/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 59 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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(0)
Obligation:
Conditional term rewrite system:
The TRS R consists of the following rules:

   pos(s(0)) -> true
   pos(0) -> false

The conditional TRS C consists of the following conditional rules:

   pos(s(x)) -> true <= pos(x) -> true
   pos(p(x)) -> false <= pos(x) -> false


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(1) CTRSToQTRSProof (SOUND)
The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC].
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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   pos(s(x)) -> U1(pos(x))
   U1(true) -> true
   pos(p(x)) -> U2(pos(x))
   U2(false) -> false
   pos(s(0)) -> true
   pos(0) -> false

Q is empty.

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

   POL(0) = 0
   POL(U1(x_1)) = x_1
   POL(U2(x_1)) = x_1
   POL(false) = 1
   POL(p(x_1)) = 1 + 2*x_1
   POL(pos(x_1)) = 2 + 2*x_1
   POL(s(x_1)) = x_1
   POL(true) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   pos(p(x)) -> U2(pos(x))
   pos(s(0)) -> true
   pos(0) -> false




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

   pos(s(x)) -> U1(pos(x))
   U1(true) -> true
   U2(false) -> false

Q is empty.

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(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:U2_1 > pos_1 > false > true > U1_1 > s_1

and weight map:

   true=1
   false=1
   pos_1=2
   s_1=1
   U1_1=1
   U2_1=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   pos(s(x)) -> U1(pos(x))
   U1(true) -> true
   U2(false) -> false




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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