/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 55 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   +(0, 0) -> 0
   +(0, 1) -> 1
   +(0, 2) -> 2
   +(0, 3) -> 3
   +(0, 4) -> 4
   +(0, 5) -> 5
   +(0, 6) -> 6
   +(0, 7) -> 7
   +(0, 8) -> 8
   +(0, 9) -> 9
   +(1, 0) -> 1
   +(1, 1) -> 2
   +(1, 2) -> 3
   +(1, 3) -> 4
   +(1, 4) -> 5
   +(1, 5) -> 6
   +(1, 6) -> 7
   +(1, 7) -> 8
   +(1, 8) -> 9
   +(1, 9) -> c(1, 0)
   +(2, 0) -> 2
   +(2, 1) -> 3
   +(2, 2) -> 4
   +(2, 3) -> 5
   +(2, 4) -> 6
   +(2, 5) -> 7
   +(2, 6) -> 8
   +(2, 7) -> 9
   +(2, 8) -> c(1, 0)
   +(2, 9) -> c(1, 1)
   +(3, 0) -> 3
   +(3, 1) -> 4
   +(3, 2) -> 5
   +(3, 3) -> 6
   +(3, 4) -> 7
   +(3, 5) -> 8
   +(3, 6) -> 9
   +(3, 7) -> c(1, 0)
   +(3, 8) -> c(1, 1)
   +(3, 9) -> c(1, 2)
   +(4, 0) -> 4
   +(4, 1) -> 5
   +(4, 2) -> 6
   +(4, 3) -> 7
   +(4, 4) -> 8
   +(4, 5) -> 9
   +(4, 6) -> c(1, 0)
   +(4, 7) -> c(1, 1)
   +(4, 8) -> c(1, 2)
   +(4, 9) -> c(1, 3)
   +(5, 0) -> 5
   +(5, 1) -> 6
   +(5, 2) -> 7
   +(5, 3) -> 8
   +(5, 4) -> 9
   +(5, 5) -> c(1, 0)
   +(5, 6) -> c(1, 1)
   +(5, 7) -> c(1, 2)
   +(5, 8) -> c(1, 3)
   +(5, 9) -> c(1, 4)
   +(6, 0) -> 6
   +(6, 1) -> 7
   +(6, 2) -> 8
   +(6, 3) -> 9
   +(6, 4) -> c(1, 0)
   +(6, 5) -> c(1, 1)
   +(6, 6) -> c(1, 2)
   +(6, 7) -> c(1, 3)
   +(6, 8) -> c(1, 4)
   +(6, 9) -> c(1, 5)
   +(7, 0) -> 7
   +(7, 1) -> 8
   +(7, 2) -> 9
   +(7, 3) -> c(1, 0)
   +(7, 4) -> c(1, 1)
   +(7, 5) -> c(1, 2)
   +(7, 6) -> c(1, 3)
   +(7, 7) -> c(1, 4)
   +(7, 8) -> c(1, 5)
   +(7, 9) -> c(1, 6)
   +(8, 0) -> 8
   +(8, 1) -> 9
   +(8, 2) -> c(1, 0)
   +(8, 3) -> c(1, 1)
   +(8, 4) -> c(1, 2)
   +(8, 5) -> c(1, 3)
   +(8, 6) -> c(1, 4)
   +(8, 7) -> c(1, 5)
   +(8, 8) -> c(1, 6)
   +(8, 9) -> c(1, 7)
   +(9, 0) -> 9
   +(9, 1) -> c(1, 0)
   +(9, 2) -> c(1, 1)
   +(9, 3) -> c(1, 2)
   +(9, 4) -> c(1, 3)
   +(9, 5) -> c(1, 4)
   +(9, 6) -> c(1, 5)
   +(9, 7) -> c(1, 6)
   +(9, 8) -> c(1, 7)
   +(9, 9) -> c(1, 8)
   +(x, c(y, z)) -> c(y, +(x, z))
   +(c(x, y), z) -> c(x, +(y, z))
   c(0, x) -> x
   c(x, c(y, z)) -> c(+(x, y), z)

Q is empty.

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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:1 > 5 > 2 > 3 > 7 > +_2 > 9 > 8 > 4 > c_2 > 0 > 6

and weight map:

   0=1
   1=9
   2=2
   3=7
   4=11
   5=8
   6=12
   7=17
   8=22
   9=27
   +_2=8
   c_2=9

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

   +(0, 0) -> 0
   +(0, 1) -> 1
   +(0, 2) -> 2
   +(0, 3) -> 3
   +(0, 4) -> 4
   +(0, 5) -> 5
   +(0, 6) -> 6
   +(0, 7) -> 7
   +(0, 8) -> 8
   +(0, 9) -> 9
   +(1, 0) -> 1
   +(1, 1) -> 2
   +(1, 2) -> 3
   +(1, 3) -> 4
   +(1, 4) -> 5
   +(1, 5) -> 6
   +(1, 6) -> 7
   +(1, 7) -> 8
   +(1, 8) -> 9
   +(1, 9) -> c(1, 0)
   +(2, 0) -> 2
   +(2, 1) -> 3
   +(2, 2) -> 4
   +(2, 3) -> 5
   +(2, 4) -> 6
   +(2, 5) -> 7
   +(2, 6) -> 8
   +(2, 7) -> 9
   +(2, 8) -> c(1, 0)
   +(2, 9) -> c(1, 1)
   +(3, 0) -> 3
   +(3, 1) -> 4
   +(3, 2) -> 5
   +(3, 3) -> 6
   +(3, 4) -> 7
   +(3, 5) -> 8
   +(3, 6) -> 9
   +(3, 7) -> c(1, 0)
   +(3, 8) -> c(1, 1)
   +(3, 9) -> c(1, 2)
   +(4, 0) -> 4
   +(4, 1) -> 5
   +(4, 2) -> 6
   +(4, 3) -> 7
   +(4, 4) -> 8
   +(4, 5) -> 9
   +(4, 6) -> c(1, 0)
   +(4, 7) -> c(1, 1)
   +(4, 8) -> c(1, 2)
   +(4, 9) -> c(1, 3)
   +(5, 0) -> 5
   +(5, 1) -> 6
   +(5, 2) -> 7
   +(5, 3) -> 8
   +(5, 4) -> 9
   +(5, 5) -> c(1, 0)
   +(5, 6) -> c(1, 1)
   +(5, 7) -> c(1, 2)
   +(5, 8) -> c(1, 3)
   +(5, 9) -> c(1, 4)
   +(6, 0) -> 6
   +(6, 1) -> 7
   +(6, 2) -> 8
   +(6, 3) -> 9
   +(6, 4) -> c(1, 0)
   +(6, 5) -> c(1, 1)
   +(6, 6) -> c(1, 2)
   +(6, 7) -> c(1, 3)
   +(6, 8) -> c(1, 4)
   +(6, 9) -> c(1, 5)
   +(7, 0) -> 7
   +(7, 1) -> 8
   +(7, 2) -> 9
   +(7, 3) -> c(1, 0)
   +(7, 4) -> c(1, 1)
   +(7, 5) -> c(1, 2)
   +(7, 6) -> c(1, 3)
   +(7, 7) -> c(1, 4)
   +(7, 8) -> c(1, 5)
   +(7, 9) -> c(1, 6)
   +(8, 0) -> 8
   +(8, 1) -> 9
   +(8, 2) -> c(1, 0)
   +(8, 3) -> c(1, 1)
   +(8, 4) -> c(1, 2)
   +(8, 5) -> c(1, 3)
   +(8, 6) -> c(1, 4)
   +(8, 7) -> c(1, 5)
   +(8, 8) -> c(1, 6)
   +(8, 9) -> c(1, 7)
   +(9, 0) -> 9
   +(9, 1) -> c(1, 0)
   +(9, 2) -> c(1, 1)
   +(9, 3) -> c(1, 2)
   +(9, 4) -> c(1, 3)
   +(9, 5) -> c(1, 4)
   +(9, 6) -> c(1, 5)
   +(9, 7) -> c(1, 6)
   +(9, 8) -> c(1, 7)
   +(9, 9) -> c(1, 8)
   +(x, c(y, z)) -> c(y, +(x, z))
   +(c(x, y), z) -> c(x, +(y, z))
   c(0, x) -> x
   c(x, c(y, z)) -> c(+(x, y), z)




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES