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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 56 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) QTRSRRRProof [EQUIVALENT, 1 ms]
(10) QTRS
(11) RisEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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

   A -> B

The conditional TRS C consists of the following conditional rules:

   f(x) -> x <= x -> a
   g(x) -> C <= A -> B


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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(x, x)
   U1(a, x) -> x
   g(x) -> U2(A)
   U2(B) -> C
   A -> B

Q is empty.

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

   POL(A) = 1
   POL(B) = 1
   POL(C) = 2
   POL(U1(x_1, x_2)) = 2 + x_1 + x_2
   POL(U2(x_1)) = 2*x_1
   POL(a) = 1
   POL(f(x_1)) = 2 + 2*x_1
   POL(g(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:

   U1(a, x) -> x




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

   f(x) -> U1(x, x)
   g(x) -> U2(A)
   U2(B) -> C
   A -> B

Q is empty.

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

   POL(A) = 1
   POL(B) = 0
   POL(C) = 0
   POL(U1(x_1, x_2)) = 1 + x_1 + x_2
   POL(U2(x_1)) = 2*x_1
   POL(f(x_1)) = 1 + 2*x_1
   POL(g(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:

   A -> B




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

   f(x) -> U1(x, x)
   g(x) -> U2(A)
   U2(B) -> C

Q is empty.

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(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
f/1(YES)
U1/2(YES,YES)
g/1(YES)
U2/1)YES(
A/0)
B/0)
C/0)

Quasi precedence:
f_1 > [U1_2, A]
g_1 > [U1_2, A]
[B, C] > [U1_2, A]


Status:
f_1: multiset status
U1_2: multiset status
g_1: multiset status
A: multiset status
B: multiset status
C: multiset status

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

   f(x) -> U1(x, x)
   g(x) -> U2(A)




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

   U2(B) -> C

Q is empty.

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(9) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:U2_1 > C > B

and weight map:

   B=1
   C=2
   U2_1=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:

   U2(B) -> C




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(10)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(11) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(12)
YES