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.

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

[terms₁, 0, first₂, nil, activate_1] > cons₂ > [s₁, n_{_{first_2]}}
[terms₁, 0, first₂, nil, activate_1] > recip₁ > [s₁, n_{_{first_2]}}
[terms₁, 0, first₂, nil, activate_1] > [sqr₁, dbl_1] > add₂ > [s₁, n_{_{first_2]}}
[terms₁, 0, first₂, nil, activate_1] > n_{_terms₁} > [s₁, n_{_{first_2]}}

Status:

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