Termination w.r.t. Q proof of TRS/SK902.27.trs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(0))) → s(0)
fib(s(s(x))) → sp(g(x))
g(0) → pair(s(0), 0)
g(s(0)) → pair(s(0), s(0))
g(s(x)) → np(g(x))
sp(pair(x, y)) → +(x, y)
np(pair(x, y)) → pair(+(x, y), x)
+(x, 0) → x
+(x, s(y)) → s(+(x, y))

Q is empty.

↳ QTRS

  ↳ DirectTerminationProof

Q restricted rewrite system:
The TRS R consists of the following rules:

fib(0) → 0
fib(s(0)) → s(0)
fib(s(s(0))) → s(0)
fib(s(s(x))) → sp(g(x))
g(0) → pair(s(0), 0)
g(s(0)) → pair(s(0), s(0))
g(s(x)) → np(g(x))
sp(pair(x, y)) → +(x, y)
np(pair(x, y)) → pair(+(x, y), x)
+(x, 0) → x
+(x, s(y)) → s(+(x, y))

Q is empty.

We use [23] with the following order to prove termination.

Lexicographic path order with status [19].
Quasi-Precedence:

[fib₁, g_1] > [sp₁, np₁, +_2] > [0, pair_2] > s₁

Status:

np₁: [1]
sp₁: [1]
+₂: [1,2]
s₁: [1]
0: multiset
fib₁: [1]
g₁: [1]
pair₂: [2,1]