7.29/2.70	YES
7.29/2.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
7.29/2.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
7.29/2.71	
7.29/2.71	
7.29/2.71	Termination of the given RelTRS could be proven:
7.29/2.71	
7.29/2.71	(0) RelTRS
7.29/2.71	(1) RelTRS S Cleaner [EQUIVALENT, 0 ms]
7.29/2.71	(2) RelTRS
7.29/2.71	(3) RelTRSRRRProof [EQUIVALENT, 52 ms]
7.29/2.71	(4) RelTRS
7.29/2.71	(5) SIsEmptyProof [EQUIVALENT, 0 ms]
7.29/2.71	(6) QTRS
7.29/2.71	(7) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
7.29/2.71	(8) YES
7.29/2.71	
7.29/2.71	
7.29/2.71	----------------------------------------
7.29/2.71	
7.29/2.71	(0)
7.29/2.71	Obligation:
7.29/2.71	Relative term rewrite system:
7.29/2.71	The relative TRS consists of the following R rules:
7.29/2.71	
7.29/2.71	   b(a(a(x1))) -> c(b(b(x1)))
7.29/2.71	   a(a(a(x1))) -> a(c(c(x1)))
7.29/2.71	   a(c(b(x1))) -> a(a(b(x1)))
7.29/2.71	   c(a(c(x1))) -> c(c(c(x1)))
7.29/2.71	
7.29/2.71	The relative TRS consists of the following S rules:
7.29/2.71	
7.29/2.71	   c(a(a(x1))) -> c(a(a(x1)))
7.29/2.71	   c(c(a(x1))) -> c(b(b(x1)))
7.29/2.71	   a(a(b(x1))) -> b(a(b(x1)))
7.29/2.71	
7.29/2.71	
7.29/2.71	----------------------------------------
7.29/2.71	
7.29/2.71	(1) RelTRS S Cleaner (EQUIVALENT)
7.29/2.71	We have deleted all rules from S that have the shape t -> t:
7.29/2.71	
7.29/2.71	   c(a(a(x1))) -> c(a(a(x1)))
7.29/2.71	
7.29/2.71	
7.29/2.71	----------------------------------------
7.29/2.71	
7.29/2.71	(2)
7.29/2.71	Obligation:
7.29/2.71	Relative term rewrite system:
7.29/2.71	The relative TRS consists of the following R rules:
7.29/2.71	
7.29/2.71	   b(a(a(x1))) -> c(b(b(x1)))
7.29/2.71	   a(a(a(x1))) -> a(c(c(x1)))
7.29/2.71	   a(c(b(x1))) -> a(a(b(x1)))
7.29/2.71	   c(a(c(x1))) -> c(c(c(x1)))
7.29/2.71	
7.29/2.71	The relative TRS consists of the following S rules:
7.29/2.71	
7.29/2.71	   c(c(a(x1))) -> c(b(b(x1)))
7.29/2.71	   a(a(b(x1))) -> b(a(b(x1)))
7.29/2.71	
7.29/2.71	
7.29/2.71	----------------------------------------
7.29/2.71	
7.29/2.71	(3) RelTRSRRRProof (EQUIVALENT)
7.29/2.71	We used the following monotonic ordering for rule removal:
7.29/2.71	Polynomial interpretation [POLO]:
7.29/2.71	
7.29/2.71	   POL(a(x_1)) = 1 + x_1
7.29/2.71	   POL(b(x_1)) = x_1
7.29/2.71	   POL(c(x_1)) = 1 + x_1
7.29/2.71	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
7.29/2.71	Rules from R:
7.29/2.71	
7.29/2.71	   b(a(a(x1))) -> c(b(b(x1)))
7.29/2.71	Rules from S:
7.29/2.71	
7.29/2.71	   c(c(a(x1))) -> c(b(b(x1)))
7.29/2.71	   a(a(b(x1))) -> b(a(b(x1)))
7.29/2.71	
7.29/2.71	
7.29/2.71	
7.29/2.71	
7.29/2.71	----------------------------------------
7.29/2.71	
7.29/2.71	(4)
7.29/2.71	Obligation:
7.29/2.71	Relative term rewrite system:
7.29/2.71	The relative TRS consists of the following R rules:
7.29/2.71	
7.29/2.71	   a(a(a(x1))) -> a(c(c(x1)))
7.29/2.71	   a(c(b(x1))) -> a(a(b(x1)))
7.29/2.71	   c(a(c(x1))) -> c(c(c(x1)))
7.29/2.71	
7.29/2.71	S is empty.
7.29/2.71	
7.29/2.71	----------------------------------------
7.29/2.71	
7.29/2.71	(5) SIsEmptyProof (EQUIVALENT)
7.29/2.71	The TRS S is empty. Hence, termination of R/S is equivalent to termination of R.
7.29/2.71	----------------------------------------
7.29/2.71	
7.29/2.71	(6)
7.29/2.71	Obligation:
7.29/2.71	Q restricted rewrite system:
7.29/2.71	The TRS R consists of the following rules:
7.29/2.71	
7.29/2.71	   a(a(a(x1))) -> a(c(c(x1)))
7.29/2.71	   a(c(b(x1))) -> a(a(b(x1)))
7.29/2.71	   c(a(c(x1))) -> c(c(c(x1)))
7.29/2.71	
7.29/2.71	Q is empty.
7.29/2.71	
7.29/2.71	----------------------------------------
7.29/2.71	
7.29/2.71	(7) RFCMatchBoundsTRSProof (EQUIVALENT)
7.29/2.71	Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
7.29/2.71	The following rules were used to construct the certificate:
7.29/2.71	
7.29/2.71	   a(a(a(x1))) -> a(c(c(x1)))
7.29/2.71	   a(c(b(x1))) -> a(a(b(x1)))
7.29/2.71	   c(a(c(x1))) -> c(c(c(x1)))
7.29/2.71	
7.29/2.71	The certificate found is represented by the following graph.
7.29/2.71	The certificate consists of the following enumerated nodes:
7.29/2.71	142, 143, 144, 145, 146, 147, 148, 149
7.29/2.71	
7.29/2.71	Node 142 is start node and node 143 is final node.
7.29/2.71	
7.29/2.71	Those nodes are connected through the following edges:
7.29/2.71	
7.29/2.71	* 142 to 144 labelled a_1(0), c_1(0)* 142 to 146 labelled a_1(0)* 143 to 143 labelled #_1(0)* 144 to 145 labelled c_1(0)* 145 to 143 labelled c_1(0)* 145 to 148 labelled c_1(1)* 146 to 147 labelled a_1(0)* 147 to 143 labelled b_1(0)* 148 to 149 labelled c_1(1)* 149 to 143 labelled c_1(1)* 149 to 148 labelled c_1(1)
7.29/2.71	
7.29/2.71	
7.29/2.71	----------------------------------------
7.29/2.71	
7.29/2.71	(8)
7.29/2.71	YES
7.55/2.74	EOF