/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 53 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 5 ms]
(8) QTRS
(9) QTRSRRRProof [EQUIVALENT, 0 ms]
(10) QTRS
(11) QTRSRRRProof [EQUIVALENT, 0 ms]
(12) QTRS
(13) RisEmptyProof [EQUIVALENT, 0 ms]
(14) 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
   d -> e
   k -> e
   l -> e
   s(c) -> t(k)
   s(c) -> t(l)
   s(e) -> t(e)
   g(x, x) -> h(x, x)

The conditional TRS C consists of the following conditional rules:

   f(x) -> pair(x, y) <= s(x) -> t(y)


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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(s(x), x)
   U1(t(y), x) -> pair(x, y)
   a -> c
   a -> d
   b -> c
   b -> d
   c -> e
   d -> e
   k -> e
   l -> e
   s(c) -> t(k)
   s(c) -> t(l)
   s(e) -> t(e)
   g(x, x) -> h(x, x)

Q is empty.

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

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

   a -> c
   a -> d
   b -> c
   d -> e




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

   f(x) -> U1(s(x), x)
   U1(t(y), x) -> pair(x, y)
   b -> d
   c -> e
   k -> e
   l -> e
   s(c) -> t(k)
   s(c) -> t(l)
   s(e) -> t(e)
   g(x, x) -> h(x, x)

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

   b -> d




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

   f(x) -> U1(s(x), x)
   U1(t(y), x) -> pair(x, y)
   c -> e
   k -> e
   l -> e
   s(c) -> t(k)
   s(c) -> t(l)
   s(e) -> t(e)
   g(x, x) -> h(x, x)

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

   g(x, x) -> h(x, x)




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

   f(x) -> U1(s(x), x)
   U1(t(y), x) -> pair(x, y)
   c -> e
   k -> e
   l -> e
   s(c) -> t(k)
   s(c) -> t(l)
   s(e) -> t(e)

Q is empty.

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

   POL(U1(x_1, x_2)) = x_1 + x_2
   POL(c) = 2
   POL(e) = 0
   POL(f(x_1)) = 2 + 2*x_1
   POL(k) = 0
   POL(l) = 0
   POL(pair(x_1, x_2)) = x_1 + 2*x_2
   POL(s(x_1)) = x_1
   POL(t(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(s(x), x)
   c -> e
   s(c) -> t(k)
   s(c) -> t(l)




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

   U1(t(y), x) -> pair(x, y)
   k -> e
   l -> e
   s(e) -> t(e)

Q is empty.

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(11) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:s_1 > l > k > e > t_1 > U1_2 > pair_2

and weight map:

   k=1
   e=1
   l=1
   t_1=1
   s_1=1
   U1_2=0
   pair_2=1

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U1(t(y), x) -> pair(x, y)
   k -> e
   l -> e
   s(e) -> t(e)




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(12)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(13) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(14)
YES