3.62/1.53	NO
3.62/1.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.62/1.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.62/1.53	
3.62/1.53	
3.62/1.53	Termination of the given CSR could be disproven:
3.62/1.53	
3.62/1.53	(0) CSR
3.62/1.53	(1) CSRRRRProof [EQUIVALENT, 44 ms]
3.62/1.53	(2) CSR
3.62/1.53	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.62/1.53	(4) CSR
3.62/1.53	(5) CSRRRRProof [EQUIVALENT, 6 ms]
3.62/1.53	(6) CSR
3.62/1.53	(7) ContextSensitiveLoopProof [COMPLETE, 1 ms]
3.62/1.53	(8) NO
3.62/1.53	
3.62/1.53	
3.62/1.53	----------------------------------------
3.62/1.53	
3.62/1.53	(0)
3.62/1.53	Obligation:
3.62/1.53	Context-sensitive rewrite system:
3.62/1.53	The TRS R consists of the following rules:
3.62/1.53	
3.62/1.53	   zeros -> cons(0, zeros)
3.62/1.53	   and(tt, X) -> X
3.62/1.53	   length(nil) -> 0
3.62/1.53	   length(cons(N, L)) -> s(length(L))
3.62/1.53	   take(0, IL) -> nil
3.62/1.53	   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))
3.62/1.53	
3.62/1.53	The replacement map contains the following entries:
3.62/1.53	
3.62/1.53	zeros: empty set
3.62/1.53	cons: {1}
3.62/1.53	0: empty set
3.62/1.53	and: {1}
3.62/1.53	tt: empty set
3.62/1.53	length: {1}
3.62/1.53	nil: empty set
3.62/1.53	s: {1}
3.62/1.53	take: {1, 2}
3.62/1.53	
3.62/1.53	----------------------------------------
3.62/1.53	
3.62/1.53	(1) CSRRRRProof (EQUIVALENT)
3.62/1.53	The following CSR is given: Context-sensitive rewrite system:
3.62/1.53	The TRS R consists of the following rules:
3.62/1.53	
3.62/1.53	   zeros -> cons(0, zeros)
3.62/1.53	   and(tt, X) -> X
3.62/1.53	   length(nil) -> 0
3.62/1.53	   length(cons(N, L)) -> s(length(L))
3.62/1.53	   take(0, IL) -> nil
3.62/1.53	   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))
3.62/1.53	
3.62/1.53	The replacement map contains the following entries:
3.62/1.53	
3.62/1.53	zeros: empty set
3.62/1.53	cons: {1}
3.62/1.53	0: empty set
3.62/1.53	and: {1}
3.62/1.53	tt: empty set
3.62/1.53	length: {1}
3.62/1.53	nil: empty set
3.62/1.53	s: {1}
3.62/1.53	take: {1, 2}
3.62/1.53	Used ordering:
3.62/1.53	Polynomial interpretation [POLO]:
3.62/1.53	
3.62/1.53	   POL(0) = 0
3.62/1.53	   POL(and(x_1, x_2)) = 2*x_1 + x_2
3.62/1.53	   POL(cons(x_1, x_2)) = x_1 + 2*x_2
3.62/1.53	   POL(length(x_1)) = 2*x_1
3.62/1.53	   POL(nil) = 0
3.62/1.53	   POL(s(x_1)) = 2*x_1
3.62/1.53	   POL(take(x_1, x_2)) = x_1 + 2*x_2
3.62/1.53	   POL(tt) = 2
3.62/1.53	   POL(zeros) = 0
3.62/1.53	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.62/1.53	
3.62/1.53	   and(tt, X) -> X
3.62/1.53	
3.62/1.53	
3.62/1.53	
3.62/1.53	
3.62/1.53	----------------------------------------
3.62/1.53	
3.62/1.53	(2)
3.62/1.53	Obligation:
3.62/1.53	Context-sensitive rewrite system:
3.62/1.53	The TRS R consists of the following rules:
3.62/1.53	
3.62/1.53	   zeros -> cons(0, zeros)
3.62/1.53	   length(nil) -> 0
3.62/1.53	   length(cons(N, L)) -> s(length(L))
3.62/1.53	   take(0, IL) -> nil
3.62/1.53	   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))
3.62/1.53	
3.62/1.53	The replacement map contains the following entries:
3.62/1.53	
3.62/1.53	zeros: empty set
3.62/1.53	cons: {1}
3.62/1.53	0: empty set
3.62/1.53	length: {1}
3.62/1.53	nil: empty set
3.62/1.53	s: {1}
3.62/1.53	take: {1, 2}
3.62/1.53	
3.62/1.53	----------------------------------------
3.62/1.53	
3.62/1.53	(3) CSRRRRProof (EQUIVALENT)
3.62/1.53	The following CSR is given: Context-sensitive rewrite system:
3.62/1.53	The TRS R consists of the following rules:
3.62/1.53	
3.62/1.53	   zeros -> cons(0, zeros)
3.62/1.53	   length(nil) -> 0
3.62/1.53	   length(cons(N, L)) -> s(length(L))
3.62/1.53	   take(0, IL) -> nil
3.62/1.53	   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))
3.62/1.53	
3.62/1.53	The replacement map contains the following entries:
3.62/1.53	
3.62/1.53	zeros: empty set
3.62/1.53	cons: {1}
3.62/1.53	0: empty set
3.62/1.53	length: {1}
3.62/1.53	nil: empty set
3.62/1.53	s: {1}
3.62/1.53	take: {1, 2}
3.62/1.53	Used ordering:
3.62/1.53	Polynomial interpretation [POLO]:
3.62/1.53	
3.62/1.53	   POL(0) = 0
3.62/1.53	   POL(cons(x_1, x_2)) = x_1 + x_2
3.62/1.53	   POL(length(x_1)) = x_1
3.62/1.53	   POL(nil) = 1
3.62/1.53	   POL(s(x_1)) = x_1
3.62/1.53	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
3.62/1.53	   POL(zeros) = 0
3.62/1.53	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.62/1.53	
3.62/1.53	   length(nil) -> 0
3.62/1.53	
3.62/1.53	
3.62/1.53	
3.62/1.53	
3.62/1.53	----------------------------------------
3.62/1.53	
3.62/1.53	(4)
3.62/1.53	Obligation:
3.62/1.53	Context-sensitive rewrite system:
3.62/1.53	The TRS R consists of the following rules:
3.62/1.53	
3.62/1.53	   zeros -> cons(0, zeros)
3.62/1.53	   length(cons(N, L)) -> s(length(L))
3.62/1.53	   take(0, IL) -> nil
3.62/1.53	   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))
3.62/1.53	
3.62/1.53	The replacement map contains the following entries:
3.62/1.53	
3.62/1.53	zeros: empty set
3.62/1.53	cons: {1}
3.62/1.53	0: empty set
3.62/1.53	length: {1}
3.62/1.53	nil: empty set
3.62/1.53	s: {1}
3.62/1.53	take: {1, 2}
3.62/1.53	
3.62/1.53	----------------------------------------
3.62/1.53	
3.62/1.53	(5) CSRRRRProof (EQUIVALENT)
3.62/1.53	The following CSR is given: Context-sensitive rewrite system:
3.62/1.53	The TRS R consists of the following rules:
3.62/1.53	
3.62/1.53	   zeros -> cons(0, zeros)
3.62/1.53	   length(cons(N, L)) -> s(length(L))
3.62/1.53	   take(0, IL) -> nil
3.62/1.53	   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))
3.62/1.53	
3.62/1.53	The replacement map contains the following entries:
3.62/1.53	
3.62/1.53	zeros: empty set
3.62/1.53	cons: {1}
3.62/1.53	0: empty set
3.62/1.53	length: {1}
3.62/1.53	nil: empty set
3.62/1.53	s: {1}
3.62/1.53	take: {1, 2}
3.62/1.53	Used ordering:
3.62/1.53	Polynomial interpretation [POLO]:
3.62/1.53	
3.62/1.53	   POL(0) = 0
3.62/1.53	   POL(cons(x_1, x_2)) = x_1 + x_2
3.62/1.53	   POL(length(x_1)) = x_1
3.62/1.53	   POL(nil) = 0
3.62/1.53	   POL(s(x_1)) = x_1
3.62/1.53	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
3.62/1.53	   POL(zeros) = 0
3.62/1.53	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.62/1.53	
3.62/1.53	   take(0, IL) -> nil
3.62/1.53	
3.62/1.53	
3.62/1.53	
3.62/1.53	
3.62/1.53	----------------------------------------
3.62/1.53	
3.62/1.53	(6)
3.62/1.53	Obligation:
3.62/1.53	Context-sensitive rewrite system:
3.62/1.53	The TRS R consists of the following rules:
3.62/1.53	
3.62/1.53	   zeros -> cons(0, zeros)
3.62/1.53	   length(cons(N, L)) -> s(length(L))
3.62/1.53	   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))
3.62/1.53	
3.62/1.53	The replacement map contains the following entries:
3.62/1.53	
3.62/1.53	zeros: empty set
3.62/1.53	cons: {1}
3.62/1.53	0: empty set
3.62/1.53	length: {1}
3.62/1.53	s: {1}
3.62/1.53	take: {1, 2}
3.62/1.53	
3.62/1.53	----------------------------------------
3.62/1.53	
3.62/1.53	(7) ContextSensitiveLoopProof (COMPLETE)
3.62/1.53	
3.62/1.53	   zeros -> cons(0, zeros)
3.62/1.53	   length(cons(N, L)) -> s(length(L))
3.62/1.53	   take(s(M), cons(N, IL)) -> cons(N, take(M, IL))
3.62/1.53	
3.62/1.53	---------- Loop: ----------
3.62/1.53	
3.62/1.53	length(zeros) -> length(cons(0, zeros)) with rule zeros -> cons(0, zeros) at position [0] and matcher [ ]
3.62/1.53	
3.62/1.53	length(cons(0, zeros)) -> s(length(zeros)) with rule length(cons(N, L)) -> s(length(L)) at position [] and matcher [N / 0, L / zeros]
3.62/1.53	
3.62/1.53	Now an instance of the first term with Matcher [ ] occurs in the last term at position [0].
3.62/1.53	
3.62/1.53	Context: s([])
3.62/1.53	
3.62/1.53	We used [[THIEMANN_LOOPS_UNDER_STRATEGIES], Theorem 1] to show that this loop is an context-sensitive loop.
3.62/1.53	----------------------------------------
3.62/1.53	
3.62/1.53	(8)
3.62/1.53	NO
3.68/1.56	EOF