/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 61 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 1 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) RisEmptyProof [EQUIVALENT, 0 ms]
(10) YES


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

   f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
   f_1(f_0(x)) -> b_0
   f_1(f_1(x)) -> b_0
   g_1(x) -> f_1(g_1(x))
   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
   f_0(g_1(x)) -> f_1(f_0(g_1(x)))
   s_0(g_1(x)) -> s_0(f_0(g_1(x)))

The replacement map contains the following entries:

f_1: empty set
g_1: empty set
a_0: empty set
f_0: {1}
s_0: {1}
b_0: empty set
*top*_0: {1}

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

   f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))
   f_1(f_0(x)) -> b_0
   f_1(f_1(x)) -> b_0
   g_1(x) -> f_1(g_1(x))
   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
   f_0(g_1(x)) -> f_1(f_0(g_1(x)))
   s_0(g_1(x)) -> s_0(f_0(g_1(x)))

The replacement map contains the following entries:

f_1: empty set
g_1: empty set
a_0: empty set
f_0: {1}
s_0: {1}
b_0: empty set
*top*_0: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(*top*_0(x_1)) = x_1
   POL(a_0) = 1
   POL(b_0) = 0
   POL(f_0(x_1)) = x_1
   POL(f_1(x_1)) = x_1
   POL(g_1(x_1)) = 2*x_1
   POL(s_0(x_1)) = 1 + x_1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f_1(g_1(a_0)) -> f_0(s_0(g_1(b_0)))




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

   f_1(f_0(x)) -> b_0
   f_1(f_1(x)) -> b_0
   g_1(x) -> f_1(g_1(x))
   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
   f_0(g_1(x)) -> f_1(f_0(g_1(x)))
   s_0(g_1(x)) -> s_0(f_0(g_1(x)))

The replacement map contains the following entries:

f_1: empty set
g_1: empty set
f_0: {1}
s_0: {1}
b_0: empty set
*top*_0: {1}

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

   f_1(f_0(x)) -> b_0
   f_1(f_1(x)) -> b_0
   g_1(x) -> f_1(g_1(x))
   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
   f_0(g_1(x)) -> f_1(f_0(g_1(x)))
   s_0(g_1(x)) -> s_0(f_0(g_1(x)))

The replacement map contains the following entries:

f_1: empty set
g_1: empty set
f_0: {1}
s_0: {1}
b_0: empty set
*top*_0: {1}
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)) = 1
   POL(g_1(x_1)) = 2
   POL(s_0(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_1(f_0(x)) -> b_0
   f_1(f_1(x)) -> b_0
   g_1(x) -> f_1(g_1(x))
   f_0(g_1(x)) -> f_1(f_0(g_1(x)))




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

   *top*_0(g_1(x)) -> *top*_0(f_0(g_1(x)))
   s_0(g_1(x)) -> s_0(f_0(g_1(x)))

The replacement map contains the following entries:

g_1: empty set
f_0: {1}
s_0: {1}
*top*_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(g_1(x)) -> *top*_0(f_0(g_1(x)))
   s_0(g_1(x)) -> s_0(f_0(g_1(x)))

The replacement map contains the following entries:

g_1: empty set
f_0: {1}
s_0: {1}
*top*_0: {1}
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

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

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

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

   <<<
 POL(s_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(x)) -> *top*_0(f_0(g_1(x)))




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

   s_0(g_1(x)) -> s_0(f_0(g_1(x)))

The replacement map contains the following entries:

g_1: empty set
f_0: {1}
s_0: {1}

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

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

   s_0(g_1(x)) -> s_0(f_0(g_1(x)))

The replacement map contains the following entries:

g_1: empty set
f_0: {1}
s_0: {1}
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(s_0(x_1)) =  	[[0]] 	 +  	[[1, 1]] 	* 	x_1
>>>

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

   <<<
 POL(f_0(x_1)) =  	[[0], [0]] 	 +  	[[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:

   s_0(g_1(x)) -> s_0(f_0(g_1(x)))




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(8)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(9) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
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(10)
YES