YES
proof of Transformed_CSR_04_Ex1_2_Luc02c_FR.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 70 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   2nd(cons(X, n__cons(Y, Z))) -> activate(Y)
   from(X) -> cons(X, n__from(n__s(X)))
   cons(X1, X2) -> n__cons(X1, X2)
   from(X) -> n__from(X)
   s(X) -> n__s(X)
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__from(X)) -> from(activate(X))
   activate(n__s(X)) -> s(activate(X))
   activate(X) -> X

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
2nd/1(YES)
cons/2(YES,YES)
n__cons/2(YES,YES)
activate/1(YES)
from/1(YES)
n__from/1(YES)
n__s/1)YES(
s/1)YES(

Quasi precedence:
[2nd_1, activate_1] > from_1 > cons_2 > n__cons_2
[2nd_1, activate_1] > from_1 > n__from_1


Status:
2nd_1: multiset status
cons_2: multiset status
n__cons_2: multiset status
activate_1: multiset status
from_1: multiset status
n__from_1: multiset status

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

   2nd(cons(X, n__cons(Y, Z))) -> activate(Y)
   from(X) -> cons(X, n__from(n__s(X)))
   cons(X1, X2) -> n__cons(X1, X2)
   from(X) -> n__from(X)
   activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
   activate(n__from(X)) -> from(activate(X))
   activate(X) -> X




----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   s(X) -> n__s(X)
   activate(n__s(X)) -> s(activate(X))

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:activate_1 > s_1 > n__s_1

and weight map:

   s_1=1
   n__s_1=1
   activate_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:

   s(X) -> n__s(X)
   activate(n__s(X)) -> s(activate(X))




----------------------------------------

(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES