YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRS Reverse [EQUIVALENT, 0 ms]
(2) RelTRS
(3) FlatCCProof [EQUIVALENT, 0 ms]
(4) RelTRS
(5) RootLabelingProof [EQUIVALENT, 0 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 243 ms]
(8) RelTRS
(9) RelTRSRRRProof [EQUIVALENT, 32 ms]
(10) RelTRS
(11) RelTRSRRRProof [EQUIVALENT, 4 ms]
(12) RelTRS
(13) RIsEmptyProof [EQUIVALENT, 0 ms]
(14) YES


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(0)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a(a(x1)) -> x1

The relative TRS consists of the following S rules:

   a(a(x1)) -> b(a(a(a(b(x1)))))


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(1) RelTRS Reverse (EQUIVALENT)
We have reversed the following relative TRS [REVERSE]:
The set of rules R is 
   a(a(x1)) -> x1

The set of rules S is 
   a(a(x1)) -> b(a(a(a(b(x1)))))

We have obtained the following relative TRS:
The set of rules R is 
   a(a(x1)) -> x1

The set of rules S is 
   a(a(x1)) -> b(a(a(a(b(x1)))))


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(2)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a(a(x1)) -> x1

The relative TRS consists of the following S rules:

   a(a(x1)) -> b(a(a(a(b(x1)))))


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(3) FlatCCProof (EQUIVALENT)
We used flat context closure [ROOTLAB]

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(4)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a(a(a(x1))) -> a(x1)
   b(a(a(x1))) -> b(x1)

The relative TRS consists of the following S rules:

   a(a(a(x1))) -> a(b(a(a(a(b(x1))))))
   b(a(a(x1))) -> b(b(a(a(a(b(x1))))))


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(5) RootLabelingProof (EQUIVALENT)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled


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(6)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1)
   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1)

The relative TRS consists of the following S rules:

   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))


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(7) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :

   <<<
 POL(a_{a_1}(x_1)) =  	[[0], [0], [1]] 	 +  	[[1, 1, 0], [0, 0, 1], [0, 0, 1]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(x1)
Rules from S:

   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))




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(8)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1)

The relative TRS consists of the following S rules:

   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))
   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))


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(9) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^3, +, *, >=, >) :

   <<<
 POL(a_{a_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 1], [0, 1, 0]] 	* 	x_1
>>>

   <<<
 POL(a_{b_1}(x_1)) =  	[[0], [1], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{a_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 1], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

   <<<
 POL(b_{b_1}(x_1)) =  	[[0], [0], [0]] 	 +  	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(x1)
Rules from S:

   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))




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(10)
Obligation:
Relative term rewrite system:
The relative TRS consists of the following R rules:

   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)

The relative TRS consists of the following S rules:

   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))


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(11) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Polynomial interpretation [POLO]:

   POL(a_{a_1}(x_1)) = 1 + x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = x_1
   POL(b_{b_1}(x_1)) = x_1
With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:

   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(x1)
   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{a_1}(x1)
Rules from S:
none




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(12)
Obligation:
Relative term rewrite system:
R is empty.
The relative TRS consists of the following S rules:

   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))))
   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))))


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(13) RIsEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(14)
YES