/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 408 ms]
(2) RelTRS
(3) SIsEmptyProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) RFCMatchBoundsTRSProof [EQUIVALENT, 8 ms]
(6) YES


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

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

The relative TRS consists of the following S rules:

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


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

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

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

   <<<
 POL(a(x_1)) =  	[[2], [0]] 	 +  	[[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(b(b(x1))) -> a(a(c(x1)))
Rules from S:

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




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

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

S is empty.

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(3) SIsEmptyProof (EQUIVALENT)
The TRS S is empty. Hence, termination of R/S is equivalent to termination of R.
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(4)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

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(5) RFCMatchBoundsTRSProof (EQUIVALENT)
Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 3. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180

Node 161 is start node and node 162 is final node.

Those nodes are connected through the following edges:

* 161 to 163 labelled c_1(0), b_1(0)* 161 to 165 labelled b_1(0), a_1(0)* 161 to 167 labelled a_1(1)* 161 to 173 labelled b_1(1)* 162 to 162 labelled #_1(0)* 163 to 164 labelled a_1(0)* 163 to 171 labelled b_1(1)* 163 to 175 labelled a_1(2)* 164 to 162 labelled b_1(0)* 164 to 167 labelled a_1(1)* 164 to 169 labelled b_1(1), b_1(2)* 164 to 163 labelled b_1(2)* 164 to 179 labelled b_1(3)* 165 to 166 labelled b_1(0)* 166 to 162 labelled c_1(0)* 166 to 169 labelled c_1(1)* 167 to 168 labelled b_1(1)* 168 to 162 labelled c_1(1)* 168 to 169 labelled c_1(1)* 169 to 170 labelled a_1(1)* 169 to 175 labelled b_1(2), a_1(2)* 169 to 177 labelled a_1(3)* 170 to 162 labelled b_1(1)* 170 to 167 labelled a_1(1)* 170 to 169 labelled b_1(1), b_1(2)* 170 to 163 labelled b_1(2)* 170 to 179 labelled b_1(3)* 171 to 172 labelled b_1(1)* 172 to 168 labelled c_1(1)* 173 to 174 labelled a_1(1)* 174 to 172 labelled b_1(1)* 175 to 176 labelled b_1(2)* 175 to 172 labelled b_1(2)* 176 to 168 labelled c_1(2)* 177 to 178 labelled b_1(3)* 178 to 168 labelled c_1(3)* 179 to 180 labelled a_1(3)* 180 to 176 labelled b_1(3)* 180 to 172 labelled b_1(3)


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(6)
YES