8.14/3.00	YES
8.37/3.10	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
8.37/3.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
8.37/3.10	
8.37/3.10	
8.37/3.10	Termination of the given RelTRS could be proven:
8.37/3.10	
8.37/3.10	(0) RelTRS
8.37/3.10	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
8.37/3.10	(2) RelTRS
8.37/3.10	(3) FlatCCProof [EQUIVALENT, 0 ms]
8.37/3.10	(4) RelTRS
8.37/3.10	(5) RootLabelingProof [EQUIVALENT, 1 ms]
8.37/3.10	(6) RelTRS
8.37/3.10	(7) RelTRSRRRProof [EQUIVALENT, 20 ms]
8.37/3.10	(8) RelTRS
8.37/3.10	(9) RelTRSRRRProof [EQUIVALENT, 156 ms]
8.37/3.10	(10) RelTRS
8.37/3.10	(11) RelTRSRRRProof [EQUIVALENT, 80 ms]
8.37/3.10	(12) RelTRS
8.37/3.10	(13) RIsEmptyProof [EQUIVALENT, 1 ms]
8.37/3.10	(14) YES
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(0)
8.37/3.10	Obligation:
8.37/3.10	Relative term rewrite system:
8.37/3.10	The relative TRS consists of the following R rules:
8.37/3.10	
8.37/3.10	   f(g(f(x))) -> f(g(g(g(f(x)))))
8.37/3.10	
8.37/3.10	The relative TRS consists of the following S rules:
8.37/3.10	
8.37/3.10	   g(x) -> g(g(x))
8.37/3.10	   g(x) -> g(g(f(g(g(x)))))
8.37/3.10	   f(x) -> g(g(f(g(g(x)))))
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(1) RelTRS Reverse (EQUIVALENT)
8.37/3.10	We have reversed the following relative TRS [REVERSE]:
8.37/3.10	The set of rules R is 
8.37/3.10	   f(g(f(x))) -> f(g(g(g(f(x)))))
8.37/3.10	
8.37/3.10	The set of rules S is 
8.37/3.10	   g(x) -> g(g(x))
8.37/3.10	   g(x) -> g(g(f(g(g(x)))))
8.37/3.10	   f(x) -> g(g(f(g(g(x)))))
8.37/3.10	
8.37/3.10	We have obtained the following relative TRS:
8.37/3.10	The set of rules R is 
8.37/3.10	   f(g(f(x))) -> f(g(g(g(f(x)))))
8.37/3.10	
8.37/3.10	The set of rules S is 
8.37/3.10	   g(x) -> g(g(x))
8.37/3.10	   g(x) -> g(g(f(g(g(x)))))
8.37/3.10	   f(x) -> g(g(f(g(g(x)))))
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(2)
8.37/3.10	Obligation:
8.37/3.10	Relative term rewrite system:
8.37/3.10	The relative TRS consists of the following R rules:
8.37/3.10	
8.37/3.10	   f(g(f(x))) -> f(g(g(g(f(x)))))
8.37/3.10	
8.37/3.10	The relative TRS consists of the following S rules:
8.37/3.10	
8.37/3.10	   g(x) -> g(g(x))
8.37/3.10	   g(x) -> g(g(f(g(g(x)))))
8.37/3.10	   f(x) -> g(g(f(g(g(x)))))
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(3) FlatCCProof (EQUIVALENT)
8.37/3.10	We used flat context closure [ROOTLAB]
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(4)
8.37/3.10	Obligation:
8.37/3.10	Relative term rewrite system:
8.37/3.10	The relative TRS consists of the following R rules:
8.37/3.10	
8.37/3.10	   f(g(f(x))) -> f(g(g(g(f(x)))))
8.37/3.10	
8.37/3.10	The relative TRS consists of the following S rules:
8.37/3.10	
8.37/3.10	   g(x) -> g(g(x))
8.37/3.10	   g(x) -> g(g(f(g(g(x)))))
8.37/3.10	   f(f(x)) -> f(g(g(f(g(g(x))))))
8.37/3.10	   g(f(x)) -> g(g(g(f(g(g(x))))))
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(5) RootLabelingProof (EQUIVALENT)
8.37/3.10	We used plain root labeling [ROOTLAB] with the following heuristic:
8.37/3.10	LabelAll: All function symbols get labeled
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(6)
8.37/3.10	Obligation:
8.37/3.10	Relative term rewrite system:
8.37/3.10	The relative TRS consists of the following R rules:
8.37/3.10	
8.37/3.10	   f_{g_1}(g_{f_1}(f_{f_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{f_1}(x)))))
8.37/3.10	   f_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(x)))))
8.37/3.10	
8.37/3.10	The relative TRS consists of the following S rules:
8.37/3.10	
8.37/3.10	   g_{f_1}(x) -> g_{g_1}(g_{f_1}(x))
8.37/3.10	   g_{g_1}(x) -> g_{g_1}(g_{g_1}(x))
8.37/3.10	   g_{f_1}(x) -> g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{f_1}(x)))))
8.37/3.10	   g_{g_1}(x) -> g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{g_1}(x)))))
8.37/3.10	   f_{f_1}(f_{f_1}(x)) -> f_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{f_1}(x))))))
8.37/3.10	   f_{f_1}(f_{g_1}(x)) -> f_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{g_1}(x))))))
8.37/3.10	   g_{f_1}(f_{f_1}(x)) -> g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{f_1}(x))))))
8.37/3.10	   g_{f_1}(f_{g_1}(x)) -> g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{g_1}(x))))))
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(7) RelTRSRRRProof (EQUIVALENT)
8.37/3.10	We used the following monotonic ordering for rule removal:
8.37/3.10	Polynomial interpretation [POLO]:
8.37/3.10	
8.37/3.10	   POL(f_{f_1}(x_1)) = 1 + x_1
8.37/3.10	   POL(f_{g_1}(x_1)) = x_1
8.37/3.10	   POL(g_{f_1}(x_1)) = x_1
8.37/3.10	   POL(g_{g_1}(x_1)) = x_1
8.37/3.10	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.37/3.10	Rules from R:
8.37/3.10	none
8.37/3.10	Rules from S:
8.37/3.10	
8.37/3.10	   f_{f_1}(f_{f_1}(x)) -> f_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{f_1}(x))))))
8.37/3.10	   f_{f_1}(f_{g_1}(x)) -> f_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{g_1}(x))))))
8.37/3.10	   g_{f_1}(f_{f_1}(x)) -> g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{f_1}(x))))))
8.37/3.10	
8.37/3.10	
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(8)
8.37/3.10	Obligation:
8.37/3.10	Relative term rewrite system:
8.37/3.10	The relative TRS consists of the following R rules:
8.37/3.10	
8.37/3.10	   f_{g_1}(g_{f_1}(f_{f_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{f_1}(x)))))
8.37/3.10	   f_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(x)))))
8.37/3.10	
8.37/3.10	The relative TRS consists of the following S rules:
8.37/3.10	
8.37/3.10	   g_{f_1}(x) -> g_{g_1}(g_{f_1}(x))
8.37/3.10	   g_{g_1}(x) -> g_{g_1}(g_{g_1}(x))
8.37/3.10	   g_{f_1}(x) -> g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{f_1}(x)))))
8.37/3.10	   g_{g_1}(x) -> g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{g_1}(x)))))
8.37/3.10	   g_{f_1}(f_{g_1}(x)) -> g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{g_1}(x))))))
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(9) RelTRSRRRProof (EQUIVALENT)
8.37/3.10	We used the following monotonic ordering for rule removal:
8.37/3.10	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.37/3.10	
8.37/3.10	   <<<
8.37/3.10	 POL(f_{g_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
8.37/3.10	>>>
8.37/3.10	
8.37/3.10	   <<<
8.37/3.10	 POL(g_{f_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
8.37/3.10	>>>
8.37/3.10	
8.37/3.10	   <<<
8.37/3.10	 POL(f_{f_1}(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
8.37/3.10	>>>
8.37/3.10	
8.37/3.10	   <<<
8.37/3.10	 POL(g_{g_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.37/3.10	>>>
8.37/3.10	
8.37/3.10	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.37/3.10	Rules from R:
8.37/3.10	
8.37/3.10	   f_{g_1}(g_{f_1}(f_{f_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{f_1}(x)))))
8.37/3.10	Rules from S:
8.37/3.10	none
8.37/3.10	
8.37/3.10	
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(10)
8.37/3.10	Obligation:
8.37/3.10	Relative term rewrite system:
8.37/3.10	The relative TRS consists of the following R rules:
8.37/3.10	
8.37/3.10	   f_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(x)))))
8.37/3.10	
8.37/3.10	The relative TRS consists of the following S rules:
8.37/3.10	
8.37/3.10	   g_{f_1}(x) -> g_{g_1}(g_{f_1}(x))
8.37/3.10	   g_{g_1}(x) -> g_{g_1}(g_{g_1}(x))
8.37/3.10	   g_{f_1}(x) -> g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{f_1}(x)))))
8.37/3.10	   g_{g_1}(x) -> g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{g_1}(x)))))
8.37/3.10	   g_{f_1}(f_{g_1}(x)) -> g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{g_1}(x))))))
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(11) RelTRSRRRProof (EQUIVALENT)
8.37/3.10	We used the following monotonic ordering for rule removal:
8.37/3.10	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
8.37/3.10	
8.37/3.10	   <<<
8.37/3.10	 POL(f_{g_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 2]] 	* 	x_1
8.37/3.10	>>>
8.37/3.10	
8.37/3.10	   <<<
8.37/3.10	 POL(g_{f_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [1, 2]] 	* 	x_1
8.37/3.10	>>>
8.37/3.10	
8.37/3.10	   <<<
8.37/3.10	 POL(g_{g_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
8.37/3.10	>>>
8.37/3.10	
8.37/3.10	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
8.37/3.10	Rules from R:
8.37/3.10	
8.37/3.10	   f_{g_1}(g_{f_1}(f_{g_1}(x))) -> f_{g_1}(g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(x)))))
8.37/3.10	Rules from S:
8.37/3.10	none
8.37/3.10	
8.37/3.10	
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(12)
8.37/3.10	Obligation:
8.37/3.10	Relative term rewrite system:
8.37/3.10	R is empty.
8.37/3.10	The relative TRS consists of the following S rules:
8.37/3.10	
8.37/3.10	   g_{f_1}(x) -> g_{g_1}(g_{f_1}(x))
8.37/3.10	   g_{g_1}(x) -> g_{g_1}(g_{g_1}(x))
8.37/3.10	   g_{f_1}(x) -> g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{f_1}(x)))))
8.37/3.10	   g_{g_1}(x) -> g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{g_1}(x)))))
8.37/3.10	   g_{f_1}(f_{g_1}(x)) -> g_{g_1}(g_{g_1}(g_{f_1}(f_{g_1}(g_{g_1}(g_{g_1}(x))))))
8.37/3.10	
8.37/3.10	
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(13) RIsEmptyProof (EQUIVALENT)
8.37/3.10	The TRS R is empty. Hence, termination is trivially proven.
8.37/3.10	----------------------------------------
8.37/3.10	
8.37/3.10	(14)
8.37/3.10	YES
8.60/3.17	EOF