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)), n__terms(s(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, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DirectTerminationProof

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

terms(N) → cons(recip(sqr(N)), n__terms(s(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, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Q is empty.

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

Recursive path order with status [2].
Quasi-Precedence:

[first₂, activate_1] > terms₁ > recip₁ > [cons₂, s₁, 0, nil]
[first₂, activate_1] > terms₁ > [sqr₁, dbl_1] > add₂ > [cons₂, s₁, 0, nil]
[first₂, activate_1] > terms₁ > n_{_terms₁} > [cons₂, s₁, 0, nil]
[first₂, activate_1] > n_{_first₂} > [cons₂, s₁, 0, nil]

Status:

dbl₁: multiset
first₂: multiset
0: multiset
nil: multiset
n_{_terms₁}: multiset
cons₂: [1,2]
activate₁: multiset
n_{_first₂}: [2,1]
sqr₁: multiset
recip₁: multiset
terms₁: multiset
add₂: multiset
s₁: multiset