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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) FlatCCProof [EQUIVALENT, 0 ms]
(2) RelTRS
(3) RootLabelingProof [EQUIVALENT, 0 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 160 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 4 ms]
(8) RelTRS
(9) SIsEmptyProof [EQUIVALENT, 0 ms]
(10) QTRS
(11) RFCMatchBoundsTRSProof [EQUIVALENT, 11 ms]
(12) YES


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

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

The relative TRS consists of the following S rules:

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


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

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

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

The relative TRS consists of the following S rules:

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


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


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

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

The relative TRS consists of the following S rules:

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


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

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

   <<<
 POL(a_{b_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
>>>

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

With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
Rules from R:
none
Rules from S:

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




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

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

The relative TRS consists of the following S rules:

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


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

   POL(a_{a_1}(x_1)) = x_1
   POL(a_{b_1}(x_1)) = x_1
   POL(b_{a_1}(x_1)) = 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:
none
Rules from S:

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




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

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

S is empty.

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(9) SIsEmptyProof (EQUIVALENT)
The TRS S is empty. Hence, termination of R/S is equivalent to termination of R.
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(10)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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(11) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
140, 141, 142, 143, 144, 145, 146, 147, 148, 149

Node 140 is start node and node 141 is final node.

Those nodes are connected through the following edges:

* 140 to 142 labelled b_{a_1}_1(0)* 141 to 141 labelled #_1(0)* 142 to 143 labelled a_{a_1}_1(0)* 143 to 144 labelled a_{a_1}_1(0)* 144 to 145 labelled a_{b_1}_1(0)* 145 to 141 labelled b_{b_1}_1(0), b_{a_1}_1(0)* 145 to 146 labelled b_{a_1}_1(1)* 146 to 147 labelled a_{a_1}_1(1)* 147 to 148 labelled a_{a_1}_1(1)* 148 to 149 labelled a_{b_1}_1(1)* 149 to 141 labelled b_{b_1}_1(1), b_{a_1}_1(1)* 149 to 146 labelled b_{a_1}_1(1)


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(12)
YES