Termination w.r.t. Q proof of TRS/TRCSREx1_GL02a

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:

a__eq(0, 0) → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0, X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0) → 0
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)

Q is empty.

↳ QTRS

  ↳ DirectTerminationProof

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

a__eq(0, 0) → true
a__eq(s(X), s(Y)) → a__eq(X, Y)
a__eq(X, Y) → false
a__inf(X) → cons(X, inf(s(X)))
a__take(0, X) → nil
a__take(s(X), cons(Y, L)) → cons(Y, take(X, L))
a__length(nil) → 0
a__length(cons(X, L)) → s(length(L))
mark(eq(X1, X2)) → a__eq(X1, X2)
mark(inf(X)) → a__inf(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(length(X)) → a__length(mark(X))
mark(0) → 0
mark(true) → true
mark(s(X)) → s(X)
mark(false) → false
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
a__eq(X1, X2) → eq(X1, X2)
a__inf(X) → inf(X)
a__take(X1, X2) → take(X1, X2)
a__length(X) → length(X)

Q is empty.

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

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

mark₁ > [a_{_eq₂}, true] > false > [0, cons₂, nil, take₂, eq_2]
mark₁ > a_{_inf₁} > s₁ > [0, cons₂, nil, take₂, eq_2]
mark₁ > a_{_inf₁} > inf₁ > [0, cons₂, nil, take₂, eq_2]
mark₁ > a_{_take₂} > [0, cons₂, nil, take₂, eq_2]
mark₁ > a_{_length₁} > s₁ > [0, cons₂, nil, take₂, eq_2]
mark₁ > a_{_length₁} > length₁ > [0, cons₂, nil, take₂, eq_2]

Status:

length₁: multiset
mark₁: [1]
a_{_length₁}: multiset
a_{_eq₂}: multiset
eq₂: multiset
0: multiset
nil: multiset
a_{_take₂}: [1,2]
cons₂: multiset
true: multiset
false: multiset
s₁: multiset
take₂: [1,2]
inf₁: multiset
a_{_inf₁}: multiset