/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Quasi decreasingness of the given CTRS could be proven:

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


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

   g(s(x)) -> x
   h(s(x)) -> x
   a -> d
   b -> d

The conditional TRS C consists of the following conditional rules:

   f(x, y) -> g(x) <= a -> d
   f(x, y) -> h(x) <= b -> d


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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:

   f(x, y) -> U1(a, x)
   U1(d, x) -> g(x)
   f(x, y) -> U2(b, x)
   U2(d, x) -> h(x)
   g(s(x)) -> x
   h(s(x)) -> x
   a -> d
   b -> d

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:d > h_1 > b > s_1 > f_2 > U2_2 > g_1 > a > U1_2

and weight map:

   a=2
   d=1
   b=2
   g_1=1
   h_1=1
   s_1=1
   f_2=2
   U1_2=1
   U2_2=1

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

   f(x, y) -> U1(a, x)
   U1(d, x) -> g(x)
   f(x, y) -> U2(b, x)
   U2(d, x) -> h(x)
   g(s(x)) -> x
   h(s(x)) -> x
   a -> d
   b -> d




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(4)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(5) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(6)
YES