3.15/1.45	YES
3.15/1.46	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.15/1.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.15/1.46	
3.15/1.46	
3.15/1.46	Termination of the given CSR could be proven:
3.15/1.46	
3.15/1.46	(0) CSR
3.15/1.46	(1) CSRRRRProof [EQUIVALENT, 57 ms]
3.15/1.46	(2) CSR
3.15/1.46	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.15/1.46	(4) CSR
3.15/1.46	(5) CSRRRRProof [EQUIVALENT, 2 ms]
3.15/1.46	(6) CSR
3.15/1.46	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.15/1.46	(8) YES
3.15/1.46	
3.15/1.46	
3.15/1.46	----------------------------------------
3.15/1.46	
3.15/1.46	(0)
3.15/1.46	Obligation:
3.15/1.46	Context-sensitive rewrite system:
3.15/1.46	The TRS R consists of the following rules:
3.15/1.46	
3.15/1.46	   eq(0, 0) -> true
3.15/1.46	   eq(s(X), s(Y)) -> eq(X, Y)
3.15/1.46	   eq(X, Y) -> false
3.15/1.46	   inf(X) -> cons(X, inf(s(X)))
3.15/1.46	   take(0, X) -> nil
3.15/1.46	   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.15/1.46	   length(nil) -> 0
3.15/1.46	   length(cons(X, L)) -> s(length(L))
3.15/1.46	
3.15/1.46	The replacement map contains the following entries:
3.15/1.46	
3.15/1.46	eq: empty set
3.15/1.46	0: empty set
3.15/1.46	true: empty set
3.15/1.46	s: empty set
3.15/1.46	false: empty set
3.15/1.46	inf: {1}
3.15/1.46	cons: empty set
3.15/1.46	take: {1, 2}
3.15/1.46	nil: empty set
3.15/1.46	length: {1}
3.15/1.46	
3.15/1.46	----------------------------------------
3.15/1.46	
3.15/1.46	(1) CSRRRRProof (EQUIVALENT)
3.15/1.46	The following CSR is given: Context-sensitive rewrite system:
3.15/1.46	The TRS R consists of the following rules:
3.15/1.46	
3.15/1.46	   eq(0, 0) -> true
3.15/1.46	   eq(s(X), s(Y)) -> eq(X, Y)
3.15/1.46	   eq(X, Y) -> false
3.15/1.46	   inf(X) -> cons(X, inf(s(X)))
3.15/1.46	   take(0, X) -> nil
3.15/1.46	   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.15/1.46	   length(nil) -> 0
3.15/1.46	   length(cons(X, L)) -> s(length(L))
3.15/1.46	
3.15/1.46	The replacement map contains the following entries:
3.15/1.46	
3.15/1.46	eq: empty set
3.15/1.46	0: empty set
3.15/1.46	true: empty set
3.15/1.46	s: empty set
3.15/1.46	false: empty set
3.15/1.46	inf: {1}
3.15/1.46	cons: empty set
3.15/1.46	take: {1, 2}
3.15/1.46	nil: empty set
3.15/1.46	length: {1}
3.15/1.46	Used ordering:
3.15/1.46	Polynomial interpretation [POLO]:
3.15/1.46	
3.15/1.46	   POL(0) = 1
3.15/1.46	   POL(cons(x_1, x_2)) = 1 + x_1
3.15/1.46	   POL(eq(x_1, x_2)) = 0
3.15/1.46	   POL(false) = 0
3.15/1.46	   POL(inf(x_1)) = 2 + x_1
3.15/1.46	   POL(length(x_1)) = 2 + x_1
3.15/1.46	   POL(nil) = 1
3.15/1.46	   POL(s(x_1)) = 0
3.15/1.46	   POL(take(x_1, x_2)) = 2*x_1 + 2*x_2
3.15/1.46	   POL(true) = 0
3.15/1.46	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.15/1.46	
3.15/1.46	   inf(X) -> cons(X, inf(s(X)))
3.15/1.46	   take(0, X) -> nil
3.15/1.46	   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.15/1.46	   length(nil) -> 0
3.15/1.46	   length(cons(X, L)) -> s(length(L))
3.15/1.46	
3.15/1.46	
3.15/1.46	
3.15/1.46	
3.15/1.46	----------------------------------------
3.15/1.46	
3.15/1.46	(2)
3.15/1.46	Obligation:
3.15/1.46	Context-sensitive rewrite system:
3.15/1.46	The TRS R consists of the following rules:
3.15/1.46	
3.15/1.46	   eq(0, 0) -> true
3.15/1.46	   eq(s(X), s(Y)) -> eq(X, Y)
3.15/1.46	   eq(X, Y) -> false
3.15/1.46	
3.15/1.46	The replacement map contains the following entries:
3.15/1.46	
3.15/1.46	eq: empty set
3.15/1.46	0: empty set
3.15/1.46	true: empty set
3.15/1.46	s: empty set
3.15/1.46	false: empty set
3.15/1.46	
3.15/1.46	----------------------------------------
3.15/1.46	
3.15/1.46	(3) CSRRRRProof (EQUIVALENT)
3.15/1.46	The following CSR is given: Context-sensitive rewrite system:
3.15/1.46	The TRS R consists of the following rules:
3.15/1.46	
3.15/1.46	   eq(0, 0) -> true
3.15/1.46	   eq(s(X), s(Y)) -> eq(X, Y)
3.15/1.46	   eq(X, Y) -> false
3.15/1.46	
3.15/1.46	The replacement map contains the following entries:
3.15/1.46	
3.15/1.46	eq: empty set
3.15/1.46	0: empty set
3.15/1.46	true: empty set
3.15/1.46	s: empty set
3.15/1.46	false: empty set
3.15/1.46	Used ordering:
3.15/1.46	Polynomial interpretation [POLO]:
3.15/1.46	
3.15/1.46	   POL(0) = 1
3.15/1.46	   POL(eq(x_1, x_2)) = x_1 + x_2
3.15/1.46	   POL(false) = 0
3.15/1.46	   POL(s(x_1)) = 1 + x_1
3.15/1.46	   POL(true) = 0
3.15/1.46	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.15/1.46	
3.15/1.46	   eq(0, 0) -> true
3.15/1.46	   eq(s(X), s(Y)) -> eq(X, Y)
3.15/1.46	
3.15/1.46	
3.15/1.46	
3.15/1.46	
3.15/1.46	----------------------------------------
3.15/1.46	
3.15/1.46	(4)
3.15/1.46	Obligation:
3.15/1.46	Context-sensitive rewrite system:
3.15/1.46	The TRS R consists of the following rules:
3.15/1.46	
3.15/1.46	   eq(X, Y) -> false
3.15/1.46	
3.15/1.46	The replacement map contains the following entries:
3.15/1.46	
3.15/1.46	eq: empty set
3.15/1.46	false: empty set
3.15/1.46	
3.15/1.46	----------------------------------------
3.15/1.46	
3.15/1.46	(5) CSRRRRProof (EQUIVALENT)
3.15/1.46	The following CSR is given: Context-sensitive rewrite system:
3.15/1.46	The TRS R consists of the following rules:
3.15/1.46	
3.15/1.46	   eq(X, Y) -> false
3.15/1.46	
3.15/1.46	The replacement map contains the following entries:
3.15/1.46	
3.15/1.46	eq: empty set
3.15/1.46	false: empty set
3.15/1.46	Used ordering:
3.15/1.46	Polynomial interpretation [POLO]:
3.15/1.46	
3.15/1.46	   POL(eq(x_1, x_2)) = 1
3.15/1.46	   POL(false) = 0
3.15/1.46	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.15/1.46	
3.15/1.46	   eq(X, Y) -> false
3.15/1.46	
3.15/1.46	
3.15/1.46	
3.15/1.46	
3.15/1.46	----------------------------------------
3.15/1.46	
3.15/1.46	(6)
3.15/1.46	Obligation:
3.15/1.46	Context-sensitive rewrite system:
3.15/1.46	R is empty.
3.15/1.46	
3.15/1.46	----------------------------------------
3.15/1.46	
3.15/1.46	(7) RisEmptyProof (EQUIVALENT)
3.15/1.46	The CSR R is empty. Hence, termination is trivially proven.
3.15/1.46	----------------------------------------
3.15/1.46	
3.15/1.46	(8)
3.15/1.46	YES
3.15/1.49	EOF