/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 38 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 16 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

   rev(nil) -> nil
   rev(rev(x)) -> x
   rev(++(x, y)) -> ++(rev(y), rev(x))
   ++(nil, y) -> y
   ++(x, nil) -> x
   ++(.(x, y), z) -> .(x, ++(y, z))
   ++(x, ++(y, z)) -> ++(++(x, y), z)
   make(x) -> .(x, nil)

Q is empty.

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

   POL(++(x_1, x_2)) = x_1 + x_2
   POL(.(x_1, x_2)) = x_1 + x_2
   POL(make(x_1)) = 2 + x_1
   POL(nil) = 0
   POL(rev(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:

   make(x) -> .(x, nil)




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

   rev(nil) -> nil
   rev(rev(x)) -> x
   rev(++(x, y)) -> ++(rev(y), rev(x))
   ++(nil, y) -> y
   ++(x, nil) -> x
   ++(.(x, y), z) -> .(x, ++(y, z))
   ++(x, ++(y, z)) -> ++(++(x, y), z)

Q is empty.

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

   POL(++(x_1, x_2)) = x_1 + x_2
   POL(.(x_1, x_2)) = 2*x_1 + x_2
   POL(nil) = 2
   POL(rev(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:

   rev(nil) -> nil
   ++(nil, y) -> y
   ++(x, nil) -> x




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

   rev(rev(x)) -> x
   rev(++(x, y)) -> ++(rev(y), rev(x))
   ++(.(x, y), z) -> .(x, ++(y, z))
   ++(x, ++(y, z)) -> ++(++(x, y), z)

Q is empty.

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(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
rev_1 > ++_2 > ._2


Status:
rev_1: [1]
++_2: [2,1]
._2: [1,2]

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

   rev(rev(x)) -> x
   rev(++(x, y)) -> ++(rev(y), rev(x))
   ++(.(x, y), z) -> .(x, ++(y, z))
   ++(x, ++(y, z)) -> ++(++(x, y), z)




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(6)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(7) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(8)
YES