191.62/49.42	YES
191.62/49.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
191.62/49.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
191.62/49.43	
191.62/49.43	
191.62/49.43	Termination of the given RelTRS could be proven:
191.62/49.43	
191.62/49.43	(0) RelTRS
191.62/49.43	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
191.62/49.43	(2) RelTRS
191.62/49.43	(3) FlatCCProof [EQUIVALENT, 0 ms]
191.62/49.43	(4) RelTRS
191.62/49.43	(5) RootLabelingProof [EQUIVALENT, 0 ms]
191.62/49.43	(6) RelTRS
191.62/49.43	(7) RelTRSRRRProof [EQUIVALENT, 1332 ms]
191.62/49.43	(8) RelTRS
191.62/49.43	(9) RelTRSRRRProof [EQUIVALENT, 1659 ms]
191.62/49.43	(10) RelTRS
191.62/49.43	(11) RelTRSRRRProof [EQUIVALENT, 8567 ms]
191.62/49.43	(12) RelTRS
191.62/49.43	(13) RelTRSRRRProof [EQUIVALENT, 7293 ms]
191.62/49.43	(14) RelTRS
191.62/49.43	(15) RelTRSRRRProof [EQUIVALENT, 5 ms]
191.62/49.43	(16) RelTRS
191.62/49.43	(17) RIsEmptyProof [EQUIVALENT, 0 ms]
191.62/49.43	(18) YES
191.62/49.43	
191.62/49.43	
191.62/49.43	----------------------------------------
191.62/49.43	
191.62/49.43	(0)
191.62/49.43	Obligation:
191.62/49.43	Relative term rewrite system:
191.62/49.43	The relative TRS consists of the following R rules:
191.62/49.43	
191.62/49.43	   a(c(c(x1))) -> b(c(c(x1)))
191.62/49.43	   c(a(c(x1))) -> b(b(a(x1)))
191.62/49.43	
191.62/49.43	The relative TRS consists of the following S rules:
191.62/49.43	
191.62/49.43	   b(b(b(x1))) -> c(c(a(x1)))
191.62/49.43	   c(b(a(x1))) -> b(a(a(x1)))
191.62/49.43	   c(c(c(x1))) -> b(c(a(x1)))
191.62/49.43	
191.62/49.43	
191.62/49.43	----------------------------------------
191.62/49.43	
191.62/49.43	(1) RelTRS Reverse (EQUIVALENT)
191.62/49.43	We have reversed the following relative TRS [REVERSE]:
191.62/49.43	The set of rules R is 
191.62/49.43	   a(c(c(x1))) -> b(c(c(x1)))
191.62/49.43	   c(a(c(x1))) -> b(b(a(x1)))
191.62/49.43	
191.62/49.43	The set of rules S is 
191.62/49.43	   b(b(b(x1))) -> c(c(a(x1)))
191.62/49.43	   c(b(a(x1))) -> b(a(a(x1)))
191.62/49.43	   c(c(c(x1))) -> b(c(a(x1)))
191.62/49.43	
191.62/49.43	We have obtained the following relative TRS:
191.62/49.43	The set of rules R is 
191.62/49.43	   c(c(a(x1))) -> c(c(b(x1)))
191.62/49.43	   c(a(c(x1))) -> a(b(b(x1)))
191.62/49.43	
191.62/49.43	The set of rules S is 
191.62/49.43	   b(b(b(x1))) -> a(c(c(x1)))
191.62/49.43	   a(b(c(x1))) -> a(a(b(x1)))
191.62/49.43	   c(c(c(x1))) -> a(c(b(x1)))
191.62/49.43	
191.62/49.43	
191.62/49.43	----------------------------------------
191.62/49.43	
191.62/49.43	(2)
191.62/49.43	Obligation:
191.62/49.43	Relative term rewrite system:
191.62/49.43	The relative TRS consists of the following R rules:
191.62/49.43	
191.62/49.43	   c(c(a(x1))) -> c(c(b(x1)))
191.62/49.43	   c(a(c(x1))) -> a(b(b(x1)))
191.62/49.43	
191.62/49.43	The relative TRS consists of the following S rules:
191.62/49.43	
191.62/49.43	   b(b(b(x1))) -> a(c(c(x1)))
191.62/49.43	   a(b(c(x1))) -> a(a(b(x1)))
191.62/49.43	   c(c(c(x1))) -> a(c(b(x1)))
191.62/49.43	
191.62/49.43	
191.62/49.43	----------------------------------------
191.62/49.43	
191.62/49.43	(3) FlatCCProof (EQUIVALENT)
191.62/49.43	We used flat context closure [ROOTLAB]
191.62/49.43	
191.62/49.43	----------------------------------------
191.62/49.43	
191.62/49.43	(4)
191.62/49.43	Obligation:
191.62/49.43	Relative term rewrite system:
191.62/49.43	The relative TRS consists of the following R rules:
191.62/49.43	
191.62/49.43	   c(c(a(x1))) -> c(c(b(x1)))
191.62/49.43	   c(c(a(c(x1)))) -> c(a(b(b(x1))))
191.62/49.43	   a(c(a(c(x1)))) -> a(a(b(b(x1))))
191.62/49.43	   b(c(a(c(x1)))) -> b(a(b(b(x1))))
191.62/49.43	
191.62/49.43	The relative TRS consists of the following S rules:
191.62/49.43	
191.62/49.43	   a(b(c(x1))) -> a(a(b(x1)))
191.62/49.43	   c(b(b(b(x1)))) -> c(a(c(c(x1))))
191.62/49.43	   a(b(b(b(x1)))) -> a(a(c(c(x1))))
191.62/49.43	   b(b(b(b(x1)))) -> b(a(c(c(x1))))
191.62/49.43	   c(c(c(c(x1)))) -> c(a(c(b(x1))))
191.62/49.43	   a(c(c(c(x1)))) -> a(a(c(b(x1))))
191.62/49.43	   b(c(c(c(x1)))) -> b(a(c(b(x1))))
191.62/49.43	
191.62/49.43	
191.62/49.43	----------------------------------------
191.62/49.43	
191.62/49.43	(5) RootLabelingProof (EQUIVALENT)
191.62/49.43	We used plain root labeling [ROOTLAB] with the following heuristic:
191.62/49.43	LabelAll: All function symbols get labeled
191.62/49.43	
191.62/49.43	
191.62/49.43	----------------------------------------
191.62/49.43	
191.62/49.43	(6)
191.62/49.43	Obligation:
191.62/49.43	Relative term rewrite system:
191.62/49.43	The relative TRS consists of the following R rules:
191.62/49.43	
191.62/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
191.62/49.43	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
191.62/49.43	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
191.62/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.62/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.62/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.62/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.62/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.62/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.62/49.43	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.62/49.43	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.62/49.43	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.62/49.43	
191.62/49.43	The relative TRS consists of the following S rules:
191.62/49.43	
191.62/49.43	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
191.62/49.43	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
191.62/49.43	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
191.62/49.43	   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.62/49.43	   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.62/49.43	   c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.62/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.62/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.62/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.62/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.62/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.62/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.62/49.43	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.62/49.43	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.62/49.43	   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.62/49.43	   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.62/49.43	   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.62/49.43	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.62/49.43	   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.62/49.43	   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.62/49.43	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.62/49.43	
191.62/49.43	
191.62/49.43	----------------------------------------
191.62/49.43	
191.62/49.43	(7) RelTRSRRRProof (EQUIVALENT)
191.62/49.43	We used the following monotonic ordering for rule removal:
191.62/49.43	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
191.62/49.43	
191.62/49.43	   <<<
191.62/49.43	 POL(c_{c_1}(x_1)) =  	[[0], [1]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
191.62/49.43	>>>
191.62/49.43	
191.62/49.43	   <<<
191.62/49.43	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.62/49.43	>>>
191.62/49.43	
191.62/49.43	   <<<
191.62/49.43	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.62/49.43	>>>
191.62/49.43	
191.62/49.43	   <<<
191.62/49.43	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.62/49.43	>>>
191.62/49.43	
191.62/49.43	   <<<
191.62/49.43	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 1]] 	* 	x_1
191.62/49.43	>>>
191.62/49.43	
191.62/49.43	   <<<
191.62/49.43	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
191.62/49.43	>>>
191.62/49.43	
191.62/49.43	   <<<
191.62/49.43	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.62/49.43	>>>
191.62/49.43	
191.62/49.43	   <<<
191.62/49.43	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
191.62/49.43	>>>
191.62/49.43	
191.62/49.43	   <<<
191.62/49.43	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.62/49.43	>>>
191.62/49.43	
191.62/49.43	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
191.62/49.43	Rules from R:
191.62/49.43	none
191.62/49.43	Rules from S:
191.62/49.43	
191.62/49.43	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
191.62/49.43	
191.62/49.43	
191.62/49.43	
191.62/49.43	
191.62/49.43	----------------------------------------
191.62/49.43	
191.62/49.43	(8)
191.62/49.43	Obligation:
191.62/49.43	Relative term rewrite system:
191.62/49.43	The relative TRS consists of the following R rules:
191.62/49.43	
191.62/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
191.62/49.43	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
191.62/49.43	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
191.62/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.62/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.62/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.62/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.62/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.62/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.62/49.43	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.62/49.43	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.62/49.43	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.62/49.43	
191.62/49.43	The relative TRS consists of the following S rules:
191.62/49.43	
191.62/49.43	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
191.62/49.43	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
191.62/49.43	   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.62/49.43	   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.87/49.43	   c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.87/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.87/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.87/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.87/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.87/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.87/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.87/49.43	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.87/49.43	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.87/49.43	   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.43	   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.87/49.43	   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.87/49.43	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.43	   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.87/49.43	   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.87/49.43	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.43	
191.87/49.43	
191.87/49.43	----------------------------------------
191.87/49.43	
191.87/49.43	(9) RelTRSRRRProof (EQUIVALENT)
191.87/49.43	We used the following monotonic ordering for rule removal:
191.87/49.43	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(c_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(b_{c_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 2]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
191.87/49.43	Rules from R:
191.87/49.43	none
191.87/49.43	Rules from S:
191.87/49.43	
191.87/49.43	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
191.87/49.43	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
191.87/49.43	
191.87/49.43	
191.87/49.43	
191.87/49.43	
191.87/49.43	----------------------------------------
191.87/49.43	
191.87/49.43	(10)
191.87/49.43	Obligation:
191.87/49.43	Relative term rewrite system:
191.87/49.43	The relative TRS consists of the following R rules:
191.87/49.43	
191.87/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
191.87/49.43	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
191.87/49.43	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
191.87/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.87/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.87/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.87/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.87/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.87/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.87/49.43	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.87/49.43	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.87/49.43	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.87/49.43	
191.87/49.43	The relative TRS consists of the following S rules:
191.87/49.43	
191.87/49.43	   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.87/49.43	   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.87/49.43	   c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.87/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.87/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.87/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.87/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.87/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.87/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.87/49.43	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.87/49.43	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.87/49.43	   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.43	   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.87/49.43	   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.87/49.43	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.43	   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.87/49.43	   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.87/49.43	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.43	
191.87/49.43	
191.87/49.43	----------------------------------------
191.87/49.43	
191.87/49.43	(11) RelTRSRRRProof (EQUIVALENT)
191.87/49.43	We used the following monotonic ordering for rule removal:
191.87/49.43	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(c_{c_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(a_{c_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(c_{b_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(b_{c_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(a_{b_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	   <<<
191.87/49.43	 POL(b_{b_1}(x_1)) =  	[[1], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
191.87/49.43	>>>
191.87/49.43	
191.87/49.43	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
191.87/49.43	Rules from R:
191.87/49.43	none
191.87/49.43	Rules from S:
191.87/49.43	
191.87/49.43	   c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.87/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.87/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.87/49.43	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.87/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.87/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.87/49.43	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
191.87/49.43	   c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.87/49.43	   c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.87/49.43	   a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.87/49.43	   a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.87/49.43	   b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
191.87/49.43	   b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
191.87/49.43	
191.87/49.43	
191.87/49.43	
191.87/49.43	
191.87/49.43	----------------------------------------
191.87/49.43	
191.87/49.43	(12)
191.87/49.43	Obligation:
191.87/49.43	Relative term rewrite system:
191.87/49.43	The relative TRS consists of the following R rules:
191.87/49.43	
191.87/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
191.87/49.43	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
191.87/49.43	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
191.87/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.87/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.87/49.43	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.87/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.87/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.87/49.43	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.87/49.43	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.87/49.44	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.87/49.44	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.87/49.44	
191.87/49.44	The relative TRS consists of the following S rules:
191.87/49.44	
191.87/49.44	   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.87/49.44	   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.87/49.44	   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.44	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.44	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.44	
191.87/49.44	
191.87/49.44	----------------------------------------
191.87/49.44	
191.87/49.44	(13) RelTRSRRRProof (EQUIVALENT)
191.87/49.44	We used the following monotonic ordering for rule removal:
191.87/49.44	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
191.87/49.44	
191.87/49.44	   <<<
191.87/49.44	 POL(c_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.87/49.44	>>>
191.87/49.44	
191.87/49.44	   <<<
191.87/49.44	 POL(c_{a_1}(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.87/49.44	>>>
191.87/49.44	
191.87/49.44	   <<<
191.87/49.44	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.87/49.44	>>>
191.87/49.44	
191.87/49.44	   <<<
191.87/49.44	 POL(c_{b_1}(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.87/49.44	>>>
191.87/49.44	
191.87/49.44	   <<<
191.87/49.44	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.87/49.44	>>>
191.87/49.44	
191.87/49.44	   <<<
191.87/49.44	 POL(a_{a_1}(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.87/49.44	>>>
191.87/49.44	
191.87/49.44	   <<<
191.87/49.44	 POL(b_{a_1}(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.87/49.44	>>>
191.87/49.44	
191.87/49.44	   <<<
191.87/49.44	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.87/49.44	>>>
191.87/49.44	
191.87/49.44	   <<<
191.87/49.44	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
191.87/49.44	>>>
191.87/49.44	
191.87/49.44	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
191.87/49.44	Rules from R:
191.87/49.44	
191.87/49.44	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
191.87/49.44	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.87/49.44	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.87/49.44	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.87/49.44	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.87/49.44	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.87/49.44	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.87/49.44	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
191.87/49.44	Rules from S:
191.87/49.44	
191.87/49.44	   c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.44	   a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.44	   b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
191.87/49.44	
191.87/49.44	
191.87/49.44	
191.87/49.44	
191.87/49.44	----------------------------------------
191.87/49.44	
191.87/49.44	(14)
191.87/49.44	Obligation:
191.87/49.44	Relative term rewrite system:
191.87/49.44	The relative TRS consists of the following R rules:
191.87/49.44	
191.87/49.44	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
191.87/49.44	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
191.87/49.44	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.87/49.44	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.87/49.44	
191.87/49.44	The relative TRS consists of the following S rules:
191.87/49.44	
191.87/49.44	   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.87/49.44	   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.87/49.44	
191.87/49.44	
191.87/49.44	----------------------------------------
191.87/49.44	
191.87/49.44	(15) RelTRSRRRProof (EQUIVALENT)
191.87/49.44	We used the following monotonic ordering for rule removal:
191.87/49.44	Knuth-Bendix order [KBO] with precedence:a_{c_1}_1 > c_{c_1}_1 > c_{a_1}_1 > b_{b_1}_1 > b_{c_1}_1 > b_{a_1}_1 > a_{a_1}_1 > a_{b_1}_1 > c_{b_1}_1
191.87/49.44	
191.87/49.44	and weight map:
191.87/49.44	
191.87/49.44	   c_{c_1}_1=8
191.87/49.44	   c_{a_1}_1=8
191.87/49.44	   a_{c_1}_1=6
191.87/49.44	   c_{b_1}_1=11
191.87/49.44	   b_{c_1}_1=3
191.87/49.44	   a_{b_1}_1=12
191.87/49.44	   b_{b_1}_1=9
191.87/49.44	   a_{a_1}_1=1
191.87/49.44	   b_{a_1}_1=6
191.87/49.44	
191.87/49.44	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
191.87/49.44	Rules from R:
191.87/49.44	
191.87/49.44	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
191.87/49.44	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
191.87/49.44	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
191.87/49.44	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
191.87/49.44	Rules from S:
191.87/49.44	
191.87/49.44	   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
191.87/49.44	   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
191.87/49.44	
191.87/49.44	
191.87/49.44	
191.87/49.44	
191.87/49.44	----------------------------------------
191.87/49.44	
191.87/49.44	(16)
191.87/49.44	Obligation:
191.87/49.44	Relative term rewrite system:
191.87/49.44	R is empty.
191.87/49.44	S is empty.
191.87/49.44	
191.87/49.44	----------------------------------------
191.87/49.44	
191.87/49.44	(17) RIsEmptyProof (EQUIVALENT)
191.87/49.44	The TRS R is empty. Hence, termination is trivially proven.
191.87/49.44	----------------------------------------
191.87/49.44	
191.87/49.44	(18)
191.87/49.44	YES
191.89/49.48	EOF