144.78/37.53	YES
145.14/37.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
145.14/37.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
145.14/37.55	
145.14/37.55	
145.14/37.55	Termination of the given RelTRS could be proven:
145.14/37.55	
145.14/37.55	(0) RelTRS
145.14/37.55	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
145.14/37.55	(2) RelTRS
145.14/37.55	(3) FlatCCProof [EQUIVALENT, 0 ms]
145.14/37.55	(4) RelTRS
145.14/37.55	(5) RootLabelingProof [EQUIVALENT, 0 ms]
145.14/37.55	(6) RelTRS
145.14/37.55	(7) RelTRSRRRProof [EQUIVALENT, 4554 ms]
145.14/37.55	(8) RelTRS
145.14/37.55	(9) RelTRSRRRProof [EQUIVALENT, 10.5 s]
145.14/37.55	(10) RelTRS
145.14/37.55	(11) RelTRSRRRProof [EQUIVALENT, 2 ms]
145.14/37.55	(12) RelTRS
145.14/37.55	(13) RelTRSRRRProof [EQUIVALENT, 2 ms]
145.14/37.55	(14) RelTRS
145.14/37.55	(15) RIsEmptyProof [EQUIVALENT, 0 ms]
145.14/37.55	(16) YES
145.14/37.55	
145.14/37.55	
145.14/37.55	----------------------------------------
145.14/37.55	
145.14/37.55	(0)
145.14/37.55	Obligation:
145.14/37.55	Relative term rewrite system:
145.14/37.55	The relative TRS consists of the following R rules:
145.14/37.55	
145.14/37.55	   c(c(c(x1))) -> b(a(a(x1)))
145.14/37.55	   a(b(a(x1))) -> a(b(b(x1)))
145.14/37.55	   a(b(a(x1))) -> b(c(c(x1)))
145.14/37.55	   a(c(b(x1))) -> a(a(c(x1)))
145.14/37.55	
145.14/37.55	The relative TRS consists of the following S rules:
145.14/37.55	
145.14/37.55	   c(c(b(x1))) -> c(c(a(x1)))
145.14/37.55	   a(b(c(x1))) -> c(a(c(x1)))
145.14/37.55	
145.14/37.55	
145.14/37.55	----------------------------------------
145.14/37.55	
145.14/37.55	(1) RelTRS Reverse (EQUIVALENT)
145.14/37.55	We have reversed the following relative TRS [REVERSE]:
145.14/37.55	The set of rules R is 
145.14/37.55	   c(c(c(x1))) -> b(a(a(x1)))
145.14/37.55	   a(b(a(x1))) -> a(b(b(x1)))
145.14/37.55	   a(b(a(x1))) -> b(c(c(x1)))
145.14/37.55	   a(c(b(x1))) -> a(a(c(x1)))
145.14/37.55	
145.14/37.55	The set of rules S is 
145.14/37.55	   c(c(b(x1))) -> c(c(a(x1)))
145.14/37.55	   a(b(c(x1))) -> c(a(c(x1)))
145.14/37.55	
145.14/37.55	We have obtained the following relative TRS:
145.14/37.55	The set of rules R is 
145.14/37.55	   c(c(c(x1))) -> a(a(b(x1)))
145.14/37.55	   a(b(a(x1))) -> b(b(a(x1)))
145.14/37.55	   a(b(a(x1))) -> c(c(b(x1)))
145.14/37.55	   b(c(a(x1))) -> c(a(a(x1)))
145.14/37.55	
145.14/37.55	The set of rules S is 
145.14/37.55	   b(c(c(x1))) -> a(c(c(x1)))
145.14/37.55	   c(b(a(x1))) -> c(a(c(x1)))
145.14/37.55	
145.14/37.55	
145.14/37.55	----------------------------------------
145.14/37.55	
145.14/37.55	(2)
145.14/37.55	Obligation:
145.14/37.55	Relative term rewrite system:
145.14/37.55	The relative TRS consists of the following R rules:
145.14/37.55	
145.14/37.55	   c(c(c(x1))) -> a(a(b(x1)))
145.14/37.55	   a(b(a(x1))) -> b(b(a(x1)))
145.14/37.55	   a(b(a(x1))) -> c(c(b(x1)))
145.14/37.55	   b(c(a(x1))) -> c(a(a(x1)))
145.14/37.55	
145.14/37.55	The relative TRS consists of the following S rules:
145.14/37.55	
145.14/37.55	   b(c(c(x1))) -> a(c(c(x1)))
145.14/37.55	   c(b(a(x1))) -> c(a(c(x1)))
145.14/37.55	
145.14/37.55	
145.14/37.55	----------------------------------------
145.14/37.55	
145.14/37.55	(3) FlatCCProof (EQUIVALENT)
145.14/37.55	We used flat context closure [ROOTLAB]
145.14/37.55	
145.14/37.55	----------------------------------------
145.14/37.55	
145.14/37.55	(4)
145.14/37.55	Obligation:
145.14/37.55	Relative term rewrite system:
145.14/37.55	The relative TRS consists of the following R rules:
145.14/37.55	
145.14/37.55	   c(c(c(c(x1)))) -> c(a(a(b(x1))))
145.14/37.55	   a(c(c(c(x1)))) -> a(a(a(b(x1))))
145.14/37.55	   b(c(c(c(x1)))) -> b(a(a(b(x1))))
145.14/37.55	   c(a(b(a(x1)))) -> c(b(b(a(x1))))
145.14/37.55	   a(a(b(a(x1)))) -> a(b(b(a(x1))))
145.14/37.55	   b(a(b(a(x1)))) -> b(b(b(a(x1))))
145.14/37.55	   c(a(b(a(x1)))) -> c(c(c(b(x1))))
145.14/37.55	   a(a(b(a(x1)))) -> a(c(c(b(x1))))
145.14/37.55	   b(a(b(a(x1)))) -> b(c(c(b(x1))))
145.14/37.55	   c(b(c(a(x1)))) -> c(c(a(a(x1))))
145.14/37.55	   a(b(c(a(x1)))) -> a(c(a(a(x1))))
145.14/37.55	   b(b(c(a(x1)))) -> b(c(a(a(x1))))
145.14/37.55	
145.14/37.55	The relative TRS consists of the following S rules:
145.14/37.55	
145.14/37.55	   c(b(a(x1))) -> c(a(c(x1)))
145.14/37.55	   c(b(c(c(x1)))) -> c(a(c(c(x1))))
145.14/37.55	   a(b(c(c(x1)))) -> a(a(c(c(x1))))
145.14/37.55	   b(b(c(c(x1)))) -> b(a(c(c(x1))))
145.14/37.55	
145.14/37.55	
145.14/37.55	----------------------------------------
145.14/37.55	
145.14/37.55	(5) RootLabelingProof (EQUIVALENT)
145.14/37.55	We used plain root labeling [ROOTLAB] with the following heuristic:
145.14/37.55	LabelAll: All function symbols get labeled
145.14/37.55	
145.14/37.55	
145.14/37.55	----------------------------------------
145.14/37.55	
145.14/37.55	(6)
145.14/37.55	Obligation:
145.14/37.55	Relative term rewrite system:
145.14/37.55	The relative TRS consists of the following R rules:
145.14/37.55	
145.14/37.55	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
145.14/37.55	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
145.14/37.55	   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
145.14/37.55	   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
145.14/37.55	   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
145.14/37.55	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
145.14/37.55	   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
145.14/37.55	   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
145.14/37.55	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
145.14/37.55	   c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1))))
145.14/37.55	   c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
145.14/37.55	   c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
145.14/37.55	   a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1))))
145.14/37.55	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
145.14/37.55	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
145.14/37.55	   b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1))))
145.14/37.55	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
145.14/37.55	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
145.14/37.55	   c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
145.14/37.55	   c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
145.14/37.55	   c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
145.14/37.55	   a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
145.14/37.55	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
145.14/37.55	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
145.14/37.55	   b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
145.14/37.55	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
145.14/37.55	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
145.14/37.55	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
145.14/37.55	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
145.14/37.55	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
145.14/37.55	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
145.14/37.55	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
145.14/37.55	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
145.14/37.55	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
145.14/37.55	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
145.14/37.55	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
145.14/37.55	
145.14/37.55	The relative TRS consists of the following S rules:
145.14/37.55	
145.14/37.55	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{c_1}(x1)))
145.14/37.55	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{a_1}(x1)))
145.14/37.55	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(x1)))
145.14/37.55	   c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
145.14/37.55	   c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
145.14/37.55	   c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
145.14/37.55	   a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
145.14/37.55	   a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
145.14/37.55	   a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
145.14/37.55	   b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
145.14/37.55	   b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
145.14/37.55	   b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
145.14/37.55	
145.14/37.55	
145.14/37.55	----------------------------------------
145.14/37.55	
145.14/37.55	(7) RelTRSRRRProof (EQUIVALENT)
145.14/37.55	We used the following monotonic ordering for rule removal:
145.14/37.55	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
145.14/37.55	
145.14/37.55	   <<<
145.14/37.55	 POL(c_{c_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
145.14/37.55	>>>
145.14/37.55	
145.14/37.55	   <<<
145.14/37.55	 POL(c_{a_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
145.14/37.55	>>>
145.14/37.55	
145.14/37.55	   <<<
145.14/37.55	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
145.14/37.55	>>>
145.14/37.55	
145.14/37.55	   <<<
145.14/37.55	 POL(a_{b_1}(x_1)) =  	[[2], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
145.14/37.55	>>>
145.14/37.55	
145.14/37.55	   <<<
145.14/37.55	 POL(b_{c_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
145.14/37.55	>>>
145.14/37.55	
145.14/37.55	   <<<
145.14/37.55	 POL(b_{a_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
145.14/37.55	>>>
145.14/37.55	
145.14/37.55	   <<<
145.14/37.55	 POL(c_{b_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
145.14/37.55	>>>
145.14/37.55	
145.14/37.55	   <<<
145.14/37.55	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
145.14/37.55	>>>
145.14/37.55	
145.14/37.55	   <<<
145.14/37.55	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
145.14/37.55	>>>
145.14/37.55	
145.14/37.55	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
145.14/37.56	Rules from R:
145.14/37.56	
145.14/37.56	   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
145.14/37.56	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
145.14/37.56	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
145.14/37.56	   c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1))))
145.14/37.56	   c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
145.14/37.56	   c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
145.14/37.56	   b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1))))
145.14/37.56	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
145.14/37.56	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
145.14/37.56	   c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
145.14/37.56	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
145.14/37.56	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
145.14/37.56	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
145.14/37.56	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
145.14/37.56	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
145.14/37.56	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
145.14/37.56	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
145.14/37.56	Rules from S:
145.14/37.56	
145.14/37.56	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(x1)))
145.14/37.56	   c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
145.14/37.56	   c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
145.14/37.56	   c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
145.14/37.56	   a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
145.14/37.56	   a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
145.14/37.56	   a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
145.14/37.56	
145.14/37.56	
145.14/37.56	
145.14/37.56	
145.14/37.56	----------------------------------------
145.14/37.56	
145.14/37.56	(8)
145.14/37.56	Obligation:
145.14/37.56	Relative term rewrite system:
145.14/37.56	The relative TRS consists of the following R rules:
145.14/37.56	
145.14/37.56	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
145.14/37.56	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
145.14/37.56	   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
145.14/37.56	   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
145.14/37.56	   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
145.14/37.56	   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
145.14/37.56	   c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
145.14/37.56	   c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
145.14/37.56	   b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
145.14/37.56	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
145.14/37.56	
145.14/37.56	The relative TRS consists of the following S rules:
145.14/37.56	
145.14/37.56	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{c_1}(x1)))
145.14/37.56	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{a_1}(x1)))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
145.14/37.56	
145.14/37.56	
145.14/37.56	----------------------------------------
145.14/37.56	
145.14/37.56	(9) RelTRSRRRProof (EQUIVALENT)
145.14/37.56	We used the following monotonic ordering for rule removal:
145.14/37.56	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
145.14/37.56	
145.14/37.56	   <<<
145.14/37.56	 POL(c_{c_1}(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
145.14/37.56	>>>
145.14/37.56	
145.14/37.56	   <<<
145.14/37.56	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
145.14/37.56	>>>
145.14/37.56	
145.14/37.56	   <<<
145.14/37.56	 POL(a_{a_1}(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
145.14/37.56	>>>
145.14/37.56	
145.14/37.56	   <<<
145.14/37.56	 POL(a_{b_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
145.14/37.56	>>>
145.14/37.56	
145.14/37.56	   <<<
145.14/37.56	 POL(b_{c_1}(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [1, 0]] 	* 	x_1
145.14/37.56	>>>
145.14/37.56	
145.14/37.56	   <<<
145.14/37.56	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [1, 0]] 	* 	x_1
145.14/37.56	>>>
145.14/37.56	
145.14/37.56	   <<<
145.14/37.56	 POL(a_{c_1}(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
145.14/37.56	>>>
145.14/37.56	
145.14/37.56	   <<<
145.14/37.56	 POL(b_{b_1}(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [1, 0]] 	* 	x_1
145.14/37.56	>>>
145.14/37.56	
145.14/37.56	   <<<
145.14/37.56	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
145.14/37.56	>>>
145.14/37.56	
145.14/37.56	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
145.14/37.56	Rules from R:
145.14/37.56	
145.14/37.56	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
145.14/37.56	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
145.14/37.56	   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
145.14/37.56	   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
145.14/37.56	   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
145.14/37.56	   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
145.14/37.56	   c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
145.14/37.56	   c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
145.14/37.56	   b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
145.14/37.56	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
145.14/37.56	Rules from S:
145.14/37.56	
145.14/37.56	   b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
145.14/37.56	
145.14/37.56	
145.14/37.56	
145.14/37.56	
145.14/37.56	----------------------------------------
145.14/37.56	
145.14/37.56	(10)
145.14/37.56	Obligation:
145.14/37.56	Relative term rewrite system:
145.14/37.56	The relative TRS consists of the following R rules:
145.14/37.56	
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
145.14/37.56	
145.14/37.56	The relative TRS consists of the following S rules:
145.14/37.56	
145.14/37.56	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{c_1}(x1)))
145.14/37.56	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{a_1}(x1)))
145.14/37.56	
145.14/37.56	
145.14/37.56	----------------------------------------
145.14/37.56	
145.14/37.56	(11) RelTRSRRRProof (EQUIVALENT)
145.14/37.56	We used the following monotonic ordering for rule removal:
145.14/37.56	Polynomial interpretation [POLO]:
145.14/37.56	
145.14/37.56	   POL(a_{a_1}(x_1)) = 1 + x_1
145.14/37.56	   POL(a_{b_1}(x_1)) = x_1
145.14/37.56	   POL(a_{c_1}(x_1)) = x_1
145.14/37.56	   POL(b_{a_1}(x_1)) = x_1
145.14/37.56	   POL(b_{b_1}(x_1)) = 1 + x_1
145.14/37.56	   POL(b_{c_1}(x_1)) = x_1
145.14/37.56	   POL(c_{a_1}(x_1)) = x_1
145.14/37.56	   POL(c_{b_1}(x_1)) = 1 + x_1
145.14/37.56	   POL(c_{c_1}(x_1)) = 1 + x_1
145.14/37.56	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
145.14/37.56	Rules from R:
145.14/37.56	none
145.14/37.56	Rules from S:
145.14/37.56	
145.14/37.56	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{a_1}(x1)))
145.14/37.56	
145.14/37.56	
145.14/37.56	
145.14/37.56	
145.14/37.56	----------------------------------------
145.14/37.56	
145.14/37.56	(12)
145.14/37.56	Obligation:
145.14/37.56	Relative term rewrite system:
145.14/37.56	The relative TRS consists of the following R rules:
145.14/37.56	
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
145.14/37.56	
145.14/37.56	The relative TRS consists of the following S rules:
145.14/37.56	
145.14/37.56	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{c_1}(x1)))
145.14/37.56	
145.14/37.56	
145.14/37.56	----------------------------------------
145.14/37.56	
145.14/37.56	(13) RelTRSRRRProof (EQUIVALENT)
145.14/37.56	We used the following monotonic ordering for rule removal:
145.14/37.56	Knuth-Bendix order [KBO] with precedence:c_{c_1}_1 > c_{b_1}_1 > c_{a_1}_1 > b_{b_1}_1 > a_{a_1}_1 > a_{c_1}_1 > b_{c_1}_1 > b_{a_1}_1 > a_{b_1}_1
145.14/37.56	
145.14/37.56	and weight map:
145.14/37.56	
145.14/37.56	   a_{a_1}_1=2
145.14/37.56	   a_{b_1}_1=1
145.14/37.56	   b_{a_1}_1=1
145.14/37.56	   a_{c_1}_1=1
145.14/37.56	   b_{b_1}_1=2
145.14/37.56	   b_{c_1}_1=1
145.14/37.56	   c_{a_1}_1=1
145.14/37.56	   c_{b_1}_1=2
145.14/37.56	   c_{c_1}_1=2
145.14/37.56	
145.14/37.56	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
145.14/37.56	Rules from R:
145.14/37.56	
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{c_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
145.14/37.56	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
145.14/37.56	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
145.14/37.56	Rules from S:
145.14/37.56	
145.14/37.56	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{c_1}(x1)))
145.14/37.56	
145.14/37.56	
145.14/37.56	
145.14/37.56	
145.14/37.56	----------------------------------------
145.14/37.56	
145.14/37.56	(14)
145.14/37.56	Obligation:
145.14/37.56	Relative term rewrite system:
145.14/37.56	R is empty.
145.14/37.56	S is empty.
145.14/37.56	
145.14/37.56	----------------------------------------
145.14/37.56	
145.14/37.56	(15) RIsEmptyProof (EQUIVALENT)
145.14/37.56	The TRS R is empty. Hence, termination is trivially proven.
145.14/37.56	----------------------------------------
145.14/37.56	
145.14/37.56	(16)
145.14/37.56	YES
145.22/37.60	EOF