/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 90 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:

   merge(nil, y) -> y
   merge(x, nil) -> x
   merge(.(x, y), .(u, v)) -> if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))
   if(true, x, y) -> x
   if(false, x, y) -> x

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
merge_2 > [._2, if_3, <_2]
nil > [._2, if_3, <_2]
++_2 > [._2, if_3, <_2]
true > [._2, if_3, <_2]
false > [._2, if_3, <_2]


Status:
merge_2: [1,2]
nil: multiset status
._2: [1,2]
if_3: multiset status
<_2: [1,2]
++_2: [1,2]
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:

   merge(nil, y) -> y
   merge(x, nil) -> x
   merge(.(x, y), .(u, v)) -> if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))
   if(true, x, y) -> x
   if(false, x, y) -> 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