11.02/3.81	YES
11.02/3.82	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
11.02/3.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.02/3.82	
11.02/3.82	
11.02/3.82	Termination of the given RelTRS could be proven:
11.02/3.82	
11.02/3.82	(0) RelTRS
11.02/3.82	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
11.02/3.82	(2) RelTRS
11.02/3.82	(3) FlatCCProof [EQUIVALENT, 0 ms]
11.02/3.82	(4) RelTRS
11.02/3.82	(5) RootLabelingProof [EQUIVALENT, 0 ms]
11.02/3.82	(6) RelTRS
11.02/3.82	(7) RelTRSRRRProof [EQUIVALENT, 16 ms]
11.02/3.82	(8) RelTRS
11.02/3.82	(9) RelTRSRRRProof [EQUIVALENT, 1 ms]
11.02/3.82	(10) RelTRS
11.02/3.82	(11) RelTRSRRRProof [EQUIVALENT, 120 ms]
11.02/3.82	(12) RelTRS
11.02/3.82	(13) RelTRSRRRProof [EQUIVALENT, 4 ms]
11.02/3.82	(14) RelTRS
11.02/3.82	(15) RIsEmptyProof [EQUIVALENT, 0 ms]
11.02/3.82	(16) YES
11.02/3.82	
11.02/3.82	
11.02/3.82	----------------------------------------
11.02/3.82	
11.02/3.82	(0)
11.02/3.82	Obligation:
11.02/3.82	Relative term rewrite system:
11.02/3.82	The relative TRS consists of the following R rules:
11.02/3.82	
11.02/3.82	   b(c(a(x1))) -> a(b(a(b(x1))))
11.02/3.82	   a(a(x1)) -> a(c(b(a(x1))))
11.02/3.82	
11.02/3.82	The relative TRS consists of the following S rules:
11.02/3.82	
11.02/3.82	   b(x1) -> c(b(x1))
11.02/3.82	
11.02/3.82	
11.02/3.82	----------------------------------------
11.02/3.82	
11.02/3.82	(1) RelTRS Reverse (EQUIVALENT)
11.02/3.82	We have reversed the following relative TRS [REVERSE]:
11.02/3.82	The set of rules R is 
11.02/3.82	   b(c(a(x1))) -> a(b(a(b(x1))))
11.02/3.82	   a(a(x1)) -> a(c(b(a(x1))))
11.02/3.82	
11.02/3.82	The set of rules S is 
11.02/3.82	   b(x1) -> c(b(x1))
11.02/3.82	
11.02/3.82	We have obtained the following relative TRS:
11.02/3.82	The set of rules R is 
11.02/3.82	   a(c(b(x1))) -> b(a(b(a(x1))))
11.02/3.82	   a(a(x1)) -> a(b(c(a(x1))))
11.02/3.82	
11.02/3.82	The set of rules S is 
11.02/3.82	   b(x1) -> b(c(x1))
11.02/3.82	
11.02/3.82	
11.02/3.82	----------------------------------------
11.02/3.82	
11.02/3.82	(2)
11.02/3.82	Obligation:
11.02/3.82	Relative term rewrite system:
11.02/3.82	The relative TRS consists of the following R rules:
11.02/3.82	
11.02/3.82	   a(c(b(x1))) -> b(a(b(a(x1))))
11.02/3.82	   a(a(x1)) -> a(b(c(a(x1))))
11.02/3.82	
11.02/3.82	The relative TRS consists of the following S rules:
11.02/3.82	
11.02/3.82	   b(x1) -> b(c(x1))
11.02/3.82	
11.02/3.82	
11.02/3.82	----------------------------------------
11.02/3.82	
11.02/3.82	(3) FlatCCProof (EQUIVALENT)
11.02/3.82	We used flat context closure [ROOTLAB]
11.02/3.82	
11.02/3.82	----------------------------------------
11.02/3.82	
11.02/3.82	(4)
11.02/3.82	Obligation:
11.02/3.82	Relative term rewrite system:
11.02/3.82	The relative TRS consists of the following R rules:
11.02/3.82	
11.02/3.82	   a(a(x1)) -> a(b(c(a(x1))))
11.02/3.82	   a(a(c(b(x1)))) -> a(b(a(b(a(x1)))))
11.02/3.82	   c(a(c(b(x1)))) -> c(b(a(b(a(x1)))))
11.02/3.82	   b(a(c(b(x1)))) -> b(b(a(b(a(x1)))))
11.02/3.82	
11.02/3.82	The relative TRS consists of the following S rules:
11.02/3.82	
11.02/3.82	   b(x1) -> b(c(x1))
11.02/3.82	
11.02/3.82	
11.02/3.82	----------------------------------------
11.02/3.82	
11.02/3.82	(5) RootLabelingProof (EQUIVALENT)
11.02/3.82	We used plain root labeling [ROOTLAB] with the following heuristic:
11.02/3.82	LabelAll: All function symbols get labeled
11.02/3.82	
11.02/3.82	
11.02/3.82	----------------------------------------
11.02/3.82	
11.02/3.82	(6)
11.02/3.82	Obligation:
11.02/3.82	Relative term rewrite system:
11.02/3.82	The relative TRS consists of the following R rules:
11.02/3.82	
11.02/3.82	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
11.02/3.82	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
11.02/3.82	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
11.02/3.82	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
11.02/3.82	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
11.02/3.82	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.82	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
11.02/3.82	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
11.02/3.82	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.82	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
11.02/3.82	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
11.02/3.82	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.82	
11.02/3.82	The relative TRS consists of the following S rules:
11.02/3.82	
11.02/3.82	   b_{a_1}(x1) -> b_{c_1}(c_{a_1}(x1))
11.02/3.82	   b_{b_1}(x1) -> b_{c_1}(c_{b_1}(x1))
11.02/3.82	   b_{c_1}(x1) -> b_{c_1}(c_{c_1}(x1))
11.02/3.82	
11.02/3.82	
11.02/3.82	----------------------------------------
11.02/3.82	
11.02/3.82	(7) RelTRSRRRProof (EQUIVALENT)
11.02/3.82	We used the following monotonic ordering for rule removal:
11.02/3.82	Polynomial interpretation [POLO]:
11.02/3.82	
11.02/3.82	   POL(a_{a_1}(x_1)) = x_1
11.02/3.82	   POL(a_{b_1}(x_1)) = x_1
11.02/3.82	   POL(a_{c_1}(x_1)) = 1 + x_1
11.02/3.82	   POL(b_{a_1}(x_1)) = x_1
11.02/3.82	   POL(b_{b_1}(x_1)) = x_1
11.02/3.82	   POL(b_{c_1}(x_1)) = x_1
11.02/3.82	   POL(c_{a_1}(x_1)) = x_1
11.02/3.82	   POL(c_{b_1}(x_1)) = x_1
11.02/3.82	   POL(c_{c_1}(x_1)) = x_1
11.02/3.82	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
11.02/3.83	Rules from R:
11.02/3.83	
11.02/3.83	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
11.02/3.83	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
11.02/3.83	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
11.02/3.83	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
11.02/3.83	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
11.02/3.83	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
11.02/3.83	Rules from S:
11.02/3.83	none
11.02/3.83	
11.02/3.83	
11.02/3.83	
11.02/3.83	
11.02/3.83	----------------------------------------
11.02/3.83	
11.02/3.83	(8)
11.02/3.83	Obligation:
11.02/3.83	Relative term rewrite system:
11.02/3.83	The relative TRS consists of the following R rules:
11.02/3.83	
11.02/3.83	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
11.02/3.83	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
11.02/3.83	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
11.02/3.83	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.83	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.83	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.83	
11.02/3.83	The relative TRS consists of the following S rules:
11.02/3.83	
11.02/3.83	   b_{a_1}(x1) -> b_{c_1}(c_{a_1}(x1))
11.02/3.83	   b_{b_1}(x1) -> b_{c_1}(c_{b_1}(x1))
11.02/3.83	   b_{c_1}(x1) -> b_{c_1}(c_{c_1}(x1))
11.02/3.83	
11.02/3.83	
11.02/3.83	----------------------------------------
11.02/3.83	
11.02/3.83	(9) RelTRSRRRProof (EQUIVALENT)
11.02/3.83	We used the following monotonic ordering for rule removal:
11.02/3.83	Polynomial interpretation [POLO]:
11.02/3.83	
11.02/3.83	   POL(a_{a_1}(x_1)) = 1 + x_1
11.02/3.83	   POL(a_{b_1}(x_1)) = x_1
11.02/3.83	   POL(a_{c_1}(x_1)) = x_1
11.02/3.83	   POL(b_{a_1}(x_1)) = x_1
11.02/3.83	   POL(b_{b_1}(x_1)) = x_1
11.02/3.83	   POL(b_{c_1}(x_1)) = x_1
11.02/3.83	   POL(c_{a_1}(x_1)) = x_1
11.02/3.83	   POL(c_{b_1}(x_1)) = x_1
11.02/3.83	   POL(c_{c_1}(x_1)) = x_1
11.02/3.83	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
11.02/3.83	Rules from R:
11.02/3.83	
11.02/3.83	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
11.02/3.83	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
11.02/3.83	   a_{a_1}(a_{c_1}(x1)) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
11.02/3.83	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.83	Rules from S:
11.02/3.83	none
11.02/3.83	
11.02/3.83	
11.02/3.83	
11.02/3.83	
11.02/3.83	----------------------------------------
11.02/3.83	
11.02/3.83	(10)
11.02/3.83	Obligation:
11.02/3.83	Relative term rewrite system:
11.02/3.83	The relative TRS consists of the following R rules:
11.02/3.83	
11.02/3.83	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.83	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.83	
11.02/3.83	The relative TRS consists of the following S rules:
11.02/3.83	
11.02/3.83	   b_{a_1}(x1) -> b_{c_1}(c_{a_1}(x1))
11.02/3.83	   b_{b_1}(x1) -> b_{c_1}(c_{b_1}(x1))
11.02/3.83	   b_{c_1}(x1) -> b_{c_1}(c_{c_1}(x1))
11.02/3.83	
11.02/3.83	
11.02/3.83	----------------------------------------
11.02/3.83	
11.02/3.83	(11) RelTRSRRRProof (EQUIVALENT)
11.02/3.83	We used the following monotonic ordering for rule removal:
11.02/3.83	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
11.02/3.83	
11.02/3.83	   <<<
11.02/3.83	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
11.02/3.83	>>>
11.02/3.83	
11.02/3.83	   <<<
11.02/3.83	 POL(a_{c_1}(x_1)) =  	[[0], [2]] 	 +  	[[2, 1], [0, 2]] 	* 	x_1
11.02/3.83	>>>
11.02/3.83	
11.02/3.83	   <<<
11.02/3.83	 POL(c_{b_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [2, 2]] 	* 	x_1
11.02/3.83	>>>
11.02/3.83	
11.02/3.83	   <<<
11.02/3.83	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 1]] 	* 	x_1
11.02/3.83	>>>
11.02/3.83	
11.02/3.83	   <<<
11.02/3.83	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
11.02/3.83	>>>
11.02/3.83	
11.02/3.83	   <<<
11.02/3.83	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
11.02/3.83	>>>
11.02/3.83	
11.02/3.83	   <<<
11.02/3.83	 POL(b_{b_1}(x_1)) =  	[[2], [2]] 	 +  	[[2, 2], [2, 2]] 	* 	x_1
11.02/3.83	>>>
11.02/3.83	
11.02/3.83	   <<<
11.02/3.83	 POL(c_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
11.02/3.83	>>>
11.02/3.83	
11.02/3.83	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
11.02/3.83	Rules from R:
11.02/3.83	
11.02/3.83	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.83	Rules from S:
11.02/3.83	
11.02/3.83	   b_{b_1}(x1) -> b_{c_1}(c_{b_1}(x1))
11.02/3.83	
11.02/3.83	
11.02/3.83	
11.02/3.83	
11.02/3.83	----------------------------------------
11.02/3.83	
11.02/3.83	(12)
11.02/3.83	Obligation:
11.02/3.83	Relative term rewrite system:
11.02/3.83	The relative TRS consists of the following R rules:
11.02/3.83	
11.02/3.83	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.83	
11.02/3.83	The relative TRS consists of the following S rules:
11.02/3.83	
11.02/3.83	   b_{a_1}(x1) -> b_{c_1}(c_{a_1}(x1))
11.02/3.83	   b_{c_1}(x1) -> b_{c_1}(c_{c_1}(x1))
11.02/3.83	
11.02/3.83	
11.02/3.83	----------------------------------------
11.02/3.83	
11.02/3.83	(13) RelTRSRRRProof (EQUIVALENT)
11.02/3.83	We used the following monotonic ordering for rule removal:
11.02/3.83	Polynomial interpretation [POLO]:
11.02/3.83	
11.02/3.83	   POL(a_{b_1}(x_1)) = x_1
11.02/3.83	   POL(a_{c_1}(x_1)) = x_1
11.02/3.83	   POL(b_{a_1}(x_1)) = 1 + x_1
11.02/3.83	   POL(b_{b_1}(x_1)) = x_1
11.02/3.83	   POL(b_{c_1}(x_1)) = 1 + x_1
11.02/3.83	   POL(c_{a_1}(x_1)) = x_1
11.02/3.83	   POL(c_{b_1}(x_1)) = 1 + x_1
11.02/3.83	   POL(c_{c_1}(x_1)) = x_1
11.02/3.83	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
11.02/3.83	Rules from R:
11.02/3.83	
11.02/3.83	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1)))))
11.02/3.83	Rules from S:
11.02/3.83	none
11.02/3.83	
11.02/3.83	
11.02/3.83	
11.02/3.83	
11.02/3.83	----------------------------------------
11.02/3.83	
11.02/3.83	(14)
11.02/3.83	Obligation:
11.02/3.83	Relative term rewrite system:
11.02/3.83	R is empty.
11.02/3.83	The relative TRS consists of the following S rules:
11.02/3.83	
11.02/3.83	   b_{a_1}(x1) -> b_{c_1}(c_{a_1}(x1))
11.02/3.83	   b_{c_1}(x1) -> b_{c_1}(c_{c_1}(x1))
11.02/3.83	
11.02/3.83	
11.02/3.83	----------------------------------------
11.02/3.83	
11.02/3.83	(15) RIsEmptyProof (EQUIVALENT)
11.02/3.83	The TRS R is empty. Hence, termination is trivially proven.
11.02/3.83	----------------------------------------
11.02/3.83	
11.02/3.83	(16)
11.02/3.83	YES
11.02/3.86	EOF