14.98/4.68	YES
15.25/4.69	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
15.25/4.69	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
15.25/4.69	
15.25/4.69	
15.25/4.69	Termination of the given RelTRS could be proven:
15.25/4.69	
15.25/4.69	(0) RelTRS
15.25/4.69	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
15.25/4.69	(2) RelTRS
15.25/4.69	(3) FlatCCProof [EQUIVALENT, 0 ms]
15.25/4.69	(4) RelTRS
15.25/4.69	(5) RootLabelingProof [EQUIVALENT, 0 ms]
15.25/4.69	(6) RelTRS
15.25/4.69	(7) RelTRSRRRProof [EQUIVALENT, 13 ms]
15.25/4.69	(8) RelTRS
15.25/4.69	(9) RelTRSRRRProof [EQUIVALENT, 9 ms]
15.25/4.69	(10) RelTRS
15.25/4.69	(11) RelTRSRRRProof [EQUIVALENT, 133 ms]
15.25/4.69	(12) RelTRS
15.25/4.69	(13) RelTRSRRRProof [EQUIVALENT, 0 ms]
15.25/4.69	(14) RelTRS
15.25/4.69	(15) RIsEmptyProof [EQUIVALENT, 0 ms]
15.25/4.69	(16) YES
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(0)
15.25/4.69	Obligation:
15.25/4.69	Relative term rewrite system:
15.25/4.69	The relative TRS consists of the following R rules:
15.25/4.69	
15.25/4.69	   a(a(b(x1))) -> b(a(x1))
15.25/4.69	   c(b(x1)) -> b(a(b(x1)))
15.25/4.69	
15.25/4.69	The relative TRS consists of the following S rules:
15.25/4.69	
15.25/4.69	   a(x1) -> a(c(a(x1)))
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(1) RelTRS Reverse (EQUIVALENT)
15.25/4.69	We have reversed the following relative TRS [REVERSE]:
15.25/4.69	The set of rules R is 
15.25/4.69	   a(a(b(x1))) -> b(a(x1))
15.25/4.69	   c(b(x1)) -> b(a(b(x1)))
15.25/4.69	
15.25/4.69	The set of rules S is 
15.25/4.69	   a(x1) -> a(c(a(x1)))
15.25/4.69	
15.25/4.69	We have obtained the following relative TRS:
15.25/4.69	The set of rules R is 
15.25/4.69	   b(a(a(x1))) -> a(b(x1))
15.25/4.69	   b(c(x1)) -> b(a(b(x1)))
15.25/4.69	
15.25/4.69	The set of rules S is 
15.25/4.69	   a(x1) -> a(c(a(x1)))
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(2)
15.25/4.69	Obligation:
15.25/4.69	Relative term rewrite system:
15.25/4.69	The relative TRS consists of the following R rules:
15.25/4.69	
15.25/4.69	   b(a(a(x1))) -> a(b(x1))
15.25/4.69	   b(c(x1)) -> b(a(b(x1)))
15.25/4.69	
15.25/4.69	The relative TRS consists of the following S rules:
15.25/4.69	
15.25/4.69	   a(x1) -> a(c(a(x1)))
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(3) FlatCCProof (EQUIVALENT)
15.25/4.69	We used flat context closure [ROOTLAB]
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(4)
15.25/4.69	Obligation:
15.25/4.69	Relative term rewrite system:
15.25/4.69	The relative TRS consists of the following R rules:
15.25/4.69	
15.25/4.69	   b(c(x1)) -> b(a(b(x1)))
15.25/4.69	   b(b(a(a(x1)))) -> b(a(b(x1)))
15.25/4.69	   a(b(a(a(x1)))) -> a(a(b(x1)))
15.25/4.69	   c(b(a(a(x1)))) -> c(a(b(x1)))
15.25/4.69	
15.25/4.69	The relative TRS consists of the following S rules:
15.25/4.69	
15.25/4.69	   a(x1) -> a(c(a(x1)))
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(5) RootLabelingProof (EQUIVALENT)
15.25/4.69	We used plain root labeling [ROOTLAB] with the following heuristic:
15.25/4.69	LabelAll: All function symbols get labeled
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(6)
15.25/4.69	Obligation:
15.25/4.69	Relative term rewrite system:
15.25/4.69	The relative TRS consists of the following R rules:
15.25/4.69	
15.25/4.69	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
15.25/4.69	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{b_1}(b_{c_1}(x1)))
15.25/4.69	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
15.25/4.69	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
15.25/4.69	   b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(x1)))
15.25/4.69	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
15.25/4.69	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
15.25/4.69	   a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
15.25/4.69	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
15.25/4.69	   c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1)))
15.25/4.69	   c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1)))
15.25/4.69	   c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1)))
15.25/4.69	
15.25/4.69	The relative TRS consists of the following S rules:
15.25/4.69	
15.25/4.69	   a_{b_1}(x1) -> a_{c_1}(c_{a_1}(a_{b_1}(x1)))
15.25/4.69	   a_{c_1}(x1) -> a_{c_1}(c_{a_1}(a_{c_1}(x1)))
15.25/4.69	   a_{a_1}(x1) -> a_{c_1}(c_{a_1}(a_{a_1}(x1)))
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(7) RelTRSRRRProof (EQUIVALENT)
15.25/4.69	We used the following monotonic ordering for rule removal:
15.25/4.69	Polynomial interpretation [POLO]:
15.25/4.69	
15.25/4.69	   POL(a_{a_1}(x_1)) = 1 + x_1
15.25/4.69	   POL(a_{b_1}(x_1)) = x_1
15.25/4.69	   POL(a_{c_1}(x_1)) = x_1
15.25/4.69	   POL(b_{a_1}(x_1)) = x_1
15.25/4.69	   POL(b_{b_1}(x_1)) = x_1
15.25/4.69	   POL(b_{c_1}(x_1)) = x_1
15.25/4.69	   POL(c_{a_1}(x_1)) = x_1
15.25/4.69	   POL(c_{b_1}(x_1)) = 1 + x_1
15.25/4.69	   POL(c_{c_1}(x_1)) = x_1
15.25/4.69	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
15.25/4.69	Rules from R:
15.25/4.69	
15.25/4.69	   b_{c_1}(c_{b_1}(x1)) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
15.25/4.69	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(x1)))
15.25/4.69	   b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(x1)))
15.25/4.69	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
15.25/4.69	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
15.25/4.69	   c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(x1)))
15.25/4.69	   c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(x1)))
15.25/4.69	   c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(x1)))
15.25/4.69	Rules from S:
15.25/4.69	none
15.25/4.69	
15.25/4.69	
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(8)
15.25/4.69	Obligation:
15.25/4.69	Relative term rewrite system:
15.25/4.69	The relative TRS consists of the following R rules:
15.25/4.69	
15.25/4.69	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{b_1}(b_{c_1}(x1)))
15.25/4.69	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
15.25/4.69	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
15.25/4.69	   a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
15.25/4.69	
15.25/4.69	The relative TRS consists of the following S rules:
15.25/4.69	
15.25/4.69	   a_{b_1}(x1) -> a_{c_1}(c_{a_1}(a_{b_1}(x1)))
15.25/4.69	   a_{c_1}(x1) -> a_{c_1}(c_{a_1}(a_{c_1}(x1)))
15.25/4.69	   a_{a_1}(x1) -> a_{c_1}(c_{a_1}(a_{a_1}(x1)))
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(9) RelTRSRRRProof (EQUIVALENT)
15.25/4.69	We used the following monotonic ordering for rule removal:
15.25/4.69	Polynomial interpretation [POLO]:
15.25/4.69	
15.25/4.69	   POL(a_{a_1}(x_1)) = x_1
15.25/4.69	   POL(a_{b_1}(x_1)) = x_1
15.25/4.69	   POL(a_{c_1}(x_1)) = x_1
15.25/4.69	   POL(b_{a_1}(x_1)) = x_1
15.25/4.69	   POL(b_{b_1}(x_1)) = x_1
15.25/4.69	   POL(b_{c_1}(x_1)) = x_1
15.25/4.69	   POL(c_{a_1}(x_1)) = x_1
15.25/4.69	   POL(c_{c_1}(x_1)) = 1 + x_1
15.25/4.69	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
15.25/4.69	Rules from R:
15.25/4.69	
15.25/4.69	   b_{c_1}(c_{c_1}(x1)) -> b_{a_1}(a_{b_1}(b_{c_1}(x1)))
15.25/4.69	Rules from S:
15.25/4.69	none
15.25/4.69	
15.25/4.69	
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(10)
15.25/4.69	Obligation:
15.25/4.69	Relative term rewrite system:
15.25/4.69	The relative TRS consists of the following R rules:
15.25/4.69	
15.25/4.69	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
15.25/4.69	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
15.25/4.69	   a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
15.25/4.69	
15.25/4.69	The relative TRS consists of the following S rules:
15.25/4.69	
15.25/4.69	   a_{b_1}(x1) -> a_{c_1}(c_{a_1}(a_{b_1}(x1)))
15.25/4.69	   a_{c_1}(x1) -> a_{c_1}(c_{a_1}(a_{c_1}(x1)))
15.25/4.69	   a_{a_1}(x1) -> a_{c_1}(c_{a_1}(a_{a_1}(x1)))
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(11) RelTRSRRRProof (EQUIVALENT)
15.25/4.69	We used the following monotonic ordering for rule removal:
15.25/4.69	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
15.25/4.69	
15.25/4.69	   <<<
15.25/4.69	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 1]] 	* 	x_1
15.25/4.69	>>>
15.25/4.69	
15.25/4.69	   <<<
15.25/4.69	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
15.25/4.69	>>>
15.25/4.69	
15.25/4.69	   <<<
15.25/4.69	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
15.25/4.69	>>>
15.25/4.69	
15.25/4.69	   <<<
15.25/4.69	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
15.25/4.69	>>>
15.25/4.69	
15.25/4.69	   <<<
15.25/4.69	 POL(a_{a_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
15.25/4.69	>>>
15.25/4.69	
15.25/4.69	   <<<
15.25/4.69	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
15.25/4.69	>>>
15.25/4.69	
15.25/4.69	   <<<
15.25/4.69	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
15.25/4.69	>>>
15.25/4.69	
15.25/4.69	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
15.25/4.69	Rules from R:
15.25/4.69	
15.25/4.69	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
15.25/4.69	   a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
15.25/4.69	Rules from S:
15.25/4.69	none
15.25/4.69	
15.25/4.69	
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(12)
15.25/4.69	Obligation:
15.25/4.69	Relative term rewrite system:
15.25/4.69	The relative TRS consists of the following R rules:
15.25/4.69	
15.25/4.69	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
15.25/4.69	
15.25/4.69	The relative TRS consists of the following S rules:
15.25/4.69	
15.25/4.69	   a_{b_1}(x1) -> a_{c_1}(c_{a_1}(a_{b_1}(x1)))
15.25/4.69	   a_{c_1}(x1) -> a_{c_1}(c_{a_1}(a_{c_1}(x1)))
15.25/4.69	   a_{a_1}(x1) -> a_{c_1}(c_{a_1}(a_{a_1}(x1)))
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(13) RelTRSRRRProof (EQUIVALENT)
15.25/4.69	We used the following monotonic ordering for rule removal:
15.25/4.69	Polynomial interpretation [POLO]:
15.25/4.69	
15.25/4.69	   POL(a_{a_1}(x_1)) = x_1
15.25/4.69	   POL(a_{b_1}(x_1)) = x_1
15.25/4.69	   POL(a_{c_1}(x_1)) = x_1
15.25/4.69	   POL(b_{a_1}(x_1)) = x_1
15.25/4.69	   POL(b_{c_1}(x_1)) = 1 + x_1
15.25/4.69	   POL(c_{a_1}(x_1)) = x_1
15.25/4.69	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
15.25/4.69	Rules from R:
15.25/4.69	
15.25/4.69	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(a_{b_1}(b_{a_1}(x1)))
15.25/4.69	Rules from S:
15.25/4.69	none
15.25/4.69	
15.25/4.69	
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(14)
15.25/4.69	Obligation:
15.25/4.69	Relative term rewrite system:
15.25/4.69	R is empty.
15.25/4.69	The relative TRS consists of the following S rules:
15.25/4.69	
15.25/4.69	   a_{b_1}(x1) -> a_{c_1}(c_{a_1}(a_{b_1}(x1)))
15.25/4.69	   a_{c_1}(x1) -> a_{c_1}(c_{a_1}(a_{c_1}(x1)))
15.25/4.69	   a_{a_1}(x1) -> a_{c_1}(c_{a_1}(a_{a_1}(x1)))
15.25/4.69	
15.25/4.69	
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(15) RIsEmptyProof (EQUIVALENT)
15.25/4.69	The TRS R is empty. Hence, termination is trivially proven.
15.25/4.69	----------------------------------------
15.25/4.69	
15.25/4.69	(16)
15.25/4.69	YES
15.25/4.72	EOF