26.37/7.54	YES
26.37/7.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
26.37/7.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.37/7.55	
26.37/7.55	
26.37/7.55	Termination of the given RelTRS could be proven:
26.37/7.55	
26.37/7.55	(0) RelTRS
26.37/7.55	(1) RelTRSRRRProof [EQUIVALENT, 1398 ms]
26.37/7.55	(2) RelTRS
26.37/7.55	(3) SIsEmptyProof [EQUIVALENT, 0 ms]
26.37/7.55	(4) QTRS
26.37/7.55	(5) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
26.37/7.55	(6) YES
26.37/7.55	
26.37/7.55	
26.37/7.55	----------------------------------------
26.37/7.55	
26.37/7.55	(0)
26.37/7.55	Obligation:
26.37/7.55	Relative term rewrite system:
26.37/7.55	The relative TRS consists of the following R rules:
26.37/7.55	
26.37/7.55	   b(a(b(x1))) -> b(a(a(a(b(x1)))))
26.37/7.55	
26.37/7.55	The relative TRS consists of the following S rules:
26.37/7.55	
26.37/7.55	   a(a(a(x1))) -> b(b(b(b(x1))))
26.37/7.55	
26.37/7.55	
26.37/7.55	----------------------------------------
26.37/7.55	
26.37/7.55	(1) RelTRSRRRProof (EQUIVALENT)
26.37/7.55	We used the following monotonic ordering for rule removal:
26.37/7.55	Matrix interpretation [MATRO] to (N^6, +, *, >=, >) :
26.37/7.55	
26.37/7.55	   <<<
26.37/7.55	 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
26.37/7.55	>>>
26.37/7.55	
26.37/7.55	   <<<
26.37/7.55	 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
26.37/7.55	>>>
26.37/7.55	
26.37/7.55	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
26.37/7.55	Rules from R:
26.37/7.55	none
26.37/7.55	Rules from S:
26.37/7.55	
26.37/7.55	   a(a(a(x1))) -> b(b(b(b(x1))))
26.37/7.55	
26.37/7.55	
26.37/7.55	
26.37/7.55	
26.37/7.55	----------------------------------------
26.37/7.55	
26.37/7.55	(2)
26.37/7.55	Obligation:
26.37/7.55	Relative term rewrite system:
26.37/7.55	The relative TRS consists of the following R rules:
26.37/7.55	
26.37/7.55	   b(a(b(x1))) -> b(a(a(a(b(x1)))))
26.37/7.55	
26.37/7.55	S is empty.
26.37/7.55	
26.37/7.55	----------------------------------------
26.37/7.55	
26.37/7.55	(3) SIsEmptyProof (EQUIVALENT)
26.37/7.55	The TRS S is empty. Hence, termination of R/S is equivalent to termination of R.
26.37/7.55	----------------------------------------
26.37/7.55	
26.37/7.55	(4)
26.37/7.55	Obligation:
26.37/7.55	Q restricted rewrite system:
26.37/7.55	The TRS R consists of the following rules:
26.37/7.55	
26.37/7.55	   b(a(b(x1))) -> b(a(a(a(b(x1)))))
26.37/7.55	
26.37/7.55	Q is empty.
26.37/7.55	
26.37/7.55	----------------------------------------
26.37/7.55	
26.37/7.55	(5) RFCMatchBoundsTRSProof (EQUIVALENT)
26.37/7.55	Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
26.37/7.55	The following rules were used to construct the certificate:
26.37/7.55	
26.37/7.55	   b(a(b(x1))) -> b(a(a(a(b(x1)))))
26.37/7.55	
26.37/7.55	The certificate found is represented by the following graph.
26.37/7.55	The certificate consists of the following enumerated nodes:
26.37/7.55	133, 134, 135, 136, 137, 138, 139, 140, 141, 142
26.37/7.55	
26.37/7.55	Node 133 is start node and node 134 is final node.
26.37/7.55	
26.37/7.55	Those nodes are connected through the following edges:
26.37/7.55	
26.37/7.55	* 133 to 135 labelled b_1(0)* 134 to 134 labelled #_1(0)* 135 to 136 labelled a_1(0)* 136 to 137 labelled a_1(0)* 137 to 138 labelled a_1(0)* 138 to 134 labelled b_1(0)* 138 to 139 labelled b_1(1)* 139 to 140 labelled a_1(1)* 140 to 141 labelled a_1(1)* 141 to 142 labelled a_1(1)* 142 to 134 labelled b_1(1)* 142 to 139 labelled b_1(1)
26.37/7.55	
26.37/7.55	
26.37/7.55	----------------------------------------
26.37/7.55	
26.37/7.55	(6)
26.37/7.55	YES
26.60/7.62	EOF