/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 63 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 7 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:

   e(0) -> true

The conditional TRS C consists of the following conditional rules:

   e(s(x)) -> true <= e(x) -> false
   e(s(x)) -> false <= e(x) -> true


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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:

   e(s(x)) -> U1(e(x))
   U1(false) -> true
   U1(true) -> false
   e(0) -> true

Q is empty.

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

   POL(0) = 0
   POL(U1(x_1)) = 2*x_1
   POL(e(x_1)) = 2 + 2*x_1
   POL(false) = 1
   POL(s(x_1)) = 1 + 2*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:

   U1(false) -> true
   U1(true) -> false
   e(0) -> true




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

   e(s(x)) -> U1(e(x))

Q is empty.

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

   POL(U1(x_1)) = 1 + x_1
   POL(e(x_1)) = 2*x_1
   POL(s(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:

   e(s(x)) -> U1(e(x))




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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