/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Quasi decreasingness of the given CTRS could be proven:

(0) CTRS
(1) CTRSToQTRSProof [SOUND, 0 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 95 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) QTRSRRRProof [EQUIVALENT, 0 ms]
(8) QTRS
(9) RisEmptyProof [EQUIVALENT, 2 ms]
(10) YES


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

   g(d, x, x) -> A
   h(x, x) -> g(x, x, f(k))
   c -> e'
   a -> c
   a -> d
   b -> c
   b -> d
   c -> e
   c -> l
   k -> l
   k -> m
   d -> m

The conditional TRS C consists of the following conditional rules:

   f(x) -> x <= x -> e, x -> e'


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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(e, x) -> U2(x, x)
   U2(e', x) -> x
   g(d, x, x) -> A
   h(x, x) -> g(x, x, f(k))
   c -> e'
   a -> c
   a -> d
   b -> c
   b -> d
   c -> e
   c -> l
   k -> l
   k -> m
   d -> m

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
[d, c, a, b] > e
[d, c, a, b] > e'
[d, c, a, b] > A
[d, c, a, b] > l
[d, c, a, b] > m
h_2 > f_1 > U1_2 > U2_2
h_2 > g_3 > A
h_2 > k > l
h_2 > k > m


Status:
f_1: [1]
U1_2: [2,1]
e: multiset status
U2_2: [2,1]
e': multiset status
g_3: multiset status
d: multiset status
A: multiset status
h_2: [1,2]
k: multiset status
c: multiset status
a: multiset status
b: multiset status
l: multiset status
m: 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)
   U1(e, x) -> U2(x, x)
   U2(e', x) -> x
   g(d, x, x) -> A
   h(x, x) -> g(x, x, f(k))
   c -> e'
   c -> e
   c -> l
   k -> l
   k -> m
   d -> m




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

   a -> c
   a -> d
   b -> c
   b -> d

Q is empty.

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

   POL(a) = 2
   POL(b) = 2
   POL(c) = 2
   POL(d) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   a -> d
   b -> d




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

   a -> c
   b -> c

Q is empty.

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(7) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:b > c > a

and weight map:

   a=2
   c=1
   b=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:

   a -> c
   b -> c




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(8)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

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(9) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(10)
YES