YES
proof of SK90_2.42.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 70 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 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:

   flatten(nil) -> nil
   flatten(unit(x)) -> flatten(x)
   flatten(++(x, y)) -> ++(flatten(x), flatten(y))
   flatten(++(unit(x), y)) -> ++(flatten(x), flatten(y))
   flatten(flatten(x)) -> flatten(x)
   rev(nil) -> nil
   rev(unit(x)) -> unit(x)
   rev(++(x, y)) -> ++(rev(y), rev(x))
   rev(rev(x)) -> x
   ++(x, nil) -> x
   ++(nil, y) -> y
   ++(++(x, y), z) -> ++(x, ++(y, z))

Q is empty.

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

   POL(++(x_1, x_2)) = 2 + x_1 + x_2
   POL(flatten(x_1)) = 2 + 2*x_1
   POL(nil) = 2
   POL(rev(x_1)) = 2 + 2*x_1
   POL(unit(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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




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

   flatten(++(x, y)) -> ++(flatten(x), flatten(y))
   rev(++(x, y)) -> ++(rev(y), rev(x))
   ++(++(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)) = 1 + x_1 + x_2
   POL(flatten(x_1)) = 2*x_1
   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:

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




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

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

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

Q is empty.

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

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

   ++(++(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