YES Termination Proof

Termination Proof

by ttt2 (version ttt2 1.15)

Input

The rewrite relation of the following TRS is considered.

f(0(x0)) → s(0(x0))
d(0(x0)) → 0(x0)
d(s(x0)) → s(s(d(p(s(x0)))))
f(s(x0)) → d(f(p(s(x0))))
p(s(x0)) → x0

Proof

1 Rule Removal

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

1.1 Bounds

The given TRS is match-bounded by 0. This is shown by the following automaton.