YES
proof of SK90_2.39.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) 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.

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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(car(x_1)) = 1 + x_1
   POL(cdr(x_1)) = 2 + x_1
   POL(false) = 0
   POL(nil) = 0
   POL(null(x_1)) = 1 + 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:

   car(.(x, y)) -> x
   cdr(.(x, y)) -> y
   null(nil) -> true
   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))
   ++(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)) = 1 + x_1 + x_2
   POL(.(x_1, x_2)) = 1 + x_1 + x_2
   POL(nil) = 0
   POL(rev(x_1)) = 2 + 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




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

   rev(.(x, y)) -> ++(rev(y), .(x, nil))
   ++(.(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)) = 1 + x_1 + x_2
   POL(nil) = 0
   POL(rev(x_1)) = 2 + 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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




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

(6)
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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(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:++_2 > ._2

and weight map:

   ._2=0
   ++_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:

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




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(8)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(9) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(10)
YES