2.61/1.34	YES
2.61/1.34	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.61/1.34	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.61/1.34	
2.61/1.34	
2.61/1.34	Termination of the given CSR could be proven:
2.61/1.34	
2.61/1.34	(0) CSR
2.61/1.34	(1) CSRRRRProof [EQUIVALENT, 26 ms]
2.61/1.34	(2) CSR
2.61/1.34	(3) CSRRRRProof [EQUIVALENT, 0 ms]
2.61/1.34	(4) CSR
2.61/1.34	(5) CSRRRRProof [EQUIVALENT, 0 ms]
2.61/1.34	(6) CSR
2.61/1.34	(7) RisEmptyProof [EQUIVALENT, 0 ms]
2.61/1.34	(8) YES
2.61/1.34	
2.61/1.34	
2.61/1.34	----------------------------------------
2.61/1.34	
2.61/1.34	(0)
2.61/1.34	Obligation:
2.61/1.34	Context-sensitive rewrite system:
2.61/1.34	The TRS R consists of the following rules:
2.61/1.34	
2.61/1.34	   f_1(x) -> g_1(f_1(x))
2.61/1.34	   g_1(f_1(x)) -> x
2.61/1.34	   g_1(x) -> a_0
2.61/1.34	
2.61/1.34	The replacement map contains the following entries:
2.61/1.34	
2.61/1.34	f_1: empty set
2.61/1.34	g_1: empty set
2.61/1.34	a_0: empty set
2.61/1.34	
2.61/1.34	----------------------------------------
2.61/1.34	
2.61/1.34	(1) CSRRRRProof (EQUIVALENT)
2.61/1.34	The following CSR is given: Context-sensitive rewrite system:
2.61/1.34	The TRS R consists of the following rules:
2.61/1.34	
2.61/1.34	   f_1(x) -> g_1(f_1(x))
2.61/1.34	   g_1(f_1(x)) -> x
2.61/1.34	   g_1(x) -> a_0
2.61/1.34	
2.61/1.34	The replacement map contains the following entries:
2.61/1.34	
2.61/1.34	f_1: empty set
2.61/1.34	g_1: empty set
2.61/1.34	a_0: empty set
2.61/1.34	Used ordering:
2.61/1.34	Polynomial interpretation [POLO]:
2.61/1.34	
2.61/1.34	   POL(a_0) = 0
2.61/1.34	   POL(f_1(x_1)) = 1 + 2*x_1
2.61/1.34	   POL(g_1(x_1)) = x_1
2.61/1.34	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.61/1.34	
2.61/1.34	   g_1(f_1(x)) -> x
2.61/1.34	
2.61/1.34	
2.61/1.34	
2.61/1.34	
2.61/1.34	----------------------------------------
2.61/1.34	
2.61/1.34	(2)
2.61/1.34	Obligation:
2.61/1.34	Context-sensitive rewrite system:
2.61/1.34	The TRS R consists of the following rules:
2.61/1.34	
2.61/1.34	   f_1(x) -> g_1(f_1(x))
2.61/1.34	   g_1(x) -> a_0
2.61/1.34	
2.61/1.34	The replacement map contains the following entries:
2.61/1.34	
2.61/1.34	f_1: empty set
2.61/1.34	g_1: empty set
2.61/1.34	a_0: empty set
2.61/1.34	
2.61/1.34	----------------------------------------
2.61/1.34	
2.61/1.34	(3) CSRRRRProof (EQUIVALENT)
2.61/1.34	The following CSR is given: Context-sensitive rewrite system:
2.61/1.34	The TRS R consists of the following rules:
2.61/1.34	
2.61/1.34	   f_1(x) -> g_1(f_1(x))
2.61/1.34	   g_1(x) -> a_0
2.61/1.34	
2.61/1.34	The replacement map contains the following entries:
2.61/1.34	
2.61/1.34	f_1: empty set
2.61/1.34	g_1: empty set
2.61/1.34	a_0: empty set
2.61/1.34	Used ordering:
2.61/1.34	Polynomial interpretation [POLO]:
2.61/1.34	
2.61/1.34	   POL(a_0) = 0
2.61/1.34	   POL(f_1(x_1)) = 1 + x_1
2.61/1.34	   POL(g_1(x_1)) = 1
2.61/1.34	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.61/1.34	
2.61/1.34	   g_1(x) -> a_0
2.61/1.34	
2.61/1.34	
2.61/1.34	
2.61/1.34	
2.61/1.34	----------------------------------------
2.61/1.34	
2.61/1.34	(4)
2.61/1.34	Obligation:
2.61/1.34	Context-sensitive rewrite system:
2.61/1.34	The TRS R consists of the following rules:
2.61/1.34	
2.61/1.34	   f_1(x) -> g_1(f_1(x))
2.61/1.34	
2.61/1.34	The replacement map contains the following entries:
2.61/1.34	
2.61/1.34	f_1: empty set
2.61/1.34	g_1: empty set
2.61/1.34	
2.61/1.34	----------------------------------------
2.61/1.34	
2.61/1.34	(5) CSRRRRProof (EQUIVALENT)
2.61/1.34	The following CSR is given: Context-sensitive rewrite system:
2.61/1.34	The TRS R consists of the following rules:
2.61/1.34	
2.61/1.34	   f_1(x) -> g_1(f_1(x))
2.61/1.34	
2.61/1.34	The replacement map contains the following entries:
2.61/1.34	
2.61/1.34	f_1: empty set
2.61/1.34	g_1: empty set
2.61/1.34	Used ordering:
2.61/1.34	Polynomial interpretation [POLO]:
2.61/1.34	
2.61/1.34	   POL(f_1(x_1)) = 1 + 2*x_1
2.61/1.34	   POL(g_1(x_1)) = 0
2.61/1.34	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.61/1.34	
2.61/1.34	   f_1(x) -> g_1(f_1(x))
2.61/1.34	
2.61/1.34	
2.61/1.34	
2.61/1.34	
2.61/1.34	----------------------------------------
2.61/1.34	
2.61/1.34	(6)
2.61/1.34	Obligation:
2.61/1.34	Context-sensitive rewrite system:
2.61/1.34	R is empty.
2.61/1.34	
2.61/1.34	----------------------------------------
2.61/1.34	
2.61/1.34	(7) RisEmptyProof (EQUIVALENT)
2.61/1.34	The CSR R is empty. Hence, termination is trivially proven.
2.61/1.34	----------------------------------------
2.61/1.34	
2.61/1.34	(8)
2.61/1.34	YES
2.61/1.36	EOF