/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 96 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 2 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:

   size(empty) -> 0
   size(push(x, y)) -> s(size(x))
   m -> s(s(s(s(0))))
   pop(empty) -> empty
   top(empty) -> eentry
   le(x, 0) -> false
   le(0, s(x)) -> true
   le(s(x), s(y)) -> le(x, y)

The conditional TRS C consists of the following conditional rules:

   pop(push(x, y)) -> x <= le(size(x), m) -> true
   top(push(x, y)) -> y <= le(size(x), m) -> true


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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:

   pop(push(x, y)) -> U1(le(size(x), m), x)
   U1(true, x) -> x
   top(push(x, y)) -> U2(le(size(x), m), y)
   U2(true, y) -> y
   size(empty) -> 0
   size(push(x, y)) -> s(size(x))
   m -> s(s(s(s(0))))
   pop(empty) -> empty
   top(empty) -> eentry
   le(x, 0) -> false
   le(0, s(x)) -> true
   le(s(x), s(y)) -> le(x, y)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
pop/1)YES(
push/2(YES,YES)
U1/2(YES,YES)
le/2(YES,YES)
size/1)YES(
m/0)
true/0)
top/1)YES(
U2/2(YES,YES)
empty/0)
0/0)
s/1)YES(
eentry/0)
false/0)

Quasi precedence:
empty > [push_2, m, true, 0] > U1_2
empty > [push_2, m, true, 0] > le_2 > false
empty > [push_2, m, true, 0] > U2_2
empty > eentry


Status:
push_2: multiset status
U1_2: multiset status
le_2: multiset status
m: multiset status
true: multiset status
U2_2: multiset status
empty: multiset status
0: multiset status
eentry: multiset status
false: multiset status

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

   pop(push(x, y)) -> U1(le(size(x), m), x)
   U1(true, x) -> x
   top(push(x, y)) -> U2(le(size(x), m), y)
   U2(true, y) -> y
   size(empty) -> 0
   size(push(x, y)) -> s(size(x))
   top(empty) -> eentry
   le(x, 0) -> false
   le(0, s(x)) -> true




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

   m -> s(s(s(s(0))))
   pop(empty) -> empty
   le(s(x), s(y)) -> le(x, y)

Q is empty.

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(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:s_1 > le_2 > empty > m > pop_1 > 0

and weight map:

   m=2
   0=1
   empty=1
   s_1=0
   pop_1=1
   le_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:

   m -> s(s(s(s(0))))
   pop(empty) -> empty
   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