/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 64 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 8 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) QTRSRRRProof [EQUIVALENT, 3 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:

   rev(nil) -> nil
   rev(.(x, y)) -> ++(rev(y), .(x, nil))
   car(.(x, y)) -> x
   cdr(.(x, y)) -> y
   null(nil) -> true
   null(.(x, y)) -> false
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))

Q is empty.

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

(1) 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(car(x_1)) = 2*x_1
   POL(cdr(x_1)) = 2*x_1
   POL(false) = 0
   POL(nil) = 0
   POL(null(x_1)) = 1 + 2*x_1
   POL(rev(x_1)) = 2*x_1
   POL(true) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   null(.(x, y)) -> false




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

   rev(nil) -> nil
   rev(.(x, y)) -> ++(rev(y), .(x, nil))
   car(.(x, y)) -> x
   cdr(.(x, y)) -> y
   null(nil) -> true
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))

Q is empty.

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

(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(car(x_1)) = 2*x_1
   POL(cdr(x_1)) = 1 + 2*x_1
   POL(nil) = 0
   POL(null(x_1)) = 2 + 2*x_1
   POL(rev(x_1)) = 2*x_1
   POL(true) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   cdr(.(x, y)) -> y




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

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

Q is empty.

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

(5) 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(car(x_1)) = 1 + 2*x_1
   POL(nil) = 0
   POL(null(x_1)) = 2 + 2*x_1
   POL(rev(x_1)) = 2*x_1
   POL(true) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   car(.(x, y)) -> x




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

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

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

Q is empty.

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

(7) 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) = 0
   POL(null(x_1)) = 1 + 2*x_1
   POL(rev(x_1)) = 2*x_1
   POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   null(nil) -> true




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

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

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

Q is empty.

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

(9) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
rev_1 > [nil, ._2, ++_2]


Status:
rev_1: [1]
nil: multiset status
._2: [1,2]
++_2: [1,2]

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

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




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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