/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 124 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 26 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 12 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 25 ms]
(8) QTRS
(9) QTRSRRRProof [EQUIVALENT, 0 ms]
(10) QTRS
(11) RisEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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

   active(2nd(cons(X, cons(Y, Z)))) -> mark(Y)
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(2nd(X)) -> 2nd(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   2nd(mark(X)) -> mark(2nd(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   proper(2nd(X)) -> 2nd(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   2nd(ok(X)) -> ok(2nd(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(2nd(x_1)) = 2*x_1
   POL(active(x_1)) = 2*x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(from(x_1)) = x_1
   POL(mark(x_1)) = x_1
   POL(ok(x_1)) = 1 + 2*x_1
   POL(proper(x_1)) = x_1
   POL(s(x_1)) = x_1
   POL(top(x_1)) = 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   2nd(ok(X)) -> ok(2nd(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   top(ok(X)) -> top(active(X))




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

   active(2nd(cons(X, cons(Y, Z)))) -> mark(Y)
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(2nd(X)) -> 2nd(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   2nd(mark(X)) -> mark(2nd(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   proper(2nd(X)) -> 2nd(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   top(mark(X)) -> top(proper(X))

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(2nd(x_1)) = 1 + 2*x_1
   POL(active(x_1)) = 2*x_1
   POL(cons(x_1, x_2)) = 2*x_1 + x_2
   POL(from(x_1)) = 2*x_1
   POL(mark(x_1)) = x_1
   POL(ok(x_1)) = 2*x_1
   POL(proper(x_1)) = x_1
   POL(s(x_1)) = x_1
   POL(top(x_1)) = 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   active(2nd(cons(X, cons(Y, Z)))) -> mark(Y)
   active(2nd(X)) -> 2nd(active(X))




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

(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(from(X)) -> mark(cons(X, from(s(X))))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   2nd(mark(X)) -> mark(2nd(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   proper(2nd(X)) -> 2nd(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   from(ok(X)) -> ok(from(X))
   s(ok(X)) -> ok(s(X))
   top(mark(X)) -> top(proper(X))

Q is empty.

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(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(2nd(x_1)) = x_1
   POL(active(x_1)) = 2*x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(from(x_1)) = 2*x_1
   POL(mark(x_1)) = x_1
   POL(ok(x_1)) = 2 + 2*x_1
   POL(proper(x_1)) = x_1
   POL(s(x_1)) = x_1
   POL(top(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   from(ok(X)) -> ok(from(X))




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(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   active(from(X)) -> mark(cons(X, from(s(X))))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   2nd(mark(X)) -> mark(2nd(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   proper(2nd(X)) -> 2nd(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   s(ok(X)) -> ok(s(X))
   top(mark(X)) -> top(proper(X))

Q is empty.

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(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
active/1(YES)
from/1(YES)
mark/1)YES(
cons/2(YES,YES)
s/1(YES)
2nd/1(YES)
proper/1)YES(
ok/1(YES)
top/1)YES(

Quasi precedence:
active_1 > cons_2 > from_1
active_1 > s_1 > ok_1 > from_1
2nd_1 > from_1


Status:
active_1: multiset status
from_1: multiset status
cons_2: multiset status
s_1: multiset status
2nd_1: multiset status
ok_1: multiset status

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

   active(from(X)) -> mark(cons(X, from(s(X))))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(from(X)) -> from(active(X))
   active(s(X)) -> s(active(X))
   s(ok(X)) -> ok(s(X))




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

(8)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   2nd(mark(X)) -> mark(2nd(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   proper(2nd(X)) -> 2nd(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   top(mark(X)) -> top(proper(X))

Q is empty.

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(9) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:proper_1 > top_1 > s_1 > 2nd_1 > from_1 > cons_2 > mark_1

and weight map:

   2nd_1=2
   mark_1=1
   from_1=2
   s_1=2
   proper_1=0
   top_1=1
   cons_2=0

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

   2nd(mark(X)) -> mark(2nd(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   from(mark(X)) -> mark(from(X))
   s(mark(X)) -> mark(s(X))
   proper(2nd(X)) -> 2nd(proper(X))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   proper(s(X)) -> s(proper(X))
   top(mark(X)) -> top(proper(X))




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(10)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(11) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(12)
YES