77.56/20.51	YES
77.56/20.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
77.56/20.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
77.56/20.52	
77.56/20.52	
77.56/20.52	Termination of the given RelTRS could be proven:
77.56/20.52	
77.56/20.52	(0) RelTRS
77.56/20.52	(1) FlatCCProof [EQUIVALENT, 0 ms]
77.56/20.52	(2) RelTRS
77.56/20.52	(3) RootLabelingProof [EQUIVALENT, 0 ms]
77.56/20.52	(4) RelTRS
77.56/20.52	(5) RelTRSRRRProof [EQUIVALENT, 1892 ms]
77.56/20.52	(6) RelTRS
77.56/20.52	(7) RelTRSRRRProof [EQUIVALENT, 2315 ms]
77.56/20.52	(8) RelTRS
77.56/20.52	(9) RelTRSRRRProof [EQUIVALENT, 1527 ms]
77.56/20.52	(10) RelTRS
77.56/20.52	(11) RelTRSRRRProof [EQUIVALENT, 758 ms]
77.56/20.52	(12) RelTRS
77.56/20.52	(13) RelTRSRRRProof [EQUIVALENT, 3 ms]
77.56/20.52	(14) RelTRS
77.56/20.52	(15) RIsEmptyProof [EQUIVALENT, 0 ms]
77.56/20.52	(16) YES
77.56/20.52	
77.56/20.52	
77.56/20.52	----------------------------------------
77.56/20.52	
77.56/20.52	(0)
77.56/20.52	Obligation:
77.56/20.52	Relative term rewrite system:
77.56/20.52	The relative TRS consists of the following R rules:
77.56/20.52	
77.56/20.52	   c(a(a(x1))) -> c(c(b(x1)))
77.56/20.52	   c(a(b(x1))) -> c(b(a(x1)))
77.56/20.52	   b(b(a(x1))) -> b(b(c(x1)))
77.56/20.52	   b(c(b(x1))) -> a(a(a(x1)))
77.56/20.52	
77.56/20.52	The relative TRS consists of the following S rules:
77.56/20.52	
77.56/20.52	   c(a(a(x1))) -> a(b(a(x1)))
77.56/20.52	   a(b(c(x1))) -> a(a(c(x1)))
77.56/20.52	
77.56/20.52	
77.56/20.52	----------------------------------------
77.56/20.52	
77.56/20.52	(1) FlatCCProof (EQUIVALENT)
77.56/20.52	We used flat context closure [ROOTLAB]
77.56/20.52	
77.56/20.52	----------------------------------------
77.56/20.52	
77.56/20.52	(2)
77.56/20.52	Obligation:
77.56/20.52	Relative term rewrite system:
77.56/20.52	The relative TRS consists of the following R rules:
77.56/20.52	
77.56/20.52	   c(a(a(x1))) -> c(c(b(x1)))
77.56/20.52	   c(a(b(x1))) -> c(b(a(x1)))
77.56/20.52	   b(b(a(x1))) -> b(b(c(x1)))
77.56/20.52	   c(b(c(b(x1)))) -> c(a(a(a(x1))))
77.56/20.52	   a(b(c(b(x1)))) -> a(a(a(a(x1))))
77.56/20.52	   b(b(c(b(x1)))) -> b(a(a(a(x1))))
77.56/20.52	
77.56/20.52	The relative TRS consists of the following S rules:
77.56/20.52	
77.56/20.52	   a(b(c(x1))) -> a(a(c(x1)))
77.56/20.52	   c(c(a(a(x1)))) -> c(a(b(a(x1))))
77.56/20.52	   a(c(a(a(x1)))) -> a(a(b(a(x1))))
77.56/20.52	   b(c(a(a(x1)))) -> b(a(b(a(x1))))
77.56/20.52	
77.56/20.52	
77.56/20.52	----------------------------------------
77.56/20.52	
77.56/20.52	(3) RootLabelingProof (EQUIVALENT)
77.56/20.52	We used plain root labeling [ROOTLAB] with the following heuristic:
77.56/20.52	LabelAll: All function symbols get labeled
77.56/20.52	
77.56/20.52	
77.56/20.52	----------------------------------------
77.56/20.52	
77.56/20.52	(4)
77.56/20.52	Obligation:
77.56/20.52	Relative term rewrite system:
77.56/20.52	The relative TRS consists of the following R rules:
77.56/20.52	
77.56/20.52	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
77.56/20.52	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
77.56/20.52	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
77.56/20.52	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1)))
77.56/20.52	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1)))
77.56/20.52	   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{b_1}(x1)))
77.56/20.52	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(x1)))
77.56/20.52	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{c_1}(c_{a_1}(x1)))
77.56/20.52	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1)))
77.56/20.52	   c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.52	   c_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.52	   c_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
77.56/20.52	   b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.52	   b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.52	   b_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
77.56/20.52	
77.56/20.52	The relative TRS consists of the following S rules:
77.56/20.52	
77.56/20.52	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{c_1}(x1)))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{a_1}(x1)))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(x1)))
77.56/20.52	   c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.52	   c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.52	   c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.52	   a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.52	   a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.52	   a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.52	   b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.52	   b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.52	   b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.52	
77.56/20.52	
77.56/20.52	----------------------------------------
77.56/20.52	
77.56/20.52	(5) RelTRSRRRProof (EQUIVALENT)
77.56/20.52	We used the following monotonic ordering for rule removal:
77.56/20.52	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(c_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(b_{b_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
77.56/20.52	Rules from R:
77.56/20.52	
77.56/20.52	   c_{a_1}(a_{b_1}(b_{b_1}(x1))) -> c_{b_1}(b_{a_1}(a_{b_1}(x1)))
77.56/20.52	Rules from S:
77.56/20.52	none
77.56/20.52	
77.56/20.52	
77.56/20.52	
77.56/20.52	
77.56/20.52	----------------------------------------
77.56/20.52	
77.56/20.52	(6)
77.56/20.52	Obligation:
77.56/20.52	Relative term rewrite system:
77.56/20.52	The relative TRS consists of the following R rules:
77.56/20.52	
77.56/20.52	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
77.56/20.52	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
77.56/20.52	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
77.56/20.52	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1)))
77.56/20.52	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1)))
77.56/20.52	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(x1)))
77.56/20.52	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{c_1}(c_{a_1}(x1)))
77.56/20.52	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1)))
77.56/20.52	   c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.52	   c_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.52	   c_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
77.56/20.52	   b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.52	   b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.52	   b_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
77.56/20.52	
77.56/20.52	The relative TRS consists of the following S rules:
77.56/20.52	
77.56/20.52	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{c_1}(x1)))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{a_1}(x1)))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(x1)))
77.56/20.52	   c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.52	   c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.52	   c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.52	   a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.52	   a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.52	   a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.52	   b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.52	   b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.52	   b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.52	
77.56/20.52	
77.56/20.52	----------------------------------------
77.56/20.52	
77.56/20.52	(7) RelTRSRRRProof (EQUIVALENT)
77.56/20.52	We used the following monotonic ordering for rule removal:
77.56/20.52	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(c_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 1]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 1], [0, 0]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	   <<<
77.56/20.52	 POL(b_{b_1}(x_1)) =  	[[0], [2]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
77.56/20.52	>>>
77.56/20.52	
77.56/20.52	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
77.56/20.52	Rules from R:
77.56/20.52	
77.56/20.52	   c_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
77.56/20.52	   b_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
77.56/20.52	Rules from S:
77.56/20.52	none
77.56/20.52	
77.56/20.52	
77.56/20.52	
77.56/20.52	
77.56/20.52	----------------------------------------
77.56/20.52	
77.56/20.52	(8)
77.56/20.52	Obligation:
77.56/20.52	Relative term rewrite system:
77.56/20.52	The relative TRS consists of the following R rules:
77.56/20.52	
77.56/20.52	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
77.56/20.52	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
77.56/20.52	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
77.56/20.52	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1)))
77.56/20.52	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1)))
77.56/20.52	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(x1)))
77.56/20.52	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{c_1}(c_{a_1}(x1)))
77.56/20.52	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1)))
77.56/20.52	   c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.52	   c_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.52	   b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.52	   b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.52	
77.56/20.52	The relative TRS consists of the following S rules:
77.56/20.52	
77.56/20.52	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{c_1}(x1)))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{a_1}(x1)))
77.56/20.52	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(x1)))
77.56/20.52	   c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.52	   c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.52	   c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.52	   a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.52	   a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.52	   a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.52	   b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.52	   b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.52	   b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.53	
77.56/20.53	
77.56/20.53	----------------------------------------
77.56/20.53	
77.56/20.53	(9) RelTRSRRRProof (EQUIVALENT)
77.56/20.53	We used the following monotonic ordering for rule removal:
77.56/20.53	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(a_{a_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(a_{c_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 2], [1, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(c_{c_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(c_{b_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(a_{b_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
77.56/20.53	Rules from R:
77.56/20.53	
77.56/20.53	   c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.53	   c_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.53	   a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.53	   a_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.53	Rules from S:
77.56/20.53	
77.56/20.53	   a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.53	   a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.53	   a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.53	
77.56/20.53	
77.56/20.53	
77.56/20.53	
77.56/20.53	----------------------------------------
77.56/20.53	
77.56/20.53	(10)
77.56/20.53	Obligation:
77.56/20.53	Relative term rewrite system:
77.56/20.53	The relative TRS consists of the following R rules:
77.56/20.53	
77.56/20.53	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
77.56/20.53	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
77.56/20.53	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
77.56/20.53	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1)))
77.56/20.53	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1)))
77.56/20.53	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(x1)))
77.56/20.53	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{c_1}(c_{a_1}(x1)))
77.56/20.53	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1)))
77.56/20.53	   b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.53	   b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.53	
77.56/20.53	The relative TRS consists of the following S rules:
77.56/20.53	
77.56/20.53	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{c_1}(x1)))
77.56/20.53	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{a_1}(x1)))
77.56/20.53	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(x1)))
77.56/20.53	   c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.53	   c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.53	   c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.53	   b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.53	   b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.53	   b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.53	
77.56/20.53	
77.56/20.53	----------------------------------------
77.56/20.53	
77.56/20.53	(11) RelTRSRRRProof (EQUIVALENT)
77.56/20.53	We used the following monotonic ordering for rule removal:
77.56/20.53	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(c_{a_1}(x_1)) =  	[[0], [2]] 	 +  	[[2, 0], [1, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(c_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 2]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	   <<<
77.56/20.53	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
77.56/20.53	>>>
77.56/20.53	
77.56/20.53	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
77.56/20.53	Rules from R:
77.56/20.53	none
77.56/20.53	Rules from S:
77.56/20.53	
77.56/20.53	   c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.53	   c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.53	   c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.53	
77.56/20.53	
77.56/20.53	
77.56/20.53	
77.56/20.53	----------------------------------------
77.56/20.53	
77.56/20.53	(12)
77.56/20.53	Obligation:
77.56/20.53	Relative term rewrite system:
77.56/20.53	The relative TRS consists of the following R rules:
77.56/20.53	
77.56/20.53	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
77.56/20.53	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
77.56/20.53	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
77.56/20.53	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1)))
77.56/20.53	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1)))
77.56/20.53	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(x1)))
77.56/20.53	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{c_1}(c_{a_1}(x1)))
77.56/20.53	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1)))
77.56/20.53	   b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.53	   b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.53	
77.56/20.53	The relative TRS consists of the following S rules:
77.56/20.53	
77.56/20.53	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{c_1}(x1)))
77.56/20.53	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{a_1}(x1)))
77.56/20.53	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(x1)))
77.56/20.53	   b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.53	   b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.53	   b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.53	
77.56/20.53	
77.56/20.53	----------------------------------------
77.56/20.53	
77.56/20.53	(13) RelTRSRRRProof (EQUIVALENT)
77.56/20.53	We used the following monotonic ordering for rule removal:
77.56/20.53	Knuth-Bendix order [KBO] with precedence:a_{b_1}_1 > a_{a_1}_1 > b_{a_1}_1 > c_{b_1}_1 > b_{c_1}_1 > a_{c_1}_1 > c_{c_1}_1 > b_{b_1}_1 > c_{a_1}_1
77.56/20.53	
77.56/20.53	and weight map:
77.56/20.53	
77.56/20.53	   c_{a_1}_1=17
77.56/20.53	   a_{a_1}_1=18
77.56/20.53	   a_{c_1}_1=10
77.56/20.53	   c_{c_1}_1=11
77.56/20.53	   c_{b_1}_1=1
77.56/20.53	   b_{c_1}_1=22
77.56/20.53	   b_{a_1}_1=23
77.56/20.53	   a_{b_1}_1=10
77.56/20.53	   b_{b_1}_1=32
77.56/20.53	
77.56/20.53	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
77.56/20.53	Rules from R:
77.56/20.53	
77.56/20.53	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
77.56/20.53	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
77.56/20.53	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
77.56/20.53	   c_{a_1}(a_{b_1}(b_{c_1}(x1))) -> c_{b_1}(b_{a_1}(a_{c_1}(x1)))
77.56/20.53	   c_{a_1}(a_{b_1}(b_{a_1}(x1))) -> c_{b_1}(b_{a_1}(a_{a_1}(x1)))
77.56/20.53	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{b_1}(b_{c_1}(c_{c_1}(x1)))
77.56/20.53	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{c_1}(c_{a_1}(x1)))
77.56/20.53	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{c_1}(c_{b_1}(x1)))
77.56/20.53	   b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
77.56/20.53	   b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
77.56/20.53	Rules from S:
77.56/20.53	
77.56/20.53	   a_{b_1}(b_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{c_1}(c_{c_1}(x1)))
77.56/20.53	   a_{b_1}(b_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{c_1}(c_{a_1}(x1)))
77.56/20.53	   a_{b_1}(b_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{c_1}(c_{b_1}(x1)))
77.56/20.53	   b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
77.56/20.53	   b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
77.56/20.53	   b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
77.56/20.53	
77.56/20.53	
77.56/20.53	
77.56/20.53	
77.56/20.53	----------------------------------------
77.56/20.53	
77.56/20.53	(14)
77.56/20.53	Obligation:
77.56/20.53	Relative term rewrite system:
77.56/20.53	R is empty.
77.56/20.53	S is empty.
77.56/20.53	
77.56/20.53	----------------------------------------
77.56/20.53	
77.56/20.53	(15) RIsEmptyProof (EQUIVALENT)
77.56/20.53	The TRS R is empty. Hence, termination is trivially proven.
77.56/20.53	----------------------------------------
77.56/20.53	
77.56/20.53	(16)
77.56/20.53	YES
77.56/20.57	EOF