5.46/2.19	YES
5.46/2.20	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.46/2.20	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.46/2.20	
5.46/2.20	
5.46/2.20	Termination of the given CSR could be proven:
5.46/2.20	
5.46/2.20	(0) CSR
5.46/2.20	(1) CSRRRRProof [EQUIVALENT, 57 ms]
5.46/2.20	(2) CSR
5.46/2.20	(3) CSRRRRProof [EQUIVALENT, 1 ms]
5.46/2.20	(4) CSR
5.46/2.20	(5) CSDependencyPairsProof [EQUIVALENT, 0 ms]
5.46/2.20	(6) QCSDP
5.46/2.20	(7) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
5.46/2.20	(8) AND
5.46/2.20	    (9) QCSDP
5.46/2.20	        (10) QCSUsableRulesProof [EQUIVALENT, 0 ms]
5.46/2.20	        (11) QCSDP
5.46/2.20	        (12) QCSDPMuMonotonicPoloProof [EQUIVALENT, 0 ms]
5.46/2.20	        (13) QCSDP
5.46/2.20	        (14) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
5.46/2.20	        (15) AND
5.46/2.20	            (16) QCSDP
5.46/2.20	                (17) QCSDPMuMonotonicPoloProof [EQUIVALENT, 0 ms]
5.46/2.20	                (18) QCSDP
5.46/2.20	                (19) PIsEmptyProof [EQUIVALENT, 0 ms]
5.46/2.20	                (20) YES
5.46/2.20	            (21) QCSDP
5.46/2.20	                (22) QCSDPMuMonotonicPoloProof [EQUIVALENT, 0 ms]
5.46/2.20	                (23) QCSDP
5.46/2.20	                (24) PIsEmptyProof [EQUIVALENT, 0 ms]
5.46/2.20	                (25) YES
5.46/2.20	    (26) QCSDP
5.46/2.20	        (27) QCSDPSubtermProof [EQUIVALENT, 0 ms]
5.46/2.20	        (28) QCSDP
5.46/2.20	        (29) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
5.46/2.20	        (30) TRUE
5.46/2.20	    (31) QCSDP
5.46/2.20	        (32) QCSDPReductionPairProof [EQUIVALENT, 75 ms]
5.46/2.20	        (33) QCSDP
5.46/2.20	        (34) QCSDependencyGraphProof [EQUIVALENT, 0 ms]
5.46/2.20	        (35) TRUE
5.46/2.20	
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(0)
5.46/2.20	Obligation:
5.46/2.20	Context-sensitive rewrite system:
5.46/2.20	The TRS R consists of the following rules:
5.46/2.20	
5.46/2.20	   and(tt, T) -> T
5.46/2.20	   isNatIList(IL) -> isNatList(IL)
5.46/2.20	   isNat(0) -> tt
5.46/2.20	   isNat(s(N)) -> isNat(N)
5.46/2.20	   isNat(length(L)) -> isNatList(L)
5.46/2.20	   isNatIList(zeros) -> tt
5.46/2.20	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   isNatList(nil) -> tt
5.46/2.20	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.20	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   zeros -> cons(0, zeros)
5.46/2.20	   take(0, IL) -> uTake1(isNatIList(IL))
5.46/2.20	   uTake1(tt) -> nil
5.46/2.20	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.20	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.20	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.20	   uLength(tt, L) -> s(length(L))
5.46/2.20	
5.46/2.20	The replacement map contains the following entries:
5.46/2.20	
5.46/2.20	and: {1, 2}
5.46/2.20	tt: empty set
5.46/2.20	isNatIList: empty set
5.46/2.20	isNatList: empty set
5.46/2.20	isNat: empty set
5.46/2.20	0: empty set
5.46/2.20	s: {1}
5.46/2.20	length: {1}
5.46/2.20	zeros: empty set
5.46/2.20	cons: {1}
5.46/2.20	nil: empty set
5.46/2.20	take: {1, 2}
5.46/2.20	uTake1: {1}
5.46/2.20	uTake2: {1}
5.46/2.20	uLength: {1}
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(1) CSRRRRProof (EQUIVALENT)
5.46/2.20	The following CSR is given: Context-sensitive rewrite system:
5.46/2.20	The TRS R consists of the following rules:
5.46/2.20	
5.46/2.20	   and(tt, T) -> T
5.46/2.20	   isNatIList(IL) -> isNatList(IL)
5.46/2.20	   isNat(0) -> tt
5.46/2.20	   isNat(s(N)) -> isNat(N)
5.46/2.20	   isNat(length(L)) -> isNatList(L)
5.46/2.20	   isNatIList(zeros) -> tt
5.46/2.20	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   isNatList(nil) -> tt
5.46/2.20	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.20	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   zeros -> cons(0, zeros)
5.46/2.20	   take(0, IL) -> uTake1(isNatIList(IL))
5.46/2.20	   uTake1(tt) -> nil
5.46/2.20	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.20	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.20	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.20	   uLength(tt, L) -> s(length(L))
5.46/2.20	
5.46/2.20	The replacement map contains the following entries:
5.46/2.20	
5.46/2.20	and: {1, 2}
5.46/2.20	tt: empty set
5.46/2.20	isNatIList: empty set
5.46/2.20	isNatList: empty set
5.46/2.20	isNat: empty set
5.46/2.20	0: empty set
5.46/2.20	s: {1}
5.46/2.20	length: {1}
5.46/2.20	zeros: empty set
5.46/2.20	cons: {1}
5.46/2.20	nil: empty set
5.46/2.20	take: {1, 2}
5.46/2.20	uTake1: {1}
5.46/2.20	uTake2: {1}
5.46/2.20	uLength: {1}
5.46/2.20	Used ordering:
5.46/2.20	Polynomial interpretation [POLO]:
5.46/2.20	
5.46/2.20	   POL(0) = 0
5.46/2.20	   POL(and(x_1, x_2)) = x_1 + x_2
5.46/2.20	   POL(cons(x_1, x_2)) = x_1 + x_2
5.46/2.20	   POL(isNat(x_1)) = 0
5.46/2.20	   POL(isNatIList(x_1)) = 0
5.46/2.20	   POL(isNatList(x_1)) = 0
5.46/2.20	   POL(length(x_1)) = 1 + x_1
5.46/2.20	   POL(nil) = 0
5.46/2.20	   POL(s(x_1)) = x_1
5.46/2.20	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
5.46/2.20	   POL(tt) = 0
5.46/2.20	   POL(uLength(x_1, x_2)) = 1 + x_1 + x_2
5.46/2.20	   POL(uTake1(x_1)) = 1 + x_1
5.46/2.20	   POL(uTake2(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
5.46/2.20	   POL(zeros) = 0
5.46/2.20	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.46/2.20	
5.46/2.20	   uTake1(tt) -> nil
5.46/2.20	
5.46/2.20	
5.46/2.20	
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(2)
5.46/2.20	Obligation:
5.46/2.20	Context-sensitive rewrite system:
5.46/2.20	The TRS R consists of the following rules:
5.46/2.20	
5.46/2.20	   and(tt, T) -> T
5.46/2.20	   isNatIList(IL) -> isNatList(IL)
5.46/2.20	   isNat(0) -> tt
5.46/2.20	   isNat(s(N)) -> isNat(N)
5.46/2.20	   isNat(length(L)) -> isNatList(L)
5.46/2.20	   isNatIList(zeros) -> tt
5.46/2.20	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   isNatList(nil) -> tt
5.46/2.20	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.20	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   zeros -> cons(0, zeros)
5.46/2.20	   take(0, IL) -> uTake1(isNatIList(IL))
5.46/2.20	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.20	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.20	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.20	   uLength(tt, L) -> s(length(L))
5.46/2.20	
5.46/2.20	The replacement map contains the following entries:
5.46/2.20	
5.46/2.20	and: {1, 2}
5.46/2.20	tt: empty set
5.46/2.20	isNatIList: empty set
5.46/2.20	isNatList: empty set
5.46/2.20	isNat: empty set
5.46/2.20	0: empty set
5.46/2.20	s: {1}
5.46/2.20	length: {1}
5.46/2.20	zeros: empty set
5.46/2.20	cons: {1}
5.46/2.20	nil: empty set
5.46/2.20	take: {1, 2}
5.46/2.20	uTake1: {1}
5.46/2.20	uTake2: {1}
5.46/2.20	uLength: {1}
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(3) CSRRRRProof (EQUIVALENT)
5.46/2.20	The following CSR is given: Context-sensitive rewrite system:
5.46/2.20	The TRS R consists of the following rules:
5.46/2.20	
5.46/2.20	   and(tt, T) -> T
5.46/2.20	   isNatIList(IL) -> isNatList(IL)
5.46/2.20	   isNat(0) -> tt
5.46/2.20	   isNat(s(N)) -> isNat(N)
5.46/2.20	   isNat(length(L)) -> isNatList(L)
5.46/2.20	   isNatIList(zeros) -> tt
5.46/2.20	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   isNatList(nil) -> tt
5.46/2.20	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.20	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   zeros -> cons(0, zeros)
5.46/2.20	   take(0, IL) -> uTake1(isNatIList(IL))
5.46/2.20	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.20	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.20	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.20	   uLength(tt, L) -> s(length(L))
5.46/2.20	
5.46/2.20	The replacement map contains the following entries:
5.46/2.20	
5.46/2.20	and: {1, 2}
5.46/2.20	tt: empty set
5.46/2.20	isNatIList: empty set
5.46/2.20	isNatList: empty set
5.46/2.20	isNat: empty set
5.46/2.20	0: empty set
5.46/2.20	s: {1}
5.46/2.20	length: {1}
5.46/2.20	zeros: empty set
5.46/2.20	cons: {1}
5.46/2.20	nil: empty set
5.46/2.20	take: {1, 2}
5.46/2.20	uTake1: {1}
5.46/2.20	uTake2: {1}
5.46/2.20	uLength: {1}
5.46/2.20	Used ordering:
5.46/2.20	Polynomial interpretation [POLO]:
5.46/2.20	
5.46/2.20	   POL(0) = 0
5.46/2.20	   POL(and(x_1, x_2)) = x_1 + x_2
5.46/2.20	   POL(cons(x_1, x_2)) = x_1 + x_2
5.46/2.20	   POL(isNat(x_1)) = 0
5.46/2.20	   POL(isNatIList(x_1)) = 0
5.46/2.20	   POL(isNatList(x_1)) = 0
5.46/2.20	   POL(length(x_1)) = 1 + x_1
5.46/2.20	   POL(nil) = 0
5.46/2.20	   POL(s(x_1)) = x_1
5.46/2.20	   POL(take(x_1, x_2)) = 1 + x_1 + x_2
5.46/2.20	   POL(tt) = 0
5.46/2.20	   POL(uLength(x_1, x_2)) = 1 + x_1 + x_2
5.46/2.20	   POL(uTake1(x_1)) = x_1
5.46/2.20	   POL(uTake2(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
5.46/2.20	   POL(zeros) = 0
5.46/2.20	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.46/2.20	
5.46/2.20	   take(0, IL) -> uTake1(isNatIList(IL))
5.46/2.20	
5.46/2.20	
5.46/2.20	
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(4)
5.46/2.20	Obligation:
5.46/2.20	Context-sensitive rewrite system:
5.46/2.20	The TRS R consists of the following rules:
5.46/2.20	
5.46/2.20	   and(tt, T) -> T
5.46/2.20	   isNatIList(IL) -> isNatList(IL)
5.46/2.20	   isNat(0) -> tt
5.46/2.20	   isNat(s(N)) -> isNat(N)
5.46/2.20	   isNat(length(L)) -> isNatList(L)
5.46/2.20	   isNatIList(zeros) -> tt
5.46/2.20	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   isNatList(nil) -> tt
5.46/2.20	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.20	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   zeros -> cons(0, zeros)
5.46/2.20	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.20	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.20	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.20	   uLength(tt, L) -> s(length(L))
5.46/2.20	
5.46/2.20	The replacement map contains the following entries:
5.46/2.20	
5.46/2.20	and: {1, 2}
5.46/2.20	tt: empty set
5.46/2.20	isNatIList: empty set
5.46/2.20	isNatList: empty set
5.46/2.20	isNat: empty set
5.46/2.20	0: empty set
5.46/2.20	s: {1}
5.46/2.20	length: {1}
5.46/2.20	zeros: empty set
5.46/2.20	cons: {1}
5.46/2.20	nil: empty set
5.46/2.20	take: {1, 2}
5.46/2.20	uTake2: {1}
5.46/2.20	uLength: {1}
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(5) CSDependencyPairsProof (EQUIVALENT)
5.46/2.20	Using Improved CS-DPs [LPAR08] we result in the following initial Q-CSDP problem.
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(6)
5.46/2.20	Obligation:
5.46/2.20	Q-restricted context-sensitive dependency pair problem:
5.46/2.20	The symbols in {and_2, s_1, length_1, take_2, AND_2, TAKE_2, LENGTH_1} are replacing on all positions.
5.46/2.20	For all symbols f in {cons_2, uTake2_4, uLength_2, UTAKE2_4, ULENGTH_2} we have mu(f) = {1}.
5.46/2.20	The symbols in {isNatIList_1, isNatList_1, isNat_1, ISNATLIST_1, ISNATILIST_1, ISNAT_1, U_1} are not replacing on any position.
5.46/2.20	
5.46/2.20	The ordinary context-sensitive dependency pairs DP_o are:
5.46/2.20	   ISNATILIST(IL) -> ISNATLIST(IL)
5.46/2.20	   ISNAT(s(N)) -> ISNAT(N)
5.46/2.20	   ISNAT(length(L)) -> ISNATLIST(L)
5.46/2.20	   ISNATILIST(cons(N, IL)) -> AND(isNat(N), isNatIList(IL))
5.46/2.20	   ISNATILIST(cons(N, IL)) -> ISNAT(N)
5.46/2.20	   ISNATILIST(cons(N, IL)) -> ISNATILIST(IL)
5.46/2.20	   ISNATLIST(cons(N, L)) -> AND(isNat(N), isNatList(L))
5.46/2.20	   ISNATLIST(cons(N, L)) -> ISNAT(N)
5.46/2.20	   ISNATLIST(cons(N, L)) -> ISNATLIST(L)
5.46/2.20	   ISNATLIST(take(N, IL)) -> AND(isNat(N), isNatIList(IL))
5.46/2.20	   ISNATLIST(take(N, IL)) -> ISNAT(N)
5.46/2.20	   ISNATLIST(take(N, IL)) -> ISNATILIST(IL)
5.46/2.20	   TAKE(s(M), cons(N, IL)) -> UTAKE2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.20	   TAKE(s(M), cons(N, IL)) -> AND(isNat(M), and(isNat(N), isNatIList(IL)))
5.46/2.20	   TAKE(s(M), cons(N, IL)) -> ISNAT(M)
5.46/2.20	   TAKE(s(M), cons(N, IL)) -> AND(isNat(N), isNatIList(IL))
5.46/2.20	   TAKE(s(M), cons(N, IL)) -> ISNAT(N)
5.46/2.20	   TAKE(s(M), cons(N, IL)) -> ISNATILIST(IL)
5.46/2.20	   LENGTH(cons(N, L)) -> ULENGTH(and(isNat(N), isNatList(L)), L)
5.46/2.20	   LENGTH(cons(N, L)) -> AND(isNat(N), isNatList(L))
5.46/2.20	   LENGTH(cons(N, L)) -> ISNAT(N)
5.46/2.20	   LENGTH(cons(N, L)) -> ISNATLIST(L)
5.46/2.20	   ULENGTH(tt, L) -> LENGTH(L)
5.46/2.20	
5.46/2.20	The collapsing dependency pairs are DP_c:
5.46/2.20	   UTAKE2(tt, M, N, IL) -> N
5.46/2.20	   ULENGTH(tt, L) -> L
5.46/2.20	
5.46/2.20	
5.46/2.20	The hidden terms of R are:
5.46/2.20	
5.46/2.20	   zeros
5.46/2.20	   take(x0, x1)
5.46/2.20	
5.46/2.20	Every hiding context is built from:
5.46/2.20	   aprove.DPFramework.CSDPProblem.QCSDPProblem$1@7d635eb2
5.46/2.20	
5.46/2.20	Hence, the new unhiding pairs DP_u are :
5.46/2.20	   UTAKE2(tt, M, N, IL) -> U(N)
5.46/2.20	   ULENGTH(tt, L) -> U(L)
5.46/2.20	   U(take(x_0, x_1)) -> U(x_0)
5.46/2.20	   U(take(x_0, x_1)) -> U(x_1)
5.46/2.20	   U(zeros) -> ZEROS
5.46/2.20	   U(take(x0, x1)) -> TAKE(x0, x1)
5.46/2.20	
5.46/2.20	The TRS R consists of the following rules:
5.46/2.20	
5.46/2.20	   and(tt, T) -> T
5.46/2.20	   isNatIList(IL) -> isNatList(IL)
5.46/2.20	   isNat(0) -> tt
5.46/2.20	   isNat(s(N)) -> isNat(N)
5.46/2.20	   isNat(length(L)) -> isNatList(L)
5.46/2.20	   isNatIList(zeros) -> tt
5.46/2.20	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   isNatList(nil) -> tt
5.46/2.20	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.20	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   zeros -> cons(0, zeros)
5.46/2.20	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.20	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.20	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.20	   uLength(tt, L) -> s(length(L))
5.46/2.20	
5.46/2.20	Q is empty.
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(7) QCSDependencyGraphProof (EQUIVALENT)
5.46/2.20	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 3 SCCs with 13 less nodes.
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(8)
5.46/2.20	Complex Obligation (AND)
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(9)
5.46/2.20	Obligation:
5.46/2.20	Q-restricted context-sensitive dependency pair problem:
5.46/2.20	The symbols in {and_2, s_1, length_1, take_2} are replacing on all positions.
5.46/2.20	For all symbols f in {cons_2, uTake2_4, uLength_2} we have mu(f) = {1}.
5.46/2.20	The symbols in {isNatIList_1, isNatList_1, isNat_1, ISNAT_1, ISNATLIST_1, ISNATILIST_1} are not replacing on any position.
5.46/2.20	
5.46/2.20	The TRS P consists of the following rules:
5.46/2.20	
5.46/2.20	   ISNATLIST(cons(N, L)) -> ISNAT(N)
5.46/2.20	   ISNAT(s(N)) -> ISNAT(N)
5.46/2.20	   ISNAT(length(L)) -> ISNATLIST(L)
5.46/2.20	   ISNATLIST(cons(N, L)) -> ISNATLIST(L)
5.46/2.20	   ISNATLIST(take(N, IL)) -> ISNAT(N)
5.46/2.20	   ISNATLIST(take(N, IL)) -> ISNATILIST(IL)
5.46/2.20	   ISNATILIST(IL) -> ISNATLIST(IL)
5.46/2.20	   ISNATILIST(cons(N, IL)) -> ISNAT(N)
5.46/2.20	   ISNATILIST(cons(N, IL)) -> ISNATILIST(IL)
5.46/2.20	
5.46/2.20	The TRS R consists of the following rules:
5.46/2.20	
5.46/2.20	   and(tt, T) -> T
5.46/2.20	   isNatIList(IL) -> isNatList(IL)
5.46/2.20	   isNat(0) -> tt
5.46/2.20	   isNat(s(N)) -> isNat(N)
5.46/2.20	   isNat(length(L)) -> isNatList(L)
5.46/2.20	   isNatIList(zeros) -> tt
5.46/2.20	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   isNatList(nil) -> tt
5.46/2.20	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.20	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.20	   zeros -> cons(0, zeros)
5.46/2.20	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.20	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.20	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.20	   uLength(tt, L) -> s(length(L))
5.46/2.20	
5.46/2.20	Q is empty.
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(10) QCSUsableRulesProof (EQUIVALENT)
5.46/2.20	The following rules are not useable [DA_EMMES] and can be deleted:
5.46/2.20	
5.46/2.20	   and(tt, x0) -> x0
5.46/2.20	   isNatIList(x0) -> isNatList(x0)
5.46/2.20	   isNat(0) -> tt
5.46/2.20	   isNat(s(x0)) -> isNat(x0)
5.46/2.20	   isNat(length(x0)) -> isNatList(x0)
5.46/2.20	   isNatIList(zeros) -> tt
5.46/2.20	   isNatIList(cons(x0, x1)) -> and(isNat(x0), isNatIList(x1))
5.46/2.20	   isNatList(nil) -> tt
5.46/2.20	   isNatList(cons(x0, x1)) -> and(isNat(x0), isNatList(x1))
5.46/2.20	   isNatList(take(x0, x1)) -> and(isNat(x0), isNatIList(x1))
5.46/2.20	   zeros -> cons(0, zeros)
5.46/2.20	   take(s(x0), cons(x1, x2)) -> uTake2(and(isNat(x0), and(isNat(x1), isNatIList(x2))), x0, x1, x2)
5.46/2.20	   uTake2(tt, x0, x1, x2) -> cons(x1, take(x0, x2))
5.46/2.20	   length(cons(x0, x1)) -> uLength(and(isNat(x0), isNatList(x1)), x1)
5.46/2.20	   uLength(tt, x0) -> s(length(x0))
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(11)
5.46/2.20	Obligation:
5.46/2.20	Q-restricted context-sensitive dependency pair problem:
5.46/2.20	The symbols in {s_1, length_1, take_2} are replacing on all positions.
5.46/2.20	For all symbols f in {cons_2} we have mu(f) = {1}.
5.46/2.20	The symbols in {ISNAT_1, ISNATLIST_1, ISNATILIST_1} are not replacing on any position.
5.46/2.20	
5.46/2.20	The TRS P consists of the following rules:
5.46/2.20	
5.46/2.20	   ISNATLIST(cons(N, L)) -> ISNAT(N)
5.46/2.20	   ISNAT(s(N)) -> ISNAT(N)
5.46/2.20	   ISNAT(length(L)) -> ISNATLIST(L)
5.46/2.20	   ISNATLIST(cons(N, L)) -> ISNATLIST(L)
5.46/2.20	   ISNATLIST(take(N, IL)) -> ISNAT(N)
5.46/2.20	   ISNATLIST(take(N, IL)) -> ISNATILIST(IL)
5.46/2.20	   ISNATILIST(IL) -> ISNATLIST(IL)
5.46/2.20	   ISNATILIST(cons(N, IL)) -> ISNAT(N)
5.46/2.20	   ISNATILIST(cons(N, IL)) -> ISNATILIST(IL)
5.46/2.20	
5.46/2.20	R is empty.
5.46/2.20	Q is empty.
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(12) QCSDPMuMonotonicPoloProof (EQUIVALENT)
5.46/2.20	By using the following mu-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted.
5.46/2.20	
5.46/2.20	Strictly oriented dependency pairs:
5.46/2.20	
5.46/2.20	   ISNAT(s(N)) -> ISNAT(N)
5.46/2.20	   ISNAT(length(L)) -> ISNATLIST(L)
5.46/2.20	   ISNATLIST(take(N, IL)) -> ISNAT(N)
5.46/2.20	   ISNATLIST(take(N, IL)) -> ISNATILIST(IL)
5.46/2.20	   ISNATILIST(IL) -> ISNATLIST(IL)
5.46/2.20	   ISNATILIST(cons(N, IL)) -> ISNAT(N)
5.46/2.20	
5.46/2.20	
5.46/2.20	Used ordering: POLO with Polynomial interpretation [POLO]:
5.46/2.20	
5.46/2.20	   POL(ISNAT(x_1)) = 1 + x_1
5.46/2.20	   POL(ISNATILIST(x_1)) = 2 + 2*x_1
5.46/2.20	   POL(ISNATLIST(x_1)) = 1 + 2*x_1
5.46/2.20	   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
5.46/2.20	   POL(length(x_1)) = 1 + 2*x_1
5.46/2.20	   POL(s(x_1)) = 2 + 2*x_1
5.46/2.20	   POL(take(x_1, x_2)) = 2 + 2*x_1 + x_2
5.46/2.20	
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(13)
5.46/2.20	Obligation:
5.46/2.20	Q-restricted context-sensitive dependency pair problem:
5.46/2.20	For all symbols f in {cons_2} we have mu(f) = {1}.
5.46/2.20	The symbols in {ISNAT_1, ISNATLIST_1, ISNATILIST_1} are not replacing on any position.
5.46/2.20	
5.46/2.20	The TRS P consists of the following rules:
5.46/2.20	
5.46/2.20	   ISNATLIST(cons(N, L)) -> ISNAT(N)
5.46/2.20	   ISNATLIST(cons(N, L)) -> ISNATLIST(L)
5.46/2.20	   ISNATILIST(cons(N, IL)) -> ISNATILIST(IL)
5.46/2.20	
5.46/2.20	R is empty.
5.46/2.20	Q is empty.
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(14) QCSDependencyGraphProof (EQUIVALENT)
5.46/2.20	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 2 SCCs with 1 less node.
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(15)
5.46/2.20	Complex Obligation (AND)
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(16)
5.46/2.20	Obligation:
5.46/2.20	Q-restricted context-sensitive dependency pair problem:
5.46/2.20	For all symbols f in {cons_2} we have mu(f) = {1}.
5.46/2.20	The symbols in {ISNATILIST_1} are not replacing on any position.
5.46/2.20	
5.46/2.20	The TRS P consists of the following rules:
5.46/2.20	
5.46/2.20	   ISNATILIST(cons(N, IL)) -> ISNATILIST(IL)
5.46/2.20	
5.46/2.20	R is empty.
5.46/2.20	Q is empty.
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(17) QCSDPMuMonotonicPoloProof (EQUIVALENT)
5.46/2.20	By using the following mu-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted.
5.46/2.20	
5.46/2.20	Strictly oriented dependency pairs:
5.46/2.20	
5.46/2.20	   ISNATILIST(cons(N, IL)) -> ISNATILIST(IL)
5.46/2.20	
5.46/2.20	
5.46/2.20	Used ordering: POLO with Polynomial interpretation [POLO]:
5.46/2.20	
5.46/2.20	   POL(ISNATILIST(x_1)) = 2*x_1
5.46/2.20	   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
5.46/2.20	
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(18)
5.46/2.20	Obligation:
5.46/2.20	Q-restricted context-sensitive dependency pair problem:
5.46/2.20	
5.46/2.20	The TRS P consists of the following rules:
5.46/2.20	none
5.46/2.20	
5.46/2.20	R is empty.
5.46/2.20	Q is empty.
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(19) PIsEmptyProof (EQUIVALENT)
5.46/2.20	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(20)
5.46/2.20	YES
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(21)
5.46/2.20	Obligation:
5.46/2.20	Q-restricted context-sensitive dependency pair problem:
5.46/2.20	For all symbols f in {cons_2} we have mu(f) = {1}.
5.46/2.20	The symbols in {ISNATLIST_1} are not replacing on any position.
5.46/2.20	
5.46/2.20	The TRS P consists of the following rules:
5.46/2.20	
5.46/2.20	   ISNATLIST(cons(N, L)) -> ISNATLIST(L)
5.46/2.20	
5.46/2.20	R is empty.
5.46/2.20	Q is empty.
5.46/2.20	
5.46/2.20	----------------------------------------
5.46/2.20	
5.46/2.20	(22) QCSDPMuMonotonicPoloProof (EQUIVALENT)
5.46/2.21	By using the following mu-monotonic polynomial ordering [POLO], at least one Dependency Pair or term rewrite system rule of this Q-CSDP problem can be strictly oriented and thus deleted.
5.46/2.21	
5.46/2.21	Strictly oriented dependency pairs:
5.46/2.21	
5.46/2.21	   ISNATLIST(cons(N, L)) -> ISNATLIST(L)
5.46/2.21	
5.46/2.21	
5.46/2.21	Used ordering: POLO with Polynomial interpretation [POLO]:
5.46/2.21	
5.46/2.21	   POL(ISNATLIST(x_1)) = 2*x_1
5.46/2.21	   POL(cons(x_1, x_2)) = 1 + x_1 + x_2
5.46/2.21	
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(23)
5.46/2.21	Obligation:
5.46/2.21	Q-restricted context-sensitive dependency pair problem:
5.46/2.21	
5.46/2.21	The TRS P consists of the following rules:
5.46/2.21	none
5.46/2.21	
5.46/2.21	R is empty.
5.46/2.21	Q is empty.
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(24) PIsEmptyProof (EQUIVALENT)
5.46/2.21	The TRS P is empty. Hence, there is no (P,Q,R,mu)-chain.
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(25)
5.46/2.21	YES
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(26)
5.46/2.21	Obligation:
5.46/2.21	Q-restricted context-sensitive dependency pair problem:
5.46/2.21	The symbols in {and_2, s_1, length_1, take_2, TAKE_2} are replacing on all positions.
5.46/2.21	For all symbols f in {cons_2, uTake2_4, uLength_2, UTAKE2_4} we have mu(f) = {1}.
5.46/2.21	The symbols in {isNatIList_1, isNatList_1, isNat_1, U_1} are not replacing on any position.
5.46/2.21	
5.46/2.21	The TRS P consists of the following rules:
5.46/2.21	
5.46/2.21	   TAKE(s(M), cons(N, IL)) -> UTAKE2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.21	   UTAKE2(tt, M, N, IL) -> U(N)
5.46/2.21	   U(take(x_0, x_1)) -> U(x_0)
5.46/2.21	   U(take(x_0, x_1)) -> U(x_1)
5.46/2.21	   U(take(x0, x1)) -> TAKE(x0, x1)
5.46/2.21	
5.46/2.21	The TRS R consists of the following rules:
5.46/2.21	
5.46/2.21	   and(tt, T) -> T
5.46/2.21	   isNatIList(IL) -> isNatList(IL)
5.46/2.21	   isNat(0) -> tt
5.46/2.21	   isNat(s(N)) -> isNat(N)
5.46/2.21	   isNat(length(L)) -> isNatList(L)
5.46/2.21	   isNatIList(zeros) -> tt
5.46/2.21	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.21	   isNatList(nil) -> tt
5.46/2.21	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.21	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.21	   zeros -> cons(0, zeros)
5.46/2.21	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.21	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.21	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.21	   uLength(tt, L) -> s(length(L))
5.46/2.21	
5.46/2.21	Q is empty.
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(27) QCSDPSubtermProof (EQUIVALENT)
5.46/2.21	We use the subterm processor [DA_EMMES].
5.46/2.21	
5.46/2.21	
5.46/2.21	The following pairs can be oriented strictly and are deleted.
5.46/2.21	
5.46/2.21	   TAKE(s(M), cons(N, IL)) -> UTAKE2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.21	   U(take(x_0, x_1)) -> U(x_0)
5.46/2.21	   U(take(x_0, x_1)) -> U(x_1)
5.46/2.21	   U(take(x0, x1)) -> TAKE(x0, x1)
5.46/2.21	The remaining pairs can at least be oriented weakly.
5.46/2.21	
5.46/2.21	   UTAKE2(tt, M, N, IL) -> U(N)
5.46/2.21	Used ordering:  Combined order from the following AFS and order.
5.46/2.21	UTAKE2(x1, x2, x3, x4)  =  x3
5.46/2.21	
5.46/2.21	TAKE(x1, x2)  =  x2
5.46/2.21	
5.46/2.21	U(x1)  =  x1
5.46/2.21	
5.46/2.21	
5.46/2.21	Subterm Order
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(28)
5.46/2.21	Obligation:
5.46/2.21	Q-restricted context-sensitive dependency pair problem:
5.46/2.21	The symbols in {and_2, s_1, length_1, take_2} are replacing on all positions.
5.46/2.21	For all symbols f in {cons_2, uTake2_4, uLength_2, UTAKE2_4} we have mu(f) = {1}.
5.46/2.21	The symbols in {isNatIList_1, isNatList_1, isNat_1, U_1} are not replacing on any position.
5.46/2.21	
5.46/2.21	The TRS P consists of the following rules:
5.46/2.21	
5.46/2.21	   UTAKE2(tt, M, N, IL) -> U(N)
5.46/2.21	
5.46/2.21	The TRS R consists of the following rules:
5.46/2.21	
5.46/2.21	   and(tt, T) -> T
5.46/2.21	   isNatIList(IL) -> isNatList(IL)
5.46/2.21	   isNat(0) -> tt
5.46/2.21	   isNat(s(N)) -> isNat(N)
5.46/2.21	   isNat(length(L)) -> isNatList(L)
5.46/2.21	   isNatIList(zeros) -> tt
5.46/2.21	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.21	   isNatList(nil) -> tt
5.46/2.21	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.21	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.21	   zeros -> cons(0, zeros)
5.46/2.21	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.21	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.21	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.21	   uLength(tt, L) -> s(length(L))
5.46/2.21	
5.46/2.21	Q is empty.
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(29) QCSDependencyGraphProof (EQUIVALENT)
5.46/2.21	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 1 less node.
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(30)
5.46/2.21	TRUE
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(31)
5.46/2.21	Obligation:
5.46/2.21	Q-restricted context-sensitive dependency pair problem:
5.46/2.21	The symbols in {and_2, s_1, length_1, take_2, LENGTH_1} are replacing on all positions.
5.46/2.21	For all symbols f in {cons_2, uTake2_4, uLength_2, ULENGTH_2} we have mu(f) = {1}.
5.46/2.21	The symbols in {isNatIList_1, isNatList_1, isNat_1} are not replacing on any position.
5.46/2.21	
5.46/2.21	The TRS P consists of the following rules:
5.46/2.21	
5.46/2.21	   ULENGTH(tt, L) -> LENGTH(L)
5.46/2.21	   LENGTH(cons(N, L)) -> ULENGTH(and(isNat(N), isNatList(L)), L)
5.46/2.21	
5.46/2.21	The TRS R consists of the following rules:
5.46/2.21	
5.46/2.21	   and(tt, T) -> T
5.46/2.21	   isNatIList(IL) -> isNatList(IL)
5.46/2.21	   isNat(0) -> tt
5.46/2.21	   isNat(s(N)) -> isNat(N)
5.46/2.21	   isNat(length(L)) -> isNatList(L)
5.46/2.21	   isNatIList(zeros) -> tt
5.46/2.21	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.21	   isNatList(nil) -> tt
5.46/2.21	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.21	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.21	   zeros -> cons(0, zeros)
5.46/2.21	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.21	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.21	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.21	   uLength(tt, L) -> s(length(L))
5.46/2.21	
5.46/2.21	Q is empty.
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(32) QCSDPReductionPairProof (EQUIVALENT)
5.46/2.21	Using the order
5.46/2.21	
5.46/2.21	Matrix interpretation [MATRO]:
5.46/2.21	
5.46/2.21	Non-tuple symbols: 
5.46/2.21	<<<
5.46/2.21	 M( isNat_1(x_1) ) =  	[[1], [1]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( isNatIList_1(x_1) ) =  	[[1], [1]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( and_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[1, 0], [0, 1]] 	* 	x_2
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( tt ) =  	[[1], [1]]
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( uLength_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1 	+ 	[[1, 1], [0, 0]] 	* 	x_2
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( nil ) =  	[[1], [1]]
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( s_1(x_1) ) =  	[[1], [0]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( uTake2_4(x_1, ..., x_4) ) =  	[[0], [0]] 	+ 	[[0, 1], [0, 1]] 	* 	x_1 	+ 	[[0, 0], [1, 0]] 	* 	x_2 	+ 	[[0, 0], [0, 0]] 	* 	x_3 	+ 	[[1, 0], [1, 1]] 	* 	x_4
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( length_1(x_1) ) =  	[[0], [0]] 	+ 	[[1, 1], [0, 0]] 	* 	x_1
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( take_2(x_1, x_2) ) =  	[[1], [0]] 	+ 	[[0, 0], [1, 0]] 	* 	x_1 	+ 	[[1, 0], [0, 1]] 	* 	x_2
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( zeros ) =  	[[0], [0]]
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( 0 ) =  	[[0], [0]]
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( cons_2(x_1, x_2) ) =  	[[0], [0]] 	+ 	[[0, 0], [0, 0]] 	* 	x_1 	+ 	[[1, 0], [1, 1]] 	* 	x_2
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( isNatList_1(x_1) ) =  	[[0], [1]] 	+ 	[[1, 0], [0, 0]] 	* 	x_1
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	Tuple symbols: 
5.46/2.21	<<<
5.46/2.21	 M( LENGTH_1(x_1) ) =  	[[1]] 	+ 	[[0, 1]] 	* 	x_1
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	<<<
5.46/2.21	 M( ULENGTH_2(x_1, x_2) ) =  	[[0]] 	+ 	[[1, 0]] 	* 	x_1 	+ 	[[0, 1]] 	* 	x_2
5.46/2.21	>>>
5.46/2.21	
5.46/2.21	
5.46/2.21	
5.46/2.21	Matrix type: 
5.46/2.21	
5.46/2.21	We used a basic matrix type which is not further parametrizeable.
5.46/2.21	
5.46/2.21	
5.46/2.21	
5.46/2.21	
5.46/2.21	
5.46/2.21	the following usable rules
5.46/2.21	
5.46/2.21	
5.46/2.21	   and(tt, T) -> T
5.46/2.21	   isNat(0) -> tt
5.46/2.21	   isNat(s(N)) -> isNat(N)
5.46/2.21	   isNat(length(L)) -> isNatList(L)
5.46/2.21	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.21	   uLength(tt, L) -> s(length(L))
5.46/2.21	   isNatList(nil) -> tt
5.46/2.21	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.21	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.21	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.21	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.21	   isNatIList(IL) -> isNatList(IL)
5.46/2.21	   isNatIList(zeros) -> tt
5.46/2.21	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.21	   zeros -> cons(0, zeros)
5.46/2.21	
5.46/2.21	
5.46/2.21	could all be oriented weakly.
5.46/2.21	
5.46/2.21	Furthermore, the pairs
5.46/2.21	
5.46/2.21	
5.46/2.21	   LENGTH(cons(N, L)) -> ULENGTH(and(isNat(N), isNatList(L)), L)
5.46/2.21	
5.46/2.21	
5.46/2.21	could be oriented strictly and thus removed by the CS-Reduction Pair Processor [LPAR08,DA_EMMES].
5.46/2.21	
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(33)
5.46/2.21	Obligation:
5.46/2.21	Q-restricted context-sensitive dependency pair problem:
5.46/2.21	The symbols in {and_2, s_1, length_1, take_2, LENGTH_1} are replacing on all positions.
5.46/2.21	For all symbols f in {cons_2, uTake2_4, uLength_2, ULENGTH_2} we have mu(f) = {1}.
5.46/2.21	The symbols in {isNatIList_1, isNatList_1, isNat_1} are not replacing on any position.
5.46/2.21	
5.46/2.21	The TRS P consists of the following rules:
5.46/2.21	
5.46/2.21	   ULENGTH(tt, L) -> LENGTH(L)
5.46/2.21	
5.46/2.21	The TRS R consists of the following rules:
5.46/2.21	
5.46/2.21	   and(tt, T) -> T
5.46/2.21	   isNatIList(IL) -> isNatList(IL)
5.46/2.21	   isNat(0) -> tt
5.46/2.21	   isNat(s(N)) -> isNat(N)
5.46/2.21	   isNat(length(L)) -> isNatList(L)
5.46/2.21	   isNatIList(zeros) -> tt
5.46/2.21	   isNatIList(cons(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.21	   isNatList(nil) -> tt
5.46/2.21	   isNatList(cons(N, L)) -> and(isNat(N), isNatList(L))
5.46/2.21	   isNatList(take(N, IL)) -> and(isNat(N), isNatIList(IL))
5.46/2.21	   zeros -> cons(0, zeros)
5.46/2.21	   take(s(M), cons(N, IL)) -> uTake2(and(isNat(M), and(isNat(N), isNatIList(IL))), M, N, IL)
5.46/2.21	   uTake2(tt, M, N, IL) -> cons(N, take(M, IL))
5.46/2.21	   length(cons(N, L)) -> uLength(and(isNat(N), isNatList(L)), L)
5.46/2.21	   uLength(tt, L) -> s(length(L))
5.46/2.21	
5.46/2.21	Q is empty.
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(34) QCSDependencyGraphProof (EQUIVALENT)
5.46/2.21	The approximation of the Context-Sensitive Dependency Graph [LPAR08] contains 0 SCCs with 1 less node.
5.46/2.21	
5.46/2.21	----------------------------------------
5.46/2.21	
5.46/2.21	(35)
5.46/2.21	TRUE
5.78/2.23	EOF