YES
proof of Various_04_06.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 63 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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

   f(x, y) -> g1(x, x, y)
   f(x, y) -> g1(y, x, x)
   f(x, y) -> g2(x, y, y)
   f(x, y) -> g2(y, y, x)
   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)
   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)
   h(x, x) -> x

Q is empty.

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

   POL(f(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(g1(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(g2(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(h(x_1, x_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(x, y) -> g1(x, x, y)
   f(x, y) -> g1(y, x, x)
   f(x, y) -> g2(x, y, y)
   f(x, y) -> g2(y, y, x)
   g1(x, x, y) -> h(x, y)
   g1(y, x, x) -> h(x, y)
   g2(x, y, y) -> h(x, y)
   g2(y, y, x) -> h(x, y)




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

   h(x, x) -> x

Q is empty.

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

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

   h(x, x) -> x




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(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES