Termination w.r.t. Q proof of TRS/TRCSREx2_Luc02a

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:

terms(N) → cons(recip(sqr(N)))
sqr(0) → 0
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y)) → cons(Y)

Q is empty.

↳ QTRS

  ↳ DirectTerminationProof

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

terms(N) → cons(recip(sqr(N)))
sqr(0) → 0
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y)) → cons(Y)

Q is empty.

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

Recursive path order with status [2].
Precedence:

terms₁ > recip₁ > nil
terms₁ > sqr₁ > add₂ > s₁ > cons₁ > nil
terms₁ > sqr₁ > dbl₁ > 0 > nil
terms₁ > sqr₁ > dbl₁ > s₁ > cons₁ > nil
first₂ > cons₁ > nil

Status:

dbl₁: multiset
first₂: multiset
terms₁: multiset
recip₁: multiset
sqr₁: multiset
add₂: [2,1]
0: multiset
s₁: [1]
cons₁: multiset
nil: multiset