/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 151 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RisEmptyProof [EQUIVALENT, 0 ms]
(6) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   D(t) -> s(h)
   D(constant) -> h
   D(b(x, y)) -> b(D(x), D(y))
   D(c(x, y)) -> b(c(y, D(x)), c(x, D(y)))
   D(m(x, y)) -> m(D(x), D(y))
   D(opp(x)) -> opp(D(x))
   D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
   D(ln(x)) -> div(D(x), x)
   D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
   b(h, x) -> x
   b(x, h) -> x
   b(s(x), s(y)) -> s(s(b(x, y)))
   b(b(x, y), z) -> b(x, b(y, z))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
D/1(YES)
t/0)
s/1)YES(
h/0)
constant/0)
b/2(YES,YES)
c/2(YES,YES)
m/2(YES,YES)
opp/1(YES)
div/2(YES,YES)
pow/2(YES,YES)
2/0)
ln/1(YES)
1/0)

Quasi precedence:
D_1 > h
D_1 > c_2 > b_2
D_1 > m_2
D_1 > opp_1
D_1 > div_2 > [pow_2, ln_1] > b_2
D_1 > 2
D_1 > 1
t > h


Status:
D_1: multiset status
t: multiset status
h: multiset status
constant: multiset status
b_2: [1,2]
c_2: multiset status
m_2: multiset status
opp_1: multiset status
div_2: multiset status
pow_2: multiset status
2: multiset status
ln_1: multiset status
1: multiset status

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

   D(t) -> s(h)
   D(constant) -> h
   D(b(x, y)) -> b(D(x), D(y))
   D(c(x, y)) -> b(c(y, D(x)), c(x, D(y)))
   D(m(x, y)) -> m(D(x), D(y))
   D(opp(x)) -> opp(D(x))
   D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
   D(ln(x)) -> div(D(x), x)
   D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
   b(h, x) -> x
   b(x, h) -> x
   b(b(x, y), z) -> b(x, b(y, z))




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(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   b(s(x), s(y)) -> s(s(b(x, y)))

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:b_2 > s_1

and weight map:

   s_1=1
   b_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:

   b(s(x), s(y)) -> s(s(b(x, y)))




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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