YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 54 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:

   if(true, x, y) -> x
   if(false, x, y) -> y
   if(x, y, y) -> y
   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
   if(x, if(x, y, z), z) -> if(x, y, z)
   if(x, y, if(x, y, z)) -> if(x, y, z)

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
true > if_3
false > if_3


Status:
if_3: [1,3,2]
true: multiset status
false: multiset status

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

   if(true, x, y) -> x
   if(false, x, y) -> y
   if(x, y, y) -> y
   if(if(x, y, z), u, v) -> if(x, if(y, u, v), if(z, u, v))
   if(x, if(x, y, z), z) -> if(x, y, z)
   if(x, y, if(x, y, z)) -> if(x, y, z)




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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