/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 130 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:

   lte(0, n) -> true
   lte(s(m), 0) -> false
   lte(s(m), s(n)) -> lte(m, n)
   insert(nil, m) -> cons(m, nil)
   ordered(nil) -> true
   ordered(cons(m, nil)) -> true

The conditional TRS C consists of the following conditional rules:

   insert(cons(n, l), m) -> cons(m, cons(n, l)) <= lte(m, n) -> true
   insert(cons(n, l), m) -> cons(n, insert(l, m)) <= lte(m, n) -> false
   ordered(cons(m, cons(n, l))) -> ordered(cons(n, l)) <= lte(m, n) -> true
   ordered(cons(m, cons(n, l))) -> false <= lte(m, n) -> false


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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:

   insert(cons(n, l), m) -> U1(lte(m, n), n, l, m)
   U1(false, n, l, m) -> cons(n, insert(l, m))
   U1(true, n, l, m) -> cons(m, cons(n, l))
   ordered(cons(m, cons(n, l))) -> U2(lte(m, n), n, l)
   U2(true, n, l) -> ordered(cons(n, l))
   U2(false, n, l) -> false
   lte(0, n) -> true
   lte(s(m), 0) -> false
   lte(s(m), s(n)) -> lte(m, n)
   insert(nil, m) -> cons(m, nil)
   ordered(nil) -> true
   ordered(cons(m, nil)) -> true

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
insert/2(YES,YES)
cons/2(YES,YES)
U1/4(YES,YES,YES,YES)
lte/2(YES,YES)
false/0)
true/0)
ordered/1)YES(
U2/3(YES,YES,YES)
0/0)
s/1)YES(
nil/0)

Quasi precedence:
[insert_2, U1_4, nil] > [true, 0] > [cons_2, lte_2, U2_3] > false


Status:
insert_2: [1,2]
cons_2: [2,1]
U1_4: [3,4,1,2]
lte_2: [2,1]
false: multiset status
true: multiset status
U2_3: [3,2,1]
0: multiset status
nil: multiset status

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

   insert(cons(n, l), m) -> U1(lte(m, n), n, l, m)
   U1(false, n, l, m) -> cons(n, insert(l, m))
   U1(true, n, l, m) -> cons(m, cons(n, l))
   ordered(cons(m, cons(n, l))) -> U2(lte(m, n), n, l)
   U2(true, n, l) -> ordered(cons(n, l))
   U2(false, n, l) -> false
   lte(0, n) -> true
   lte(s(m), 0) -> false
   insert(nil, m) -> cons(m, nil)
   ordered(nil) -> true
   ordered(cons(m, nil)) -> true




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

   lte(s(m), s(n)) -> lte(m, n)

Q is empty.

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(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:s_1 > lte_2

and weight map:

   s_1=0
   lte_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:

   lte(s(m), s(n)) -> lte(m, n)




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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