/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 78 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 28 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) QTRSRRRProof [EQUIVALENT, 10 ms]
(10) QTRS
(11) QTRSRRRProof [EQUIVALENT, 0 ms]
(12) QTRS
(13) QTRSRRRProof [EQUIVALENT, 0 ms]
(14) QTRS
(15) RisEmptyProof [EQUIVALENT, 0 ms]
(16) YES


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

   active(and(tt, X)) -> mark(X)
   active(plus(N, 0)) -> mark(N)
   active(plus(N, s(M))) -> mark(s(plus(N, M)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(0) -> active(0)
   mark(s(X)) -> active(s(mark(X)))
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)

Q is empty.

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

   POL(0) = 1
   POL(active(x_1)) = x_1
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(mark(x_1)) = x_1
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   active(plus(N, 0)) -> mark(N)




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

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

   active(and(tt, X)) -> mark(X)
   active(plus(N, s(M))) -> mark(s(plus(N, M)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(0) -> active(0)
   mark(s(X)) -> active(s(mark(X)))
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)

Q is empty.

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

   POL(0) = 0
   POL(active(x_1)) = x_1
   POL(and(x_1, x_2)) = x_1 + 2*x_2
   POL(mark(x_1)) = x_1
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = x_1
   POL(tt) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   active(and(tt, X)) -> mark(X)




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

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

   active(plus(N, s(M))) -> mark(s(plus(N, M)))
   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(0) -> active(0)
   mark(s(X)) -> active(s(mark(X)))
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)

Q is empty.

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

   POL(0) = 0
   POL(active(x_1)) = x_1
   POL(and(x_1, x_2)) = x_1 + x_2
   POL(mark(x_1)) = x_1
   POL(plus(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(s(x_1)) = 2 + x_1
   POL(tt) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   active(plus(N, s(M))) -> mark(s(plus(N, M)))




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

   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(0) -> active(0)
   mark(s(X)) -> active(s(mark(X)))
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   s(mark(X)) -> s(X)
   s(active(X)) -> s(X)

Q is empty.

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

   POL(0) = 2
   POL(active(x_1)) = 2 + x_1
   POL(and(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(mark(x_1)) = 2*x_1
   POL(plus(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(s(x_1)) = 2 + 2*x_1
   POL(tt) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   and(active(X1), X2) -> and(X1, X2)
   and(X1, active(X2)) -> and(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   s(active(X)) -> s(X)




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

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

   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(0) -> active(0)
   mark(s(X)) -> active(s(mark(X)))
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   s(mark(X)) -> s(X)

Q is empty.

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

   POL(0) = 1
   POL(active(x_1)) = x_1
   POL(and(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(mark(x_1)) = 2*x_1
   POL(plus(x_1, x_2)) = 1 + 2*x_1 + x_2
   POL(s(x_1)) = 2*x_1
   POL(tt) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   mark(and(X1, X2)) -> active(and(mark(X1), X2))
   mark(tt) -> active(tt)
   mark(plus(X1, X2)) -> active(plus(mark(X1), mark(X2)))
   mark(0) -> active(0)




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

(10)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   mark(s(X)) -> active(s(mark(X)))
   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   s(mark(X)) -> s(X)

Q is empty.

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

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

   and(mark(X1), X2) -> and(X1, X2)
   and(X1, mark(X2)) -> and(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   s(mark(X)) -> s(X)




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

(12)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   mark(s(X)) -> active(s(mark(X)))

Q is empty.

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(13) QTRSRRRProof (EQUIVALENT)
Used ordering:
mark/1(YES)
s/1(YES)
active/1)YES(

Quasi precedence:
mark_1 > s_1


Status:
mark_1: multiset status
s_1: multiset status

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

   mark(s(X)) -> active(s(mark(X)))




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(14)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(15) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(16)
YES