/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 53 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:

   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))

The conditional TRS C consists of the following conditional rules:

   f(x, y) -> plus(y, z) <= plus(x, y) -> plus(z, 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, y) -> U1(plus(x, y), y)
   U1(plus(z, z), y) -> plus(y, z)
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))

Q is empty.

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

   POL(0) = 2
   POL(U1(x_1, x_2)) = x_1 + x_2
   POL(f(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = 2 + x_1
   POL(z) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f(x, y) -> U1(plus(x, y), y)
   plus(0, y) -> y




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

   U1(plus(z, z), y) -> plus(y, z)
   plus(s(x), y) -> plus(x, s(y))

Q is empty.

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(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:s_1 > z > plus_2 > U1_2

and weight map:

   z=2
   s_1=1
   U1_2=0
   plus_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:

   U1(plus(z, z), y) -> plus(y, z)
   plus(s(x), y) -> plus(x, s(y))




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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