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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 149 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:

   dx(X) -> one
   dx(a) -> zero
   dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA))
   dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
   dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA))
   dx(neg(ALPHA)) -> neg(dx(ALPHA))
   dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
   dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA)
   dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
dx/1(YES)
one/0)
a/0)
zero/0)
plus/2(YES,YES)
times/2(YES,YES)
minus/2(YES,YES)
neg/1)YES(
div/2(YES,YES)
exp/2(YES,YES)
two/0)
ln/1(YES)

Quasi precedence:
[dx_1, one] > [a, zero]
[dx_1, one] > two
[dx_1, one] > ln_1 > [div_2, exp_2] > [plus_2, times_2]
[dx_1, one] > ln_1 > [div_2, exp_2] > minus_2


Status:
dx_1: multiset status
one: multiset status
a: multiset status
zero: multiset status
plus_2: [2,1]
times_2: multiset status
minus_2: multiset status
div_2: multiset status
exp_2: multiset status
two: multiset status
ln_1: multiset status

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

   dx(X) -> one
   dx(a) -> zero
   dx(plus(ALPHA, BETA)) -> plus(dx(ALPHA), dx(BETA))
   dx(times(ALPHA, BETA)) -> plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
   dx(minus(ALPHA, BETA)) -> minus(dx(ALPHA), dx(BETA))
   dx(div(ALPHA, BETA)) -> minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
   dx(ln(ALPHA)) -> div(dx(ALPHA), ALPHA)
   dx(exp(ALPHA, BETA)) -> plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))




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

   dx(neg(ALPHA)) -> neg(dx(ALPHA))

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Polynomial interpretation [POLO]:

   POL(dx(x_1)) = 2*x_1
   POL(neg(x_1)) = 1 + 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   dx(neg(ALPHA)) -> neg(dx(ALPHA))




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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