/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination of the given RelTRS could be proven:

(0) RelTRS
(1) RelTRSRRRProof [EQUIVALENT, 1459 ms]
(2) RelTRS
(3) SIsEmptyProof [EQUIVALENT, 0 ms]
(4) QTRS
(5) QTRSRoofMatchBoundsTAProof [EQUIVALENT, 0 ms]
(6) 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) RelTRSRRRProof (EQUIVALENT)
We used the following monotonic ordering for rule removal:
Matrix interpretation [MATRO] to (N^6, +, *, >=, >) :

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

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

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

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

Q is empty.

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(5) QTRSRoofMatchBoundsTAProof (EQUIVALENT)
The TRS R could be shown to be Match-Bounded [TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] by  1. Therefore it terminates.

The compatible tree automaton used to show the Match-Boundedness is represented by: 
final states : [0, 1]
transitions: 
b0(0) -> 0
b0(1) -> 0
a0(0) -> 1
a0(1) -> 1
b1(0) -> 5
a1(5) -> 4
a1(4) -> 3
a1(3) -> 2
b1(2) -> 0
b1(1) -> 5
0 -> 5

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(6)
YES