/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 62 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(x) -> h(x, x)
   h(x, y) -> i(x)

The conditional TRS C consists of the following conditional rules:

   f(x) -> x <= x -> a


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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) -> U1(x, x)
   U1(a, x) -> x
   g(x) -> h(x, x)
   h(x, y) -> i(x)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
f/1(YES)
U1/2(YES,YES)
a/0)
g/1(YES)
h/2(YES,YES)
i/1)YES(

Quasi precedence:
f_1 > U1_2 > h_2
a > h_2
g_1 > h_2


Status:
f_1: multiset status
U1_2: multiset status
a: multiset status
g_1: multiset status
h_2: multiset status

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

   f(x) -> U1(x, x)
   U1(a, x) -> x
   g(x) -> h(x, x)
   h(x, y) -> i(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