YES
proof of SK90_2.20.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 65 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:

   sum(0) -> 0
   sum(s(x)) -> +(sqr(s(x)), sum(x))
   sqr(x) -> *(x, x)
   sum(s(x)) -> +(*(s(x), s(x)), sum(x))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
s_1 > sum_1 > 0 > *_2
s_1 > sum_1 > +_2 > *_2
s_1 > sum_1 > sqr_1 > *_2


Status:
sum_1: [1]
0: multiset status
s_1: multiset status
+_2: [1,2]
sqr_1: [1]
*_2: [2,1]

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

   sum(0) -> 0
   sum(s(x)) -> +(sqr(s(x)), sum(x))
   sqr(x) -> *(x, x)
   sum(s(x)) -> +(*(s(x), s(x)), sum(x))




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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