/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 153 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:

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)
   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
f/2(YES,YES)
nil/0)
g/2(YES,YES)
++/2(YES,YES)
null/1(YES)
true/0)
false/0)
mem/2(YES,YES)
or/2(YES,YES)
=/2(YES,YES)
max/1)YES(
not/1(YES)
max'/2(YES,YES)
u/0)

Quasi precedence:
f_2 > nil
f_2 > [g_2, max'_2] > false
f_2 > [g_2, max'_2] > or_2
++_2 > [g_2, max'_2] > false
++_2 > [g_2, max'_2] > or_2
[mem_2, =_2] > [null_1, true] > false
[mem_2, =_2] > or_2
[mem_2, =_2] > not_1
u > [g_2, max'_2] > false
u > [g_2, max'_2] > or_2


Status:
f_2: multiset status
nil: multiset status
g_2: multiset status
++_2: [1,2]
null_1: multiset status
true: multiset status
false: multiset status
mem_2: multiset status
or_2: [1,2]
=_2: multiset status
not_1: multiset status
max'_2: multiset status
u: multiset status

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

   f(x, nil) -> g(nil, x)
   f(x, g(y, z)) -> g(f(x, y), z)
   ++(x, nil) -> x
   ++(x, g(y, z)) -> g(++(x, y), z)
   null(nil) -> true
   null(g(x, y)) -> false
   mem(nil, y) -> false
   mem(g(x, y), z) -> or(=(y, z), mem(x, z))
   mem(x, max(x)) -> not(null(x))
   max(g(g(nil, x), y)) -> max'(x, y)




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

   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:max_1 > max'_2 > u > g_2

and weight map:

   u=1
   max_1=2
   g_2=0
   max'_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:

   max(g(g(g(x, y), z), u)) -> max'(max(g(g(x, y), z)), u)




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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