/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 62 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 11 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:

   le(0, s(x)) -> true
   le(x, 0) -> false
   le(s(x), s(y)) -> le(x, y)
   min(cons(x, nil)) -> x

The conditional TRS C consists of the following conditional rules:

   min(cons(x, l)) -> x <= le(x, min(l)) -> true
   min(cons(x, l)) -> min(l) <= le(x, min(l)) -> false
   min(cons(x, l)) -> min(l) <= min(l) -> x


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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:

   min(cons(x, l)) -> U1(le(x, min(l)), x, l)
   U1(true, x, l) -> x
   U1(false, x, l) -> min(l)
   min(cons(x, l)) -> U2(min(l), l)
   U2(x, l) -> min(l)
   le(0, s(x)) -> true
   le(x, 0) -> false
   le(s(x), s(y)) -> le(x, y)
   min(cons(x, nil)) -> x

Q is empty.

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

   POL(0) = 0
   POL(U1(x_1, x_2, x_3)) = 2 + x_1 + x_2 + 2*x_3
   POL(U2(x_1, x_2)) = x_1 + 2*x_2
   POL(cons(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(false) = 1
   POL(le(x_1, x_2)) = 2 + x_1 + x_2
   POL(min(x_1)) = 2*x_1
   POL(nil) = 0
   POL(s(x_1)) = 2*x_1
   POL(true) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U1(true, x, l) -> x
   U1(false, x, l) -> min(l)
   min(cons(x, l)) -> U2(min(l), l)
   le(0, s(x)) -> true
   le(x, 0) -> false
   min(cons(x, nil)) -> x




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

   min(cons(x, l)) -> U1(le(x, min(l)), x, l)
   U2(x, l) -> min(l)
   le(s(x), s(y)) -> le(x, y)

Q is empty.

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

   POL(U1(x_1, x_2, x_3)) = x_1 + 2*x_2 + 2*x_3
   POL(U2(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(cons(x_1, x_2)) = 2 + 2*x_1 + 2*x_2
   POL(le(x_1, x_2)) = x_1 + x_2
   POL(min(x_1)) = 2*x_1
   POL(s(x_1)) = 1 + 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   min(cons(x, l)) -> U1(le(x, min(l)), x, l)
   U2(x, l) -> min(l)
   le(s(x), s(y)) -> le(x, 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