2.68/1.39	YES
2.68/1.40	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
2.68/1.40	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
2.68/1.40	
2.68/1.40	
2.68/1.40	Termination of the given CSR could be proven:
2.68/1.40	
2.68/1.40	(0) CSR
2.68/1.40	(1) CSRRRRProof [EQUIVALENT, 56 ms]
2.68/1.40	(2) CSR
2.68/1.40	(3) CSRRRRProof [EQUIVALENT, 0 ms]
2.68/1.40	(4) CSR
2.68/1.40	(5) CSRRRRProof [EQUIVALENT, 0 ms]
2.68/1.40	(6) CSR
2.68/1.40	(7) RisEmptyProof [EQUIVALENT, 0 ms]
2.68/1.40	(8) YES
2.68/1.40	
2.68/1.40	
2.68/1.40	----------------------------------------
2.68/1.40	
2.68/1.40	(0)
2.68/1.40	Obligation:
2.68/1.40	Context-sensitive rewrite system:
2.68/1.40	The TRS R consists of the following rules:
2.68/1.40	
2.68/1.40	   f_1(h_0(x)) -> f_1(i_1(x))
2.68/1.40	   f_1(i_1(x)) -> a_0
2.68/1.40	   i_1(x) -> h_0(x)
2.68/1.40	
2.68/1.40	The replacement map contains the following entries:
2.68/1.40	
2.68/1.40	f_1: empty set
2.68/1.40	h_0: {1}
2.68/1.40	i_1: empty set
2.68/1.40	a_0: empty set
2.68/1.40	
2.68/1.40	----------------------------------------
2.68/1.40	
2.68/1.40	(1) CSRRRRProof (EQUIVALENT)
2.68/1.40	The following CSR is given: Context-sensitive rewrite system:
2.68/1.40	The TRS R consists of the following rules:
2.68/1.40	
2.68/1.40	   f_1(h_0(x)) -> f_1(i_1(x))
2.68/1.40	   f_1(i_1(x)) -> a_0
2.68/1.40	   i_1(x) -> h_0(x)
2.68/1.40	
2.68/1.40	The replacement map contains the following entries:
2.68/1.40	
2.68/1.40	f_1: empty set
2.68/1.40	h_0: {1}
2.68/1.40	i_1: empty set
2.68/1.40	a_0: empty set
2.68/1.40	Used ordering:
2.68/1.40	Polynomial interpretation [POLO]:
2.68/1.40	
2.68/1.40	   POL(a_0) = 1
2.68/1.40	   POL(f_1(x_1)) = 1 + 2*x_1
2.68/1.40	   POL(h_0(x_1)) = 1 + 2*x_1
2.68/1.40	   POL(i_1(x_1)) = 1 + 2*x_1
2.68/1.40	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.68/1.40	
2.68/1.40	   f_1(i_1(x)) -> a_0
2.68/1.40	
2.68/1.40	
2.68/1.40	
2.68/1.40	
2.68/1.40	----------------------------------------
2.68/1.40	
2.68/1.40	(2)
2.68/1.40	Obligation:
2.68/1.40	Context-sensitive rewrite system:
2.68/1.40	The TRS R consists of the following rules:
2.68/1.40	
2.68/1.40	   f_1(h_0(x)) -> f_1(i_1(x))
2.68/1.40	   i_1(x) -> h_0(x)
2.68/1.40	
2.68/1.40	The replacement map contains the following entries:
2.68/1.40	
2.68/1.40	f_1: empty set
2.68/1.40	h_0: {1}
2.68/1.40	i_1: empty set
2.68/1.40	
2.68/1.40	----------------------------------------
2.68/1.40	
2.68/1.40	(3) CSRRRRProof (EQUIVALENT)
2.68/1.40	The following CSR is given: Context-sensitive rewrite system:
2.68/1.40	The TRS R consists of the following rules:
2.68/1.40	
2.68/1.40	   f_1(h_0(x)) -> f_1(i_1(x))
2.68/1.40	   i_1(x) -> h_0(x)
2.68/1.40	
2.68/1.40	The replacement map contains the following entries:
2.68/1.40	
2.68/1.40	f_1: empty set
2.68/1.40	h_0: {1}
2.68/1.40	i_1: empty set
2.68/1.40	Used ordering:
2.68/1.40	Polynomial interpretation [POLO]:
2.68/1.40	
2.68/1.40	   POL(f_1(x_1)) = 0
2.68/1.40	   POL(h_0(x_1)) = x_1
2.68/1.40	   POL(i_1(x_1)) = 1 + x_1
2.68/1.40	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.68/1.40	
2.68/1.40	   i_1(x) -> h_0(x)
2.68/1.40	
2.68/1.40	
2.68/1.40	
2.68/1.40	
2.68/1.40	----------------------------------------
2.68/1.40	
2.68/1.40	(4)
2.68/1.40	Obligation:
2.68/1.40	Context-sensitive rewrite system:
2.68/1.40	The TRS R consists of the following rules:
2.68/1.40	
2.68/1.40	   f_1(h_0(x)) -> f_1(i_1(x))
2.68/1.40	
2.68/1.40	The replacement map contains the following entries:
2.68/1.40	
2.68/1.40	f_1: empty set
2.68/1.40	h_0: {1}
2.68/1.40	i_1: empty set
2.68/1.40	
2.68/1.40	----------------------------------------
2.68/1.40	
2.68/1.40	(5) CSRRRRProof (EQUIVALENT)
2.68/1.40	The following CSR is given: Context-sensitive rewrite system:
2.68/1.40	The TRS R consists of the following rules:
2.68/1.40	
2.68/1.40	   f_1(h_0(x)) -> f_1(i_1(x))
2.68/1.40	
2.68/1.40	The replacement map contains the following entries:
2.68/1.40	
2.68/1.40	f_1: empty set
2.68/1.40	h_0: {1}
2.68/1.40	i_1: empty set
2.68/1.40	Used ordering:
2.68/1.40	Polynomial interpretation [POLO]:
2.68/1.40	
2.68/1.40	   POL(f_1(x_1)) = x_1
2.68/1.40	   POL(h_0(x_1)) = 2 + 2*x_1
2.68/1.40	   POL(i_1(x_1)) = 1 + 2*x_1
2.68/1.40	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
2.68/1.40	
2.68/1.40	   f_1(h_0(x)) -> f_1(i_1(x))
2.68/1.40	
2.68/1.40	
2.68/1.40	
2.68/1.40	
2.68/1.40	----------------------------------------
2.68/1.40	
2.68/1.40	(6)
2.68/1.40	Obligation:
2.68/1.40	Context-sensitive rewrite system:
2.68/1.40	R is empty.
2.68/1.40	
2.68/1.40	----------------------------------------
2.68/1.40	
2.68/1.40	(7) RisEmptyProof (EQUIVALENT)
2.68/1.40	The CSR R is empty. Hence, termination is trivially proven.
2.68/1.40	----------------------------------------
2.68/1.40	
2.68/1.40	(8)
2.68/1.40	YES
2.90/1.42	EOF