/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRS Reverse [EQUIVALENT, 0 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 1250 ms]
(4) RelTRS
(5) RelTRSRRRProof [EQUIVALENT, 57 ms]
(6) RelTRS
(7) RelTRSRRRProof [EQUIVALENT, 0 ms]
(8) RelTRS
(9) RelTRSRRRProof [EQUIVALENT, 33 ms]
(10) RelTRS
(11) SIsEmptyProof [EQUIVALENT, 0 ms]
(12) QTRS
(13) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(14) YES


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

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

The relative TRS consists of the following S rules:

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


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

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

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

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


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

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

The relative TRS consists of the following S rules:

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


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(3) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^6, +, *, >=, >) :

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

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

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

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

   a(a(a(x1))) -> b(c(c(x1)))
Rules from S:
none




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

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

The relative TRS consists of the following S rules:

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


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

(5) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(a(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [2, 0]] 	* 	x_1
>>>

   <<<
 POL(c(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
>>>

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

   b(a(c(x1))) -> b(b(c(x1)))
Rules from S:
none




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

   c(c(b(x1))) -> c(c(c(x1)))

The relative TRS consists of the following S rules:

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


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

   POL(a(x_1)) = x_1
   POL(b(x_1)) = 1 + x_1
   POL(c(x_1)) = 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:

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




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

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

   c(c(b(x1))) -> c(c(c(x1)))

The relative TRS consists of the following S rules:

   b(b(c(x1))) -> c(c(c(x1)))
   b(c(b(x1))) -> b(b(b(x1)))


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(9) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(c(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [2, 0]] 	* 	x_1
>>>

   <<<
 POL(b(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 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(b(c(x1))) -> c(c(c(x1)))
   b(c(b(x1))) -> b(b(b(x1)))




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

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

   c(c(b(x1))) -> c(c(c(x1)))

S is empty.

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(11) SIsEmptyProof (EQUIVALENT)
The TRS S is empty. Hence, termination of R/S is equivalent to termination of R.
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(12)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   c(c(b(x1))) -> c(c(c(x1)))

Q is empty.

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(13) 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:

   c(c(b(x1))) -> c(c(c(x1)))

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
153, 154, 155, 156, 157, 158

Node 153 is start node and node 154 is final node.

Those nodes are connected through the following edges:

* 153 to 155 labelled c_1(0)* 154 to 154 labelled #_1(0)* 155 to 156 labelled c_1(0)* 155 to 157 labelled c_1(1)* 156 to 154 labelled c_1(0)* 156 to 157 labelled c_1(1)* 157 to 158 labelled c_1(1)* 157 to 157 labelled c_1(1)* 158 to 154 labelled c_1(1)* 158 to 157 labelled c_1(1)


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(14)
YES