YES
proof of Der95_08.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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 jera 20211004 unpublished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, 67 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:

   D(t) -> 1
   D(constant) -> 0
   D(+(x, y)) -> +(D(x), D(y))
   D(*(x, y)) -> +(*(y, D(x)), *(x, D(y)))
   D(-(x, y)) -> -(D(x), D(y))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
[D_1, t, 1, *_2] > [+_2, -_2]
[constant, 0] > [+_2, -_2]


Status:
D_1: multiset status
t: multiset status
1: multiset status
constant: multiset status
0: multiset status
+_2: [2,1]
*_2: multiset status
-_2: multiset status

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

   D(t) -> 1
   D(constant) -> 0
   D(+(x, y)) -> +(D(x), D(y))
   D(*(x, y)) -> +(*(y, D(x)), *(x, D(y)))
   D(-(x, y)) -> -(D(x), D(y))




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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