3.27/1.43	YES
3.27/1.43	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.27/1.43	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.27/1.43	
3.27/1.43	
3.27/1.43	Termination of the given CSR could be proven:
3.27/1.43	
3.27/1.43	(0) CSR
3.27/1.43	(1) CSRRRRProof [EQUIVALENT, 37 ms]
3.27/1.43	(2) CSR
3.27/1.43	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.27/1.43	(4) CSR
3.27/1.43	(5) CSRRRRProof [EQUIVALENT, 2 ms]
3.27/1.43	(6) CSR
3.27/1.43	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.27/1.43	(8) YES
3.27/1.43	
3.27/1.43	
3.27/1.43	----------------------------------------
3.27/1.43	
3.27/1.43	(0)
3.27/1.43	Obligation:
3.27/1.43	Context-sensitive rewrite system:
3.27/1.43	The TRS R consists of the following rules:
3.27/1.43	
3.27/1.43	   from(X) -> cons(X, from(s(X)))
3.27/1.43	   length(nil) -> 0
3.27/1.43	   length(cons(X, Y)) -> s(length1(Y))
3.27/1.43	   length1(X) -> length(X)
3.27/1.43	
3.27/1.43	The replacement map contains the following entries:
3.27/1.43	
3.27/1.43	from: {1}
3.27/1.43	cons: {1}
3.27/1.43	s: {1}
3.27/1.43	length: empty set
3.27/1.43	nil: empty set
3.27/1.43	0: empty set
3.27/1.43	length1: empty set
3.27/1.43	
3.27/1.43	----------------------------------------
3.27/1.43	
3.27/1.43	(1) CSRRRRProof (EQUIVALENT)
3.27/1.43	The following CSR is given: Context-sensitive rewrite system:
3.27/1.43	The TRS R consists of the following rules:
3.27/1.43	
3.27/1.43	   from(X) -> cons(X, from(s(X)))
3.27/1.43	   length(nil) -> 0
3.27/1.43	   length(cons(X, Y)) -> s(length1(Y))
3.27/1.43	   length1(X) -> length(X)
3.27/1.43	
3.27/1.43	The replacement map contains the following entries:
3.27/1.43	
3.27/1.43	from: {1}
3.27/1.43	cons: {1}
3.27/1.43	s: {1}
3.27/1.43	length: empty set
3.27/1.43	nil: empty set
3.27/1.43	0: empty set
3.27/1.43	length1: empty set
3.27/1.43	Used ordering:
3.27/1.43	Polynomial interpretation [POLO]:
3.27/1.43	
3.27/1.43	   POL(0) = 1
3.27/1.43	   POL(cons(x_1, x_2)) = x_1
3.27/1.43	   POL(from(x_1)) = 1 + x_1
3.27/1.43	   POL(length(x_1)) = 1
3.27/1.43	   POL(length1(x_1)) = 1
3.27/1.43	   POL(nil) = 0
3.27/1.43	   POL(s(x_1)) = x_1
3.27/1.43	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.27/1.43	
3.27/1.43	   from(X) -> cons(X, from(s(X)))
3.27/1.43	
3.27/1.43	
3.27/1.43	
3.27/1.43	
3.27/1.43	----------------------------------------
3.27/1.43	
3.27/1.43	(2)
3.27/1.43	Obligation:
3.27/1.43	Context-sensitive rewrite system:
3.27/1.43	The TRS R consists of the following rules:
3.27/1.43	
3.27/1.43	   length(nil) -> 0
3.27/1.43	   length(cons(X, Y)) -> s(length1(Y))
3.27/1.43	   length1(X) -> length(X)
3.27/1.43	
3.27/1.43	The replacement map contains the following entries:
3.27/1.43	
3.27/1.43	cons: {1}
3.27/1.43	s: {1}
3.27/1.43	length: empty set
3.27/1.43	nil: empty set
3.27/1.43	0: empty set
3.27/1.43	length1: empty set
3.27/1.43	
3.27/1.43	----------------------------------------
3.27/1.43	
3.27/1.43	(3) CSRRRRProof (EQUIVALENT)
3.27/1.43	The following CSR is given: Context-sensitive rewrite system:
3.27/1.43	The TRS R consists of the following rules:
3.27/1.43	
3.27/1.43	   length(nil) -> 0
3.27/1.43	   length(cons(X, Y)) -> s(length1(Y))
3.27/1.43	   length1(X) -> length(X)
3.27/1.43	
3.27/1.43	The replacement map contains the following entries:
3.27/1.43	
3.27/1.43	cons: {1}
3.27/1.43	s: {1}
3.27/1.43	length: empty set
3.27/1.43	nil: empty set
3.27/1.43	0: empty set
3.27/1.43	length1: empty set
3.27/1.43	Used ordering:
3.27/1.43	Polynomial interpretation [POLO]:
3.27/1.43	
3.27/1.43	   POL(0) = 1
3.27/1.43	   POL(cons(x_1, x_2)) = 2 + x_1 + 2*x_2
3.27/1.43	   POL(length(x_1)) = 1 + 2*x_1
3.27/1.43	   POL(length1(x_1)) = 2 + 2*x_1
3.27/1.43	   POL(nil) = 0
3.27/1.43	   POL(s(x_1)) = x_1
3.27/1.43	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.27/1.43	
3.27/1.43	   length(cons(X, Y)) -> s(length1(Y))
3.27/1.43	   length1(X) -> length(X)
3.27/1.43	
3.27/1.43	
3.27/1.43	
3.27/1.43	
3.27/1.43	----------------------------------------
3.27/1.43	
3.27/1.43	(4)
3.27/1.43	Obligation:
3.27/1.43	Context-sensitive rewrite system:
3.27/1.43	The TRS R consists of the following rules:
3.27/1.43	
3.27/1.43	   length(nil) -> 0
3.27/1.43	
3.27/1.43	The replacement map contains the following entries:
3.27/1.43	
3.27/1.43	length: empty set
3.27/1.43	nil: empty set
3.27/1.43	0: empty set
3.27/1.43	
3.27/1.43	----------------------------------------
3.27/1.43	
3.27/1.43	(5) CSRRRRProof (EQUIVALENT)
3.27/1.43	The following CSR is given: Context-sensitive rewrite system:
3.27/1.43	The TRS R consists of the following rules:
3.27/1.43	
3.27/1.43	   length(nil) -> 0
3.27/1.43	
3.27/1.43	The replacement map contains the following entries:
3.27/1.43	
3.27/1.43	length: empty set
3.27/1.43	nil: empty set
3.27/1.43	0: empty set
3.27/1.43	Used ordering:
3.27/1.43	Polynomial interpretation [POLO]:
3.27/1.43	
3.27/1.43	   POL(0) = 0
3.27/1.43	   POL(length(x_1)) = x_1
3.27/1.43	   POL(nil) = 1
3.27/1.43	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.27/1.43	
3.27/1.43	   length(nil) -> 0
3.27/1.43	
3.27/1.43	
3.27/1.43	
3.27/1.43	
3.27/1.43	----------------------------------------
3.27/1.43	
3.27/1.43	(6)
3.27/1.43	Obligation:
3.27/1.43	Context-sensitive rewrite system:
3.27/1.43	R is empty.
3.27/1.43	
3.27/1.43	----------------------------------------
3.27/1.43	
3.27/1.43	(7) RisEmptyProof (EQUIVALENT)
3.27/1.43	The CSR R is empty. Hence, termination is trivially proven.
3.27/1.43	----------------------------------------
3.27/1.43	
3.27/1.43	(8)
3.27/1.43	YES
3.33/1.45	EOF