YES
proof of Transformed_CSR_04_Ex1_Luc02b_Z.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 73 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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

   from(X) -> cons(X, n__from(s(X)))
   first(0, Z) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   sel(0, cons(X, Z)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
sel_2 > [from_1, first_2, activate_1] > cons_2 > n__first_2 > [n__from_1, s_1]
sel_2 > [from_1, first_2, activate_1] > [0, nil] > [n__from_1, s_1]


Status:
from_1: multiset status
cons_2: multiset status
n__from_1: multiset status
s_1: [1]
first_2: multiset status
0: multiset status
nil: multiset status
n__first_2: multiset status
activate_1: multiset status
sel_2: [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)))
   first(0, Z) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   sel(0, cons(X, Z)) -> X
   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
   from(X) -> n__from(X)
   first(X1, X2) -> n__first(X1, X2)
   activate(n__from(X)) -> from(X)
   activate(n__first(X1, X2)) -> first(X1, X2)
   activate(X) -> X




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES