YES
proof of Mixed_TRS_jones6.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 68 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:

   f(a, empty) -> g(a, empty)
   f(a, cons(x, k)) -> f(cons(x, a), k)
   g(empty, d) -> d
   g(cons(x, k), d) -> g(k, cons(x, d))

Q is empty.

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

   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
   POL(empty) = 0
   POL(f(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(g(x_1, x_2)) = 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:

   f(a, empty) -> g(a, empty)
   g(cons(x, k), d) -> g(k, cons(x, d))




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

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

   f(a, cons(x, k)) -> f(cons(x, a), k)
   g(empty, d) -> d

Q is empty.

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

   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(empty) = 0
   POL(f(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(g(x_1, x_2)) = 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:

   g(empty, d) -> d




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

   f(a, cons(x, k)) -> f(cons(x, a), k)

Q is empty.

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

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

   f(a, cons(x, k)) -> f(cons(x, a), k)




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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