/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 44 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 15 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) RisEmptyProof [EQUIVALENT, 2 ms]
(10) YES


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

   h(a, a) -> i(b)
   h(a, b) -> i(c)
   h(b, b) -> i(d)

The conditional TRS C consists of the following conditional rules:

   f(x) -> g(x, y, z) <= h(a, x) -> i(y), h(a, y) -> i(z)


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

Q is empty.

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

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

   f(x) -> U1(h(a, x), x)




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

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

Q is empty.

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

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

   h(a, b) -> i(c)
   h(b, b) -> i(d)




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

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

Q is empty.

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(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
U1_2 > U2_3 > [i_1, g_3]
U1_2 > [h_2, a, b] > [i_1, g_3]


Status:
U1_2: multiset status
i_1: multiset status
U2_3: multiset status
h_2: multiset status
a: multiset status
g_3: 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:

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




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(8)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(9) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(10)
YES