/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 29 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:

   a -> c
   a -> d
   b -> c
   b -> d
   s(k) -> t(a)
   s(l) -> t(a)
   g(x, x) -> h(x, x)

The conditional TRS C consists of the following conditional rules:

   f(x, y) -> y <= s(x) -> t(y)


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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(s(x), y)
   U1(t(y), y) -> y
   a -> c
   a -> d
   b -> c
   b -> d
   s(k) -> t(a)
   s(l) -> t(a)
   g(x, x) -> h(x, x)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:g_2 > f_2 > U1_2 > h_2 > l > k > b > a > d > c > s_1 > t_1

and weight map:

   a=2
   c=2
   d=2
   b=2
   k=1
   l=1
   s_1=2
   t_1=1
   f_2=2
   U1_2=0
   g_2=0
   h_2=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:

   f(x, y) -> U1(s(x), y)
   U1(t(y), y) -> y
   a -> c
   a -> d
   b -> c
   b -> d
   s(k) -> t(a)
   s(l) -> t(a)
   g(x, x) -> h(x, x)




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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