/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 60 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

   g(s(x)) -> x
   h(s(x)) -> x

The conditional TRS C consists of the following conditional rules:

   f(x, y) -> g(s(x)) <= c(g(x)) -> c(a)
   f(x, y) -> h(s(x)) <= c(h(x)) -> c(a)


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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, y) -> U1(c(g(x)), x)
   U1(c(a), x) -> g(s(x))
   f(x, y) -> U2(c(h(x)), x)
   U2(c(a), x) -> h(s(x))
   g(s(x)) -> x
   h(s(x)) -> x

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_1 + x_2
   POL(a) = 0
   POL(c(x_1)) = x_1
   POL(f(x_1, x_2)) = 1 + 2*x_1 + x_2
   POL(g(x_1)) = x_1
   POL(h(x_1)) = x_1
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f(x, y) -> U1(c(g(x)), x)
   f(x, y) -> U2(c(h(x)), x)




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

   U1(c(a), x) -> g(s(x))
   U2(c(a), x) -> h(s(x))
   g(s(x)) -> x
   h(s(x)) -> x

Q is empty.

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(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:h_1 > s_1 > U2_2 > g_1 > a > U1_2 > c_1

and weight map:

   a=1
   c_1=1
   g_1=1
   s_1=1
   h_1=1
   U1_2=1
   U2_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(c(a), x) -> g(s(x))
   U2(c(a), x) -> h(s(x))
   g(s(x)) -> x
   h(s(x)) -> x




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(6)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(7) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(8)
YES