YES
proof of TCT_12_polycounter-5.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 59 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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

   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5)
   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5)
   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5)
   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5)
   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5)
   f(0, 0, 0, 0, 0) -> 0

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
s_1 > [f_5, 0]


Status:
f_5: [4,3,2,1,5]
s_1: [1]
0: multiset status

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

   f(s(x1), x2, x3, x4, x5) -> f(x1, x2, x3, x4, x5)
   f(0, s(x2), x3, x4, x5) -> f(x2, x2, x3, x4, x5)
   f(0, 0, s(x3), x4, x5) -> f(x3, x3, x3, x4, x5)
   f(0, 0, 0, s(x4), x5) -> f(x4, x4, x4, x4, x5)
   f(0, 0, 0, 0, s(x5)) -> f(x5, x5, x5, x5, x5)
   f(0, 0, 0, 0, 0) -> 0




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES