/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 70 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 7 ms]
(6) CSR
(7) CSRInnermostProof [EQUIVALENT, 0 ms]
(8) CSR
(9) CSDependencyPairsProof [EQUIVALENT, 1 ms]
(10) QCSDP
(11) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
(12) TRUE


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(0)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   *top*_0(a_1) -> *top*_0(f_0(a_1))
   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
   f_0(f_0(f_0(a_1))) -> f_1(f_0(f_0(f_0(a_1))))
   f_1(f_1(f_0(f_0(x)))) -> c_0
   f_1(f_0(f_0(f_0(x)))) -> c_0
   f_1(f_1(f_1(f_0(x)))) -> c_0
   f_1(f_1(f_1(f_1(x)))) -> c_0

The replacement map contains the following entries:

*top*_0: {1}
a_1: empty set
f_0: {1}
f_1: empty set
c_0: empty set

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(1) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   *top*_0(a_1) -> *top*_0(f_0(a_1))
   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
   f_0(f_0(f_0(a_1))) -> f_1(f_0(f_0(f_0(a_1))))
   f_1(f_1(f_0(f_0(x)))) -> c_0
   f_1(f_0(f_0(f_0(x)))) -> c_0
   f_1(f_1(f_1(f_0(x)))) -> c_0
   f_1(f_1(f_1(f_1(x)))) -> c_0

The replacement map contains the following entries:

*top*_0: {1}
a_1: empty set
f_0: {1}
f_1: empty set
c_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(*top*_0(x_1)) = 2*x_1
   POL(a_1) = 2
   POL(c_0) = 1
   POL(f_0(x_1)) = x_1
   POL(f_1(x_1)) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f_1(f_1(f_0(f_0(x)))) -> c_0
   f_1(f_0(f_0(f_0(x)))) -> c_0
   f_1(f_1(f_1(f_0(x)))) -> c_0
   f_1(f_1(f_1(f_1(x)))) -> c_0




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

(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   *top*_0(a_1) -> *top*_0(f_0(a_1))
   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
   f_0(f_0(f_0(a_1))) -> f_1(f_0(f_0(f_0(a_1))))

The replacement map contains the following entries:

*top*_0: {1}
a_1: empty set
f_0: {1}
f_1: empty set

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(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   *top*_0(a_1) -> *top*_0(f_0(a_1))
   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))
   f_0(f_0(f_0(a_1))) -> f_1(f_0(f_0(f_0(a_1))))

The replacement map contains the following entries:

*top*_0: {1}
a_1: empty set
f_0: {1}
f_1: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(*top*_0(x_1)) = x_1
   POL(a_1) = 1
   POL(f_0(x_1)) = x_1
   POL(f_1(x_1)) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f_0(f_0(f_0(a_1))) -> f_1(f_0(f_0(f_0(a_1))))




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

(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   *top*_0(a_1) -> *top*_0(f_0(a_1))
   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))

The replacement map contains the following entries:

*top*_0: {1}
a_1: empty set
f_0: {1}

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(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   *top*_0(a_1) -> *top*_0(f_0(a_1))
   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))

The replacement map contains the following entries:

*top*_0: {1}
a_1: empty set
f_0: {1}
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(*top*_0(x_1)) =  	[[0]] 	 +  	[[1, 1]] 	* 	x_1
>>>

   <<<
 POL(a_1) =  	[[0], [1]]
>>>

   <<<
 POL(f_0(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

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

   *top*_0(a_1) -> *top*_0(f_0(a_1))




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

(6)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))

The replacement map contains the following entries:

*top*_0: {1}
a_1: empty set
f_0: {1}

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(7) CSRInnermostProof (EQUIVALENT)
The CSR is orthogonal. By [CS_Inn] we can switch to innermost.
----------------------------------------

(8)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))

The replacement map contains the following entries:

*top*_0: {1}
a_1: empty set
f_0: {1}


Innermost Strategy.

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(9) CSDependencyPairsProof (EQUIVALENT)
Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
----------------------------------------

(10)
Obligation:
Q-restricted context-sensitive dependency pair problem:
The symbols in {*top*_0_1, f_0_1, *TOP*_0_1} are replacing on all positions.

The ordinary context-sensitive dependency pairs DP_o are:
   *TOP*_0(f_0(a_1)) -> *TOP*_0(f_0(f_0(a_1)))
   *TOP*_0(f_0(f_0(a_1))) -> *TOP*_0(f_0(f_0(f_0(a_1))))

The TRS R consists of the following rules:

   *top*_0(f_0(a_1)) -> *top*_0(f_0(f_0(a_1)))
   *top*_0(f_0(f_0(a_1))) -> *top*_0(f_0(f_0(f_0(a_1))))

The set Q consists of the following terms:

   *top*_0(f_0(a_1))
   *top*_0(f_0(f_0(a_1)))


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(11) QCSDependencyGraphProof (EQUIVALENT)
The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 2 less nodes.
The rules *TOP*_0(f_0(a_1)) -> *TOP*_0(f_0(f_0(a_1))) and *TOP*_0(f_0(a_1)) -> *TOP*_0(f_0(f_0(a_1))) form no chain, because ECap^mu(*TOP*_0(f_0(f_0(a_1)))) = *TOP*_0(f_0(f_0(a_1))) does not unify with *TOP*_0(f_0(a_1)). The rules *TOP*_0(f_0(f_0(a_1))) -> *TOP*_0(f_0(f_0(f_0(a_1)))) and *TOP*_0(f_0(a_1)) -> *TOP*_0(f_0(f_0(a_1))) form no chain, because ECap^mu(*TOP*_0(f_0(f_0(f_0(a_1))))) = *TOP*_0(f_0(f_0(f_0(a_1)))) does not unify with *TOP*_0(f_0(a_1)). The rules *TOP*_0(f_0(f_0(a_1))) -> *TOP*_0(f_0(f_0(f_0(a_1)))) and *TOP*_0(f_0(f_0(a_1))) -> *TOP*_0(f_0(f_0(f_0(a_1)))) form no chain, because ECap^mu(*TOP*_0(f_0(f_0(f_0(a_1))))) = *TOP*_0(f_0(f_0(f_0(a_1)))) does not unify with *TOP*_0(f_0(f_0(a_1))). 
----------------------------------------

(12)
TRUE