/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished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, 86 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 0 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) RisEmptyProof [EQUIVALENT, 0 ms]
(8) YES


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

   f_1(f_1(x, y), z) -> c_0
   f_1(f_0(x, y), z) -> c_0
   f_1(x, f_1(y, z)) -> f_1(f_0(x, y), z)
   f_1(x, f_1(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_0(x, y), z)
   *top*_0(a_1) -> *top*_0(f_0(a_1, a_1))
   f_0(a_1, x0) -> f_1(f_0(a_1, a_1), x0)
   f_0(x0, a_1) -> f_1(x0, f_0(a_1, a_1))

The replacement map contains the following entries:

f_1: empty set
c_0: empty set
f_0: {1, 2}
*top*_0: {1}
a_1: 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:

   f_1(f_1(x, y), z) -> c_0
   f_1(f_0(x, y), z) -> c_0
   f_1(x, f_1(y, z)) -> f_1(f_0(x, y), z)
   f_1(x, f_1(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_0(x, y), z)
   *top*_0(a_1) -> *top*_0(f_0(a_1, a_1))
   f_0(a_1, x0) -> f_1(f_0(a_1, a_1), x0)
   f_0(x0, a_1) -> f_1(x0, f_0(a_1, a_1))

The replacement map contains the following entries:

f_1: empty set
c_0: empty set
f_0: {1, 2}
*top*_0: {1}
a_1: empty set
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(f_1(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(c_0) =  	[[0], [0]]
>>>

   <<<
 POL(f_0(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [1, 0]] 	* 	x_2
>>>

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

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




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

   f_1(f_1(x, y), z) -> c_0
   f_1(f_0(x, y), z) -> c_0
   f_1(x, f_1(y, z)) -> f_1(f_0(x, y), z)
   f_1(x, f_1(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_0(x, y), z)
   f_0(a_1, x0) -> f_1(f_0(a_1, a_1), x0)
   f_0(x0, a_1) -> f_1(x0, f_0(a_1, a_1))

The replacement map contains the following entries:

f_1: empty set
c_0: empty set
f_0: {1, 2}
a_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:

   f_1(f_1(x, y), z) -> c_0
   f_1(f_0(x, y), z) -> c_0
   f_1(x, f_1(y, z)) -> f_1(f_0(x, y), z)
   f_1(x, f_1(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_0(x, y), z)
   f_0(a_1, x0) -> f_1(f_0(a_1, a_1), x0)
   f_0(x0, a_1) -> f_1(x0, f_0(a_1, a_1))

The replacement map contains the following entries:

f_1: empty set
c_0: empty set
f_0: {1, 2}
a_1: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(a_1) = 1
   POL(c_0) = 0
   POL(f_0(x_1, x_2)) = 2 + x_1 + x_2
   POL(f_1(x_1, x_2)) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   f_0(a_1, x0) -> f_1(f_0(a_1, a_1), x0)
   f_0(x0, a_1) -> f_1(x0, f_0(a_1, a_1))




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

   f_1(f_1(x, y), z) -> c_0
   f_1(f_0(x, y), z) -> c_0
   f_1(x, f_1(y, z)) -> f_1(f_0(x, y), z)
   f_1(x, f_1(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_0(x, y), z)

The replacement map contains the following entries:

f_1: empty set
c_0: empty set
f_0: {1, 2}

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

   f_1(f_1(x, y), z) -> c_0
   f_1(f_0(x, y), z) -> c_0
   f_1(x, f_1(y, z)) -> f_1(f_0(x, y), z)
   f_1(x, f_1(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_0(x, y), z)

The replacement map contains the following entries:

f_1: empty set
c_0: empty set
f_0: {1, 2}
Used ordering:
Polynomial interpretation [POLO]:

   POL(c_0) = 0
   POL(f_0(x_1, x_2)) = 1 + x_1 + x_2
   POL(f_1(x_1, x_2)) = 1 + x_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(x, y), z) -> c_0
   f_1(f_0(x, y), z) -> c_0
   f_1(x, f_1(y, z)) -> f_1(f_0(x, y), z)
   f_1(x, f_1(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_1(x, y), z)
   f_1(x, f_0(y, z)) -> f_1(f_0(x, y), z)




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(6)
Obligation:
Context-sensitive rewrite system:
R is empty.

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(7) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
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(8)
YES