/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 46 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 9 ms]
(8) QTRS
(9) QTRSRRRProof [EQUIVALENT, 0 ms]
(10) QTRS
(11) QTRSRRRProof [EQUIVALENT, 0 ms]
(12) QTRS
(13) QTRSRRRProof [EQUIVALENT, 4 ms]
(14) QTRS
(15) QTRSRRRProof [EQUIVALENT, 0 ms]
(16) QTRS
(17) RisEmptyProof [EQUIVALENT, 0 ms]
(18) YES


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(0)
Obligation:
Conditional term rewrite system:
The TRS R consists of the following rules:

   a -> c
   a -> d
   b -> c
   b -> d
   c -> e
   c -> l
   k -> l
   k -> m
   d -> m

The conditional TRS C consists of the following conditional rules:

   f(x) -> x <= x -> e
   g(d, x, y) -> A <= y -> x
   h(x, y) -> g(x, y, f(k)) <= y -> 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(x, x)
   U1(e, x) -> x
   g(d, x, y) -> U2(y)
   U2(x) -> A
   h(x, y) -> U3(y, x, y)
   U3(x, x, y) -> g(x, y, f(k))
   a -> c
   a -> d
   b -> c
   b -> d
   c -> e
   c -> l
   k -> l
   k -> m
   d -> m

Q is empty.

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

   POL(A) = 0
   POL(U1(x_1, x_2)) = x_1 + x_2
   POL(U2(x_1)) = 1 + x_1
   POL(U3(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3
   POL(a) = 2
   POL(b) = 2
   POL(c) = 0
   POL(d) = 1
   POL(e) = 0
   POL(f(x_1)) = 2*x_1
   POL(g(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
   POL(h(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(k) = 1
   POL(l) = 0
   POL(m) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   g(d, x, y) -> U2(y)
   U2(x) -> A
   a -> c
   a -> d
   b -> c
   b -> d
   k -> l
   k -> m
   d -> m




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(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(x) -> U1(x, x)
   U1(e, x) -> x
   h(x, y) -> U3(y, x, y)
   U3(x, x, y) -> g(x, y, f(k))
   c -> e
   c -> l

Q is empty.

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

   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
   POL(U3(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3
   POL(c) = 1
   POL(e) = 1
   POL(f(x_1)) = 2 + 2*x_1
   POL(g(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
   POL(h(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(k) = 0
   POL(l) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U1(e, x) -> x




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(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(x) -> U1(x, x)
   h(x, y) -> U3(y, x, y)
   U3(x, x, y) -> g(x, y, f(k))
   c -> e
   c -> l

Q is empty.

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

   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
   POL(U3(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3
   POL(c) = 2
   POL(e) = 2
   POL(f(x_1)) = 2 + 2*x_1
   POL(g(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
   POL(h(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(k) = 0
   POL(l) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   c -> l




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(8)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(x) -> U1(x, x)
   h(x, y) -> U3(y, x, y)
   U3(x, x, y) -> g(x, y, f(k))
   c -> e

Q is empty.

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

   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
   POL(U3(x_1, x_2, x_3)) = 2 + x_1 + 2*x_2 + x_3
   POL(c) = 2
   POL(e) = 0
   POL(f(x_1)) = 2 + 2*x_1
   POL(g(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
   POL(h(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(k) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   c -> e




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(10)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f(x) -> U1(x, x)
   h(x, y) -> U3(y, x, y)
   U3(x, x, y) -> g(x, y, f(k))

Q is empty.

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

   POL(U1(x_1, x_2)) = 1 + x_1 + x_2
   POL(U3(x_1, x_2, x_3)) = 2 + x_1 + x_2 + x_3
   POL(f(x_1)) = 2 + 2*x_1
   POL(g(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
   POL(h(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(k) = 0
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)




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(12)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   h(x, y) -> U3(y, x, y)
   U3(x, x, y) -> g(x, y, f(k))

Q is empty.

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

   POL(U3(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3
   POL(f(x_1)) = 2*x_1
   POL(g(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + 2*x_3
   POL(h(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(k) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   h(x, y) -> U3(y, x, y)




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(14)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   U3(x, x, y) -> g(x, y, f(k))

Q is empty.

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(15) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:k > f_1 > U3_3 > g_3

and weight map:

   k=1
   f_1=1
   U3_3=1
   g_3=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:

   U3(x, x, y) -> g(x, y, f(k))




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(16)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(17) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(18)
YES