YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

g(h(x0)) → g(f(s(x0)))
f(s(s(s(x0)))) → h(f(s(h(x0))))
f(h(x0)) → h(f(s(h(x0))))
h(x0) → x0
f(f(s(s(x0)))) → s(s(s(f(f(x0)))))
b(a(x0)) → a(b(x0))
a(a(a(x0))) → b(a(a(b(x0))))
b(b(b(b(x0)))) → a(x0)

Proof

1 Rule Removal

Using the linear polynomial interpretation over the arctic semiring over the integers
[a(x1)] = 7 · x1 + -∞
[g(x1)] = 0 · x1 + -∞
[f(x1)] = 0 · x1 + -∞
[b(x1)] = 2 · x1 + -∞
[h(x1)] = 0 · x1 + -∞
[s(x1)] = 0 · x1 + -∞
the rules
g(h(x0)) → g(f(s(x0)))
f(s(s(s(x0)))) → h(f(s(h(x0))))
f(h(x0)) → h(f(s(h(x0))))
h(x0) → x0
f(f(s(s(x0)))) → s(s(s(f(f(x0)))))
b(a(x0)) → a(b(x0))
remain.

1.1 Bounds

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