3.48/1.61	NO
3.48/1.63	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.48/1.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.48/1.63	
3.48/1.63	
3.48/1.63	Termination of the given CSR could be disproven:
3.48/1.63	
3.48/1.63	(0) CSR
3.48/1.63	(1) CSRRRRProof [EQUIVALENT, 39 ms]
3.48/1.63	(2) CSR
3.48/1.63	(3) CSRRRRProof [EQUIVALENT, 0 ms]
3.48/1.63	(4) CSR
3.48/1.63	(5) ContextSensitiveLoopProof [COMPLETE, 39 ms]
3.48/1.63	(6) NO
3.48/1.63	
3.48/1.63	
3.48/1.63	----------------------------------------
3.48/1.63	
3.48/1.63	(0)
3.48/1.63	Obligation:
3.48/1.63	Context-sensitive rewrite system:
3.48/1.63	The TRS R consists of the following rules:
3.48/1.63	
3.48/1.63	   zeros -> cons(0, zeros)
3.48/1.63	   U11(tt, L) -> U12(tt, L)
3.48/1.63	   U12(tt, L) -> s(length(L))
3.48/1.63	   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
3.48/1.63	   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
3.48/1.63	   U23(tt, IL, M, N) -> cons(N, take(M, IL))
3.48/1.63	   length(nil) -> 0
3.48/1.63	   length(cons(N, L)) -> U11(tt, L)
3.48/1.63	   take(0, IL) -> nil
3.48/1.63	   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)
3.48/1.63	
3.48/1.63	The replacement map contains the following entries:
3.48/1.63	
3.48/1.63	zeros: empty set
3.48/1.63	cons: {1}
3.48/1.63	0: empty set
3.48/1.63	U11: {1}
3.48/1.63	tt: empty set
3.48/1.63	U12: {1}
3.48/1.63	s: {1}
3.48/1.63	length: {1}
3.48/1.63	U21: {1}
3.48/1.63	U22: {1}
3.48/1.63	U23: {1}
3.48/1.63	take: {1, 2}
3.48/1.63	nil: empty set
3.48/1.63	
3.48/1.63	----------------------------------------
3.48/1.63	
3.48/1.63	(1) CSRRRRProof (EQUIVALENT)
3.48/1.63	The following CSR is given: Context-sensitive rewrite system:
3.48/1.63	The TRS R consists of the following rules:
3.48/1.63	
3.48/1.63	   zeros -> cons(0, zeros)
3.48/1.63	   U11(tt, L) -> U12(tt, L)
3.48/1.63	   U12(tt, L) -> s(length(L))
3.48/1.63	   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
3.48/1.63	   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
3.48/1.63	   U23(tt, IL, M, N) -> cons(N, take(M, IL))
3.48/1.63	   length(nil) -> 0
3.48/1.63	   length(cons(N, L)) -> U11(tt, L)
3.48/1.63	   take(0, IL) -> nil
3.48/1.63	   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)
3.48/1.63	
3.48/1.63	The replacement map contains the following entries:
3.48/1.63	
3.48/1.63	zeros: empty set
3.48/1.63	cons: {1}
3.48/1.63	0: empty set
3.48/1.63	U11: {1}
3.48/1.63	tt: empty set
3.48/1.63	U12: {1}
3.48/1.63	s: {1}
3.48/1.63	length: {1}
3.48/1.63	U21: {1}
3.48/1.63	U22: {1}
3.48/1.63	U23: {1}
3.48/1.63	take: {1, 2}
3.48/1.63	nil: empty set
3.48/1.63	Used ordering:
3.48/1.63	Polynomial interpretation [POLO]:
3.48/1.63	
3.48/1.63	   POL(0) = 0
3.48/1.63	   POL(U11(x_1, x_2)) = x_1 + x_2
3.48/1.63	   POL(U12(x_1, x_2)) = x_1 + x_2
3.48/1.63	   POL(U21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
3.48/1.63	   POL(U22(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
3.48/1.63	   POL(U23(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
3.48/1.63	   POL(cons(x_1, x_2)) = x_1 + x_2
3.48/1.63	   POL(length(x_1)) = 1 + x_1
3.48/1.63	   POL(nil) = 1
3.48/1.63	   POL(s(x_1)) = x_1
3.48/1.63	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
3.48/1.63	   POL(tt) = 1
3.48/1.63	   POL(zeros) = 0
3.48/1.63	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.48/1.63	
3.48/1.63	   length(nil) -> 0
3.48/1.63	
3.48/1.63	
3.48/1.63	
3.48/1.63	
3.48/1.63	----------------------------------------
3.48/1.63	
3.48/1.63	(2)
3.48/1.63	Obligation:
3.48/1.63	Context-sensitive rewrite system:
3.48/1.63	The TRS R consists of the following rules:
3.48/1.63	
3.48/1.63	   zeros -> cons(0, zeros)
3.48/1.63	   U11(tt, L) -> U12(tt, L)
3.48/1.63	   U12(tt, L) -> s(length(L))
3.48/1.63	   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
3.48/1.63	   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
3.48/1.63	   U23(tt, IL, M, N) -> cons(N, take(M, IL))
3.48/1.63	   length(cons(N, L)) -> U11(tt, L)
3.48/1.63	   take(0, IL) -> nil
3.48/1.63	   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)
3.48/1.63	
3.48/1.63	The replacement map contains the following entries:
3.48/1.63	
3.48/1.63	zeros: empty set
3.48/1.63	cons: {1}
3.48/1.63	0: empty set
3.48/1.63	U11: {1}
3.48/1.63	tt: empty set
3.48/1.63	U12: {1}
3.48/1.63	s: {1}
3.48/1.63	length: {1}
3.48/1.63	U21: {1}
3.48/1.63	U22: {1}
3.48/1.63	U23: {1}
3.48/1.63	take: {1, 2}
3.48/1.63	nil: empty set
3.48/1.63	
3.48/1.63	----------------------------------------
3.48/1.63	
3.48/1.63	(3) CSRRRRProof (EQUIVALENT)
3.48/1.63	The following CSR is given: Context-sensitive rewrite system:
3.48/1.63	The TRS R consists of the following rules:
3.48/1.63	
3.48/1.63	   zeros -> cons(0, zeros)
3.48/1.63	   U11(tt, L) -> U12(tt, L)
3.48/1.63	   U12(tt, L) -> s(length(L))
3.48/1.63	   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
3.48/1.63	   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
3.48/1.63	   U23(tt, IL, M, N) -> cons(N, take(M, IL))
3.48/1.63	   length(cons(N, L)) -> U11(tt, L)
3.48/1.63	   take(0, IL) -> nil
3.48/1.63	   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)
3.48/1.63	
3.48/1.63	The replacement map contains the following entries:
3.48/1.63	
3.48/1.63	zeros: empty set
3.48/1.63	cons: {1}
3.48/1.63	0: empty set
3.48/1.63	U11: {1}
3.48/1.63	tt: empty set
3.48/1.63	U12: {1}
3.48/1.63	s: {1}
3.48/1.63	length: {1}
3.48/1.63	U21: {1}
3.48/1.63	U22: {1}
3.48/1.63	U23: {1}
3.48/1.63	take: {1, 2}
3.48/1.63	nil: empty set
3.48/1.63	Used ordering:
3.48/1.63	Polynomial interpretation [POLO]:
3.48/1.63	
3.48/1.63	   POL(0) = 0
3.48/1.63	   POL(U11(x_1, x_2)) = x_1 + x_2
3.48/1.63	   POL(U12(x_1, x_2)) = x_1 + x_2
3.48/1.63	   POL(U21(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
3.48/1.63	   POL(U22(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
3.48/1.63	   POL(U23(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
3.48/1.63	   POL(cons(x_1, x_2)) = x_1 + x_2
3.48/1.63	   POL(length(x_1)) = x_1
3.48/1.63	   POL(nil) = 0
3.48/1.63	   POL(s(x_1)) = x_1
3.48/1.63	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
3.48/1.63	   POL(tt) = 0
3.48/1.63	   POL(zeros) = 0
3.48/1.63	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.48/1.63	
3.48/1.63	   take(0, IL) -> nil
3.48/1.63	
3.48/1.63	
3.48/1.63	
3.48/1.63	
3.48/1.63	----------------------------------------
3.48/1.63	
3.48/1.63	(4)
3.48/1.63	Obligation:
3.48/1.63	Context-sensitive rewrite system:
3.48/1.63	The TRS R consists of the following rules:
3.48/1.63	
3.48/1.63	   zeros -> cons(0, zeros)
3.48/1.63	   U11(tt, L) -> U12(tt, L)
3.48/1.63	   U12(tt, L) -> s(length(L))
3.48/1.63	   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
3.48/1.63	   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
3.48/1.63	   U23(tt, IL, M, N) -> cons(N, take(M, IL))
3.48/1.63	   length(cons(N, L)) -> U11(tt, L)
3.48/1.63	   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)
3.48/1.63	
3.48/1.63	The replacement map contains the following entries:
3.48/1.63	
3.48/1.63	zeros: empty set
3.48/1.63	cons: {1}
3.48/1.63	0: empty set
3.48/1.63	U11: {1}
3.48/1.63	tt: empty set
3.48/1.63	U12: {1}
3.48/1.63	s: {1}
3.48/1.63	length: {1}
3.48/1.63	U21: {1}
3.48/1.63	U22: {1}
3.48/1.63	U23: {1}
3.48/1.63	take: {1, 2}
3.48/1.63	
3.48/1.63	----------------------------------------
3.48/1.63	
3.48/1.63	(5) ContextSensitiveLoopProof (COMPLETE)
3.48/1.63	
3.48/1.63	   zeros -> cons(0, zeros)
3.48/1.63	   U11(tt, L) -> U12(tt, L)
3.48/1.63	   U12(tt, L) -> s(length(L))
3.48/1.63	   U21(tt, IL, M, N) -> U22(tt, IL, M, N)
3.48/1.63	   U22(tt, IL, M, N) -> U23(tt, IL, M, N)
3.48/1.63	   U23(tt, IL, M, N) -> cons(N, take(M, IL))
3.48/1.63	   length(cons(N, L)) -> U11(tt, L)
3.48/1.63	   take(s(M), cons(N, IL)) -> U21(tt, IL, M, N)
3.48/1.63	
3.48/1.63	---------- Loop: ----------
3.48/1.63	
3.48/1.63	U12(tt, zeros) -> s(length(zeros)) with rule U12(tt, L) -> s(length(L)) at position [] and matcher [L / zeros]
3.48/1.63	
3.48/1.63	s(length(zeros)) -> s(length(cons(0, zeros))) with rule zeros -> cons(0, zeros) at position [0,0] and matcher [ ]
3.48/1.63	
3.48/1.63	s(length(cons(0, zeros))) -> s(U11(tt, zeros)) with rule length(cons(N, L)) -> U11(tt, L) at position [0] and matcher [N / 0, L / zeros]
3.48/1.63	
3.48/1.63	s(U11(tt, zeros)) -> s(U12(tt, zeros)) with rule U11(tt, L) -> U12(tt, L) at position [0] and matcher [L / zeros]
3.48/1.63	
3.48/1.63	Now an instance of the first term with Matcher [ ] occurs in the last term at position [0].
3.48/1.63	
3.48/1.63	Context: s([])
3.48/1.63	
3.48/1.63	We used [[THIEMANN_LOOPS_UNDER_STRATEGIES], Theorem 1] to show that this loop is an context-sensitive loop.
3.48/1.63	----------------------------------------
3.48/1.63	
3.48/1.63	(6)
3.48/1.63	NO
3.48/1.66	EOF