YES Termination w.r.t. Q proof of Transformed_CSR_04_Ex7_BLR02_Z.ari

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRSRRRProof (⇔, 66 ms)
↳2 QTRS
↳3 QTRSRRRProof (⇔, 0 ms)
↳4 QTRS
↳5 RisEmptyProof (⇔, 0 ms)
↳6 YES

(0) Obligation:

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

from(X) → cons(X, n__from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(activate(XS))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
from(x1)  =  from(x1)
cons(x1, x2)  =  cons(x1, x2)
n__from(x1)  =  x1
s(x1)  =  s(x1)
head(x1)  =  x1
2nd(x1)  =  2nd(x1)
activate(x1)  =  activate(x1)
take(x1, x2)  =  take(x1, x2)
0  =  0
nil  =  nil
n__take(x1, x2)  =  n__take(x1, x2)
sel(x1, x2)  =  sel(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:
sel2 > [from1, 2nd1, activate1, take2] > cons2 > ntake2
sel2 > [from1, 2nd1, activate1, take2] > s1 > ntake2
sel2 > [from1, 2nd1, activate1, take2] > [0, nil] > ntake2

Status:
from1: multiset
cons2: multiset
s1: [1]
2nd1: multiset
activate1: multiset
take2: multiset
0: multiset
nil: multiset
ntake2: multiset
sel2: [1,2]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

from(X) → cons(X, n__from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(activate(XS))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
take(X1, X2) → n__take(X1, X2)
activate(n__take(X1, X2)) → take(X1, X2)
activate(X) → X


(2) Obligation:

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

activate(n__from(X)) → from(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Knuth-Bendix order [KBO] with precedence:
activate1 > from1 > nfrom1

and weight map:

activate_1=1
n__from_1=1
from_1=2

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

activate(n__from(X)) → from(X)


(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) YES