/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RelTRSRRRProof [EQUIVALENT, 1107 ms]
(2) RelTRS
(3) RelTRSRRRProof [EQUIVALENT, 62 ms]
(4) RelTRS
(5) SIsEmptyProof [EQUIVALENT, 0 ms]
(6) QTRS
(7) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
(8) YES


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

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

The relative TRS consists of the following S rules:

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


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

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

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

   <<<
 POL(b(x_1)) =  	[[0], [1], [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, 1, 0, 0], [0, 0, 0, 0, 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:
none
Rules from S:

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




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

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

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

The relative TRS consists of the following S rules:

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


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

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

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

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

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

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

   c(c(a(x1))) -> b(a(c(x1)))
Rules from S:

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




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

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

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

S is empty.

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(5) SIsEmptyProof (EQUIVALENT)
The TRS S is empty. Hence, termination of R/S is equivalent to termination of R.
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(6)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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(7) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 0. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
159, 160, 161, 162

Node 159 is start node and node 160 is final node.

Those nodes are connected through the following edges:

* 159 to 161 labelled a_1(0)* 160 to 160 labelled #_1(0)* 161 to 162 labelled a_1(0)* 162 to 160 labelled a_1(0)


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(8)
YES