3.26/1.54	YES
3.26/1.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.26/1.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.26/1.55	
3.26/1.55	
3.26/1.55	Termination of the given CSR could be proven:
3.26/1.55	
3.26/1.55	(0) CSR
3.26/1.55	(1) CSRRRRProof [EQUIVALENT, 41 ms]
3.26/1.55	(2) CSR
3.26/1.55	(3) CSRRRRProof [EQUIVALENT, 7 ms]
3.26/1.55	(4) CSR
3.26/1.55	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.26/1.55	(6) CSR
3.26/1.55	(7) CSRRRRProof [EQUIVALENT, 0 ms]
3.26/1.55	(8) CSR
3.26/1.55	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.26/1.55	(10) YES
3.26/1.55	
3.26/1.55	
3.26/1.55	----------------------------------------
3.26/1.55	
3.26/1.55	(0)
3.26/1.55	Obligation:
3.26/1.55	Context-sensitive rewrite system:
3.26/1.55	The TRS R consists of the following rules:
3.26/1.55	
3.26/1.55	   from_2(x) -> cons_0(x, from_1(s_0(x)))
3.26/1.55	   from_3(x) -> cons_0(x, from_2(s_0(x)))
3.26/1.55	   from_4(x) -> cons_0(x, from_2(s_0(x)))
3.26/1.55	   from_1(x) -> cons_1(x, from_1(s_0(x)))
3.26/1.55	   cons_1(s_0(s_0(x)), xs) -> nil_0
3.26/1.55	
3.26/1.55	The replacement map contains the following entries:
3.26/1.55	
3.26/1.55	from_2: empty set
3.26/1.55	cons_0: {1, 2}
3.26/1.55	from_1: empty set
3.26/1.55	s_0: {1}
3.26/1.55	from_3: empty set
3.26/1.55	from_4: empty set
3.26/1.55	cons_1: empty set
3.26/1.55	nil_0: empty set
3.26/1.55	
3.26/1.55	----------------------------------------
3.26/1.55	
3.26/1.55	(1) CSRRRRProof (EQUIVALENT)
3.26/1.55	The following CSR is given: Context-sensitive rewrite system:
3.26/1.55	The TRS R consists of the following rules:
3.26/1.55	
3.26/1.55	   from_2(x) -> cons_0(x, from_1(s_0(x)))
3.26/1.55	   from_3(x) -> cons_0(x, from_2(s_0(x)))
3.26/1.55	   from_4(x) -> cons_0(x, from_2(s_0(x)))
3.26/1.55	   from_1(x) -> cons_1(x, from_1(s_0(x)))
3.26/1.55	   cons_1(s_0(s_0(x)), xs) -> nil_0
3.26/1.55	
3.26/1.55	The replacement map contains the following entries:
3.26/1.55	
3.26/1.55	from_2: empty set
3.26/1.55	cons_0: {1, 2}
3.26/1.55	from_1: empty set
3.26/1.55	s_0: {1}
3.26/1.55	from_3: empty set
3.26/1.55	from_4: empty set
3.26/1.55	cons_1: empty set
3.26/1.55	nil_0: empty set
3.26/1.55	Used ordering:
3.26/1.55	Polynomial interpretation [POLO]:
3.26/1.55	
3.26/1.55	   POL(cons_0(x_1, x_2)) = x_1 + x_2
3.26/1.55	   POL(cons_1(x_1, x_2)) = 0
3.26/1.55	   POL(from_1(x_1)) = 0
3.26/1.55	   POL(from_2(x_1)) = x_1
3.26/1.55	   POL(from_3(x_1)) = 2 + 2*x_1
3.26/1.55	   POL(from_4(x_1)) = 2 + 2*x_1
3.26/1.55	   POL(nil_0) = 0
3.26/1.55	   POL(s_0(x_1)) = x_1
3.26/1.55	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.26/1.55	
3.26/1.55	   from_3(x) -> cons_0(x, from_2(s_0(x)))
3.26/1.55	   from_4(x) -> cons_0(x, from_2(s_0(x)))
3.26/1.55	
3.26/1.55	
3.26/1.55	
3.26/1.55	
3.26/1.55	----------------------------------------
3.26/1.55	
3.26/1.55	(2)
3.26/1.55	Obligation:
3.26/1.55	Context-sensitive rewrite system:
3.26/1.55	The TRS R consists of the following rules:
3.26/1.55	
3.26/1.55	   from_2(x) -> cons_0(x, from_1(s_0(x)))
3.26/1.55	   from_1(x) -> cons_1(x, from_1(s_0(x)))
3.26/1.55	   cons_1(s_0(s_0(x)), xs) -> nil_0
3.26/1.55	
3.26/1.55	The replacement map contains the following entries:
3.26/1.55	
3.26/1.55	from_2: empty set
3.26/1.55	cons_0: {1, 2}
3.26/1.55	from_1: empty set
3.26/1.55	s_0: {1}
3.26/1.55	cons_1: empty set
3.26/1.55	nil_0: empty set
3.26/1.55	
3.26/1.55	----------------------------------------
3.26/1.55	
3.26/1.55	(3) CSRRRRProof (EQUIVALENT)
3.26/1.55	The following CSR is given: Context-sensitive rewrite system:
3.26/1.55	The TRS R consists of the following rules:
3.26/1.55	
3.26/1.55	   from_2(x) -> cons_0(x, from_1(s_0(x)))
3.26/1.55	   from_1(x) -> cons_1(x, from_1(s_0(x)))
3.26/1.55	   cons_1(s_0(s_0(x)), xs) -> nil_0
3.26/1.55	
3.26/1.55	The replacement map contains the following entries:
3.26/1.55	
3.26/1.55	from_2: empty set
3.26/1.55	cons_0: {1, 2}
3.26/1.55	from_1: empty set
3.26/1.55	s_0: {1}
3.26/1.55	cons_1: empty set
3.26/1.55	nil_0: empty set
3.26/1.55	Used ordering:
3.26/1.55	Polynomial interpretation [POLO]:
3.26/1.55	
3.26/1.55	   POL(cons_0(x_1, x_2)) = x_1 + x_2
3.26/1.55	   POL(cons_1(x_1, x_2)) = 1
3.26/1.55	   POL(from_1(x_1)) = 1
3.26/1.55	   POL(from_2(x_1)) = 1 + x_1
3.26/1.55	   POL(nil_0) = 0
3.26/1.55	   POL(s_0(x_1)) = 1 + x_1
3.26/1.55	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.26/1.55	
3.26/1.55	   cons_1(s_0(s_0(x)), xs) -> nil_0
3.26/1.55	
3.26/1.55	
3.26/1.55	
3.26/1.55	
3.26/1.55	----------------------------------------
3.26/1.55	
3.26/1.55	(4)
3.26/1.55	Obligation:
3.26/1.55	Context-sensitive rewrite system:
3.26/1.55	The TRS R consists of the following rules:
3.26/1.55	
3.26/1.55	   from_2(x) -> cons_0(x, from_1(s_0(x)))
3.26/1.55	   from_1(x) -> cons_1(x, from_1(s_0(x)))
3.26/1.55	
3.26/1.55	The replacement map contains the following entries:
3.26/1.55	
3.26/1.55	from_2: empty set
3.26/1.55	cons_0: {1, 2}
3.26/1.55	from_1: empty set
3.26/1.55	s_0: {1}
3.26/1.55	cons_1: empty set
3.26/1.55	
3.26/1.55	----------------------------------------
3.26/1.55	
3.26/1.55	(5) CSRRRRProof (EQUIVALENT)
3.26/1.55	The following CSR is given: Context-sensitive rewrite system:
3.26/1.55	The TRS R consists of the following rules:
3.26/1.55	
3.26/1.55	   from_2(x) -> cons_0(x, from_1(s_0(x)))
3.26/1.55	   from_1(x) -> cons_1(x, from_1(s_0(x)))
3.26/1.55	
3.26/1.55	The replacement map contains the following entries:
3.26/1.55	
3.26/1.55	from_2: empty set
3.26/1.55	cons_0: {1, 2}
3.26/1.55	from_1: empty set
3.26/1.55	s_0: {1}
3.26/1.55	cons_1: empty set
3.26/1.55	Used ordering:
3.26/1.55	Polynomial interpretation [POLO]:
3.26/1.55	
3.26/1.55	   POL(cons_0(x_1, x_2)) = x_1 + x_2
3.26/1.55	   POL(cons_1(x_1, x_2)) = x_2
3.26/1.55	   POL(from_1(x_1)) = 0
3.26/1.55	   POL(from_2(x_1)) = 1 + x_1
3.26/1.55	   POL(s_0(x_1)) = 1 + x_1
3.26/1.55	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.26/1.55	
3.26/1.55	   from_2(x) -> cons_0(x, from_1(s_0(x)))
3.26/1.55	
3.26/1.55	
3.26/1.55	
3.26/1.55	
3.26/1.55	----------------------------------------
3.26/1.55	
3.26/1.55	(6)
3.26/1.55	Obligation:
3.26/1.55	Context-sensitive rewrite system:
3.26/1.55	The TRS R consists of the following rules:
3.26/1.55	
3.26/1.55	   from_1(x) -> cons_1(x, from_1(s_0(x)))
3.26/1.55	
3.26/1.55	The replacement map contains the following entries:
3.26/1.55	
3.26/1.55	from_1: empty set
3.26/1.55	s_0: {1}
3.26/1.55	cons_1: empty set
3.26/1.55	
3.26/1.55	----------------------------------------
3.26/1.55	
3.26/1.55	(7) CSRRRRProof (EQUIVALENT)
3.26/1.55	The following CSR is given: Context-sensitive rewrite system:
3.26/1.55	The TRS R consists of the following rules:
3.26/1.55	
3.26/1.55	   from_1(x) -> cons_1(x, from_1(s_0(x)))
3.26/1.55	
3.26/1.55	The replacement map contains the following entries:
3.26/1.55	
3.26/1.55	from_1: empty set
3.26/1.55	s_0: {1}
3.26/1.55	cons_1: empty set
3.26/1.55	Used ordering:
3.26/1.55	Polynomial interpretation [POLO]:
3.26/1.55	
3.26/1.55	   POL(cons_1(x_1, x_2)) = 2*x_1
3.26/1.55	   POL(from_1(x_1)) = 1 + 2*x_1
3.26/1.55	   POL(s_0(x_1)) = x_1
3.26/1.55	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.26/1.55	
3.26/1.55	   from_1(x) -> cons_1(x, from_1(s_0(x)))
3.26/1.55	
3.26/1.55	
3.26/1.55	
3.26/1.55	
3.26/1.55	----------------------------------------
3.26/1.55	
3.26/1.55	(8)
3.26/1.55	Obligation:
3.26/1.55	Context-sensitive rewrite system:
3.26/1.55	R is empty.
3.26/1.55	
3.26/1.55	----------------------------------------
3.26/1.55	
3.26/1.55	(9) RisEmptyProof (EQUIVALENT)
3.26/1.55	The CSR R is empty. Hence, termination is trivially proven.
3.26/1.55	----------------------------------------
3.26/1.55	
3.26/1.55	(10)
3.26/1.55	YES
3.30/1.66	EOF