YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 36 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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> =(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
or/2(YES,YES)
true/0)
false/0)
mem/2(YES,YES)
nil/0)
set/1)YES(
=/2(YES,YES)
union/2(YES,YES)

Quasi precedence:
[mem_2, =_2] > or_2 > [true, false, nil]
union_2 > or_2 > [true, false, nil]


Status:
or_2: [2,1]
true: multiset status
false: multiset status
mem_2: [1,2]
nil: multiset status
=_2: [1,2]
union_2: [2,1]

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

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))




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

   mem(x, set(y)) -> =(x, y)

Q is empty.

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

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

   mem(x, set(y)) -> =(x, y)




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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