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


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YES
proof of /export/starexec/sandbox/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 63 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 13 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) 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))

Q is empty.

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

   POL(false) = 2
   POL(if(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
   POL(true) = 2
   POL(u) = 0
   POL(v) = 0
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




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

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

Q is empty.

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

   POL(if(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + x_3
   POL(u) = 0
   POL(v) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   if(x, y, y) -> y




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(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
[u, v] > if_3


Status:
if_3: [1,2,3]
u: multiset status
v: multiset status

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

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




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(6)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(7) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(8)
YES