/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 72 ms]
(2) QTRS
(3) QTRSRRRProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRRRProof [EQUIVALENT, 12 ms]
(6) QTRS
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))
   double(x) -> +(x, x)

Q is empty.

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

   POL(+(x_1, x_2)) = x_1 + x_2
   POL(0) = 1
   POL(double(x_1)) = 2*x_1
   POL(s(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   double(0) -> 0
   +(x, 0) -> x




----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   double(s(x)) -> s(s(double(x)))
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))
   double(x) -> +(x, x)

Q is empty.

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(3) QTRSRRRProof (EQUIVALENT)
Used ordering:
double/1(YES)
s/1)YES(
+/2(YES,YES)

Quasi precedence:
double_1 > +_2


Status:
double_1: [1]
+_2: multiset status

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

   double(x) -> +(x, x)




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

   double(s(x)) -> s(s(double(x)))
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))

Q is empty.

----------------------------------------

(5) QTRSRRRProof (EQUIVALENT)
Used ordering:
Quasi precedence:
double_1 > s_1
+_2 > s_1


Status:
double_1: multiset status
s_1: multiset status
+_2: multiset status

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

   double(s(x)) -> s(s(double(x)))
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))




----------------------------------------

(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