0
0 0
· x1 +
-∞ -∞
-∞ -∞
the rules
c(c(x0)) → a(b(x0))
b(x0) → a(a(x0))
a(c(a(x0))) → a(c(c(x0)))
remain.

1.1 String Reversal

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

1.1.1 Bounds

The given TRS is match-bounded by 4. This is shown by the following automaton.