/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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 of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 103 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 10 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) CSRRRRProof [EQUIVALENT, 0 ms]
(10) CSR
(11) RisEmptyProof [EQUIVALENT, 0 ms]
(12) YES


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

   *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x)
   f_0(f_1(f_0(g_1(x)))) -> f_0(x)
   g_0(f_1(f_0(g_1(x)))) -> g_1(x)
   *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x))
   f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x))
   g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x))
   f_1(f_0(g_0(x))) -> x
   *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x)
   f_0(f_1(f_0(g_0(x)))) -> f_1(x)
   g_0(f_1(f_0(g_0(x)))) -> g_0(x)
   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))

The replacement map contains the following entries:

*top*_0: {1}
f_1: empty set
f_0: {1}
g_1: empty set
g_0: {1}
b_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(f_1(f_0(g_1(x)))) -> *top*_0(x)
   f_0(f_1(f_0(g_1(x)))) -> f_0(x)
   g_0(f_1(f_0(g_1(x)))) -> g_1(x)
   *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x))
   f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x))
   g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x))
   f_1(f_0(g_0(x))) -> x
   *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x)
   f_0(f_1(f_0(g_0(x)))) -> f_1(x)
   g_0(f_1(f_0(g_0(x)))) -> g_0(x)
   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))

The replacement map contains the following entries:

*top*_0: {1}
f_1: empty set
f_0: {1}
g_1: empty set
g_0: {1}
b_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(*top*_0(x_1)) = 2*x_1
   POL(b_0) = 0
   POL(f_0(x_1)) = 2*x_1
   POL(f_1(x_1)) = x_1
   POL(g_0(x_1)) = 2 + x_1
   POL(g_1(x_1)) = 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   g_0(f_1(f_0(g_1(x)))) -> g_1(x)
   *top*_0(f_0(f_1(f_0(g_0(x))))) -> *top*_0(f_0(x))
   f_0(f_0(f_1(f_0(g_0(x))))) -> f_1(f_0(x))
   g_0(f_0(f_1(f_0(g_0(x))))) -> g_0(f_0(x))
   f_1(f_0(g_0(x))) -> x
   *top*_0(f_1(f_0(g_0(x)))) -> *top*_0(x)
   f_0(f_1(f_0(g_0(x)))) -> f_1(x)
   g_0(f_1(f_0(g_0(x)))) -> g_0(x)




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

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

   *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x)
   f_0(f_1(f_0(g_1(x)))) -> f_0(x)
   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))

The replacement map contains the following entries:

*top*_0: {1}
f_1: empty set
f_0: {1}
g_1: empty set
g_0: {1}
b_0: empty set

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

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

   *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x)
   f_0(f_1(f_0(g_1(x)))) -> f_0(x)
   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))

The replacement map contains the following entries:

*top*_0: {1}
f_1: empty set
f_0: {1}
g_1: empty set
g_0: {1}
b_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(*top*_0(x_1)) = x_1
   POL(b_0) = 0
   POL(f_0(x_1)) = x_1
   POL(f_1(x_1)) = x_1
   POL(g_0(x_1)) = 2*x_1
   POL(g_1(x_1)) = 2 + 2*x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   *top*_0(f_1(f_0(g_1(x)))) -> *top*_0(x)
   f_0(f_1(f_0(g_1(x)))) -> f_0(x)




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

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

   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))

The replacement map contains the following entries:

*top*_0: {1}
f_1: empty set
f_0: {1}
g_1: empty set
g_0: {1}
b_0: empty set

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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(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))
   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))

The replacement map contains the following entries:

*top*_0: {1}
f_1: empty set
f_0: {1}
g_1: empty set
g_0: {1}
b_0: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(*top*_0(x_1)) = x_1
   POL(b_0) = 1
   POL(f_0(x_1)) = x_1
   POL(f_1(x_1)) = 0
   POL(g_0(x_1)) = x_1
   POL(g_1(x_1)) = x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f_0(g_1(b_0)) -> f_1(f_0(g_1(b_0)))




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

   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))

The replacement map contains the following entries:

*top*_0: {1}
f_0: {1}
g_1: empty set
g_0: {1}
b_0: empty set

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

   *top*_0(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))
   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))

The replacement map contains the following entries:

*top*_0: {1}
f_0: {1}
g_1: empty set
g_0: {1}
b_0: empty set
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

   <<<
 POL(g_1(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(b_0) =  	[[0], [1]]
>>>

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

   <<<
 POL(g_0(x_1)) =  	[[0]] 	 +  	[[1, 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(g_1(b_0)) -> *top*_0(f_0(g_1(b_0)))




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

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

   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))

The replacement map contains the following entries:

f_0: {1}
g_1: empty set
g_0: {1}
b_0: empty set

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

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

   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))

The replacement map contains the following entries:

f_0: {1}
g_1: empty set
g_0: {1}
b_0: empty set
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(g_0(x_1)) =  	[[0]] 	 +  	[[1, 1]] 	* 	x_1
>>>

   <<<
 POL(g_1(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(b_0) =  	[[1], [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:

   g_0(g_1(b_0)) -> g_0(f_0(g_1(b_0)))




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(10)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(11) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
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(12)
YES