/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 63 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 -> b
   g(a) -> h(b)

The conditional TRS C consists of the following conditional rules:

   f(x) -> c(x, y) <= g(x) -> y, y -> h(x)


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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(g(x), x)
   U1(y, x) -> U2(y, x, y)
   U2(h(x), x, y) -> c(x, y)
   a -> b
   g(a) -> h(b)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
f/1(YES)
U1/2(YES,YES)
g/1)YES(
U2/3(YES,YES,YES)
h/1(YES)
c/2(YES,YES)
a/0)
b/0)

Quasi precedence:
f_1 > U1_2 > U2_3 > c_2
a > h_1 > c_2
a > b > c_2


Status:
f_1: [1]
U1_2: [2,1]
U2_3: [2,1,3]
h_1: multiset status
c_2: [2,1]
a: multiset status
b: 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(g(x), x)
   U1(y, x) -> U2(y, x, y)
   U2(h(x), x, y) -> c(x, y)
   a -> b
   g(a) -> h(b)




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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