1146.13/291.64	WORST_CASE(Omega(n^1), ?)
1151.74/293.08	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1151.74/293.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1151.74/293.08	
1151.74/293.08	
1151.74/293.08	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1151.74/293.08	
1151.74/293.08	(0) CpxTRS
1151.74/293.08	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1151.74/293.08	(2) CpxTRS
1151.74/293.08	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1151.74/293.08	(4) typed CpxTrs
1151.74/293.08	(5) OrderProof [LOWER BOUND(ID), 8 ms]
1151.74/293.08	(6) typed CpxTrs
1151.74/293.08	(7) RewriteLemmaProof [LOWER BOUND(ID), 474 ms]
1151.74/293.08	(8) BEST
1151.74/293.08	    (9) proven lower bound
1151.74/293.08	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1151.74/293.08	        (11) BOUNDS(n^1, INF)
1151.74/293.08	    (12) typed CpxTrs
1151.74/293.08	        (13) RewriteLemmaProof [LOWER BOUND(ID), 154 ms]
1151.74/293.08	        (14) typed CpxTrs
1151.74/293.08	        (15) RewriteLemmaProof [LOWER BOUND(ID), 107 ms]
1151.74/293.08	        (16) typed CpxTrs
1151.74/293.08	        (17) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
1151.74/293.08	        (18) typed CpxTrs
1151.74/293.08	        (19) RewriteLemmaProof [LOWER BOUND(ID), 84 ms]
1151.74/293.08	        (20) typed CpxTrs
1151.74/293.08	        (21) RewriteLemmaProof [LOWER BOUND(ID), 86 ms]
1151.74/293.08	        (22) typed CpxTrs
1151.74/293.08	        (23) RewriteLemmaProof [LOWER BOUND(ID), 114 ms]
1151.74/293.08	        (24) typed CpxTrs
1151.74/293.08	        (25) RewriteLemmaProof [LOWER BOUND(ID), 218 ms]
1151.74/293.08	        (26) typed CpxTrs
1151.74/293.08	        (27) RewriteLemmaProof [LOWER BOUND(ID), 217 ms]
1151.74/293.08	        (28) typed CpxTrs
1151.74/293.08	        (29) RewriteLemmaProof [LOWER BOUND(ID), 110 ms]
1151.74/293.08	        (30) typed CpxTrs
1151.74/293.08	        (31) RewriteLemmaProof [LOWER BOUND(ID), 92 ms]
1151.74/293.08	        (32) typed CpxTrs
1151.74/293.08	        (33) RewriteLemmaProof [LOWER BOUND(ID), 114 ms]
1151.74/293.08	        (34) typed CpxTrs
1151.74/293.08	        (35) RewriteLemmaProof [LOWER BOUND(ID), 87 ms]
1151.74/293.08	        (36) typed CpxTrs
1151.74/293.08	        (37) RewriteLemmaProof [LOWER BOUND(ID), 147 ms]
1151.74/293.08	        (38) typed CpxTrs
1151.74/293.08	        (39) RewriteLemmaProof [LOWER BOUND(ID), 147 ms]
1151.74/293.08	        (40) typed CpxTrs
1151.74/293.08	        (41) RewriteLemmaProof [LOWER BOUND(ID), 155 ms]
1151.74/293.08	        (42) typed CpxTrs
1151.74/293.08	        (43) RewriteLemmaProof [LOWER BOUND(ID), 176 ms]
1151.74/293.08	        (44) typed CpxTrs
1151.74/293.08	        (45) RewriteLemmaProof [LOWER BOUND(ID), 104 ms]
1151.74/293.08	        (46) typed CpxTrs
1151.74/293.08	        (47) RewriteLemmaProof [LOWER BOUND(ID), 201 ms]
1151.74/293.08	        (48) typed CpxTrs
1151.74/293.08	
1151.74/293.08	
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(0)
1151.74/293.08	Obligation:
1151.74/293.08	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1151.74/293.08	
1151.74/293.08	
1151.74/293.08	The TRS R consists of the following rules:
1151.74/293.08	
1151.74/293.08	   active(zeros) -> mark(cons(0, zeros))
1151.74/293.08	   active(U11(tt)) -> mark(tt)
1151.74/293.08	   active(U21(tt)) -> mark(tt)
1151.74/293.08	   active(U31(tt)) -> mark(tt)
1151.74/293.08	   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.08	   active(U42(tt)) -> mark(tt)
1151.74/293.08	   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.08	   active(U52(tt)) -> mark(tt)
1151.74/293.08	   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.08	   active(U62(tt)) -> mark(tt)
1151.74/293.08	   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.08	   active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.08	   active(U81(tt)) -> mark(nil)
1151.74/293.08	   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.08	   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.08	   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.08	   active(isNat(0)) -> mark(tt)
1151.74/293.08	   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.08	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.08	   active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.08	   active(isNatIList(zeros)) -> mark(tt)
1151.74/293.08	   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.08	   active(isNatList(nil)) -> mark(tt)
1151.74/293.08	   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.08	   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.08	   active(length(nil)) -> mark(0)
1151.74/293.08	   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.08	   active(take(0, IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.08	   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.08	   active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.08	   active(U11(X)) -> U11(active(X))
1151.74/293.08	   active(U21(X)) -> U21(active(X))
1151.74/293.08	   active(U31(X)) -> U31(active(X))
1151.74/293.08	   active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.08	   active(U42(X)) -> U42(active(X))
1151.74/293.08	   active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.08	   active(U52(X)) -> U52(active(X))
1151.74/293.08	   active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.08	   active(U62(X)) -> U62(active(X))
1151.74/293.08	   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.08	   active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.08	   active(s(X)) -> s(active(X))
1151.74/293.08	   active(length(X)) -> length(active(X))
1151.74/293.08	   active(U81(X)) -> U81(active(X))
1151.74/293.08	   active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.08	   active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.08	   active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.08	   active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.08	   active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.08	   cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.08	   U11(mark(X)) -> mark(U11(X))
1151.74/293.08	   U21(mark(X)) -> mark(U21(X))
1151.74/293.08	   U31(mark(X)) -> mark(U31(X))
1151.74/293.08	   U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.08	   U42(mark(X)) -> mark(U42(X))
1151.74/293.08	   U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.08	   U52(mark(X)) -> mark(U52(X))
1151.74/293.08	   U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.08	   U62(mark(X)) -> mark(U62(X))
1151.74/293.08	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.08	   U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.08	   s(mark(X)) -> mark(s(X))
1151.74/293.08	   length(mark(X)) -> mark(length(X))
1151.74/293.08	   U81(mark(X)) -> mark(U81(X))
1151.74/293.08	   U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.08	   U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.08	   U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.08	   take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.08	   take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.08	   proper(zeros) -> ok(zeros)
1151.74/293.08	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.08	   proper(0) -> ok(0)
1151.74/293.08	   proper(U11(X)) -> U11(proper(X))
1151.74/293.08	   proper(tt) -> ok(tt)
1151.74/293.08	   proper(U21(X)) -> U21(proper(X))
1151.74/293.08	   proper(U31(X)) -> U31(proper(X))
1151.74/293.08	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.08	   proper(U42(X)) -> U42(proper(X))
1151.74/293.08	   proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.08	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.08	   proper(U52(X)) -> U52(proper(X))
1151.74/293.08	   proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.08	   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.08	   proper(U62(X)) -> U62(proper(X))
1151.74/293.08	   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.08	   proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.08	   proper(isNat(X)) -> isNat(proper(X))
1151.74/293.08	   proper(s(X)) -> s(proper(X))
1151.74/293.08	   proper(length(X)) -> length(proper(X))
1151.74/293.08	   proper(U81(X)) -> U81(proper(X))
1151.74/293.08	   proper(nil) -> ok(nil)
1151.74/293.08	   proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	   proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	   proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.08	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.08	   U11(ok(X)) -> ok(U11(X))
1151.74/293.08	   U21(ok(X)) -> ok(U21(X))
1151.74/293.08	   U31(ok(X)) -> ok(U31(X))
1151.74/293.08	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.08	   U42(ok(X)) -> ok(U42(X))
1151.74/293.08	   isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.08	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.08	   U52(ok(X)) -> ok(U52(X))
1151.74/293.08	   isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.08	   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.08	   U62(ok(X)) -> ok(U62(X))
1151.74/293.08	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.08	   U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.08	   isNat(ok(X)) -> ok(isNat(X))
1151.74/293.08	   s(ok(X)) -> ok(s(X))
1151.74/293.08	   length(ok(X)) -> ok(length(X))
1151.74/293.08	   U81(ok(X)) -> ok(U81(X))
1151.74/293.08	   U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.08	   U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.08	   U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.08	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.08	   top(mark(X)) -> top(proper(X))
1151.74/293.08	   top(ok(X)) -> top(active(X))
1151.74/293.08	
1151.74/293.08	S is empty.
1151.74/293.08	Rewrite Strategy: FULL
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1151.74/293.08	Renamed function symbols to avoid clashes with predefined symbol.
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(2)
1151.74/293.08	Obligation:
1151.74/293.08	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1151.74/293.08	
1151.74/293.08	
1151.74/293.08	The TRS R consists of the following rules:
1151.74/293.08	
1151.74/293.08	   active(zeros) -> mark(cons(0', zeros))
1151.74/293.08	   active(U11(tt)) -> mark(tt)
1151.74/293.08	   active(U21(tt)) -> mark(tt)
1151.74/293.08	   active(U31(tt)) -> mark(tt)
1151.74/293.08	   active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.08	   active(U42(tt)) -> mark(tt)
1151.74/293.08	   active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.08	   active(U52(tt)) -> mark(tt)
1151.74/293.08	   active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.08	   active(U62(tt)) -> mark(tt)
1151.74/293.08	   active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.08	   active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.08	   active(U81(tt)) -> mark(nil)
1151.74/293.08	   active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.08	   active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.08	   active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.08	   active(isNat(0')) -> mark(tt)
1151.74/293.08	   active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.08	   active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.08	   active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.08	   active(isNatIList(zeros)) -> mark(tt)
1151.74/293.08	   active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.08	   active(isNatList(nil)) -> mark(tt)
1151.74/293.08	   active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.08	   active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.08	   active(length(nil)) -> mark(0')
1151.74/293.08	   active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.08	   active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.08	   active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.08	   active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.08	   active(U11(X)) -> U11(active(X))
1151.74/293.08	   active(U21(X)) -> U21(active(X))
1151.74/293.08	   active(U31(X)) -> U31(active(X))
1151.74/293.08	   active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.08	   active(U42(X)) -> U42(active(X))
1151.74/293.08	   active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.08	   active(U52(X)) -> U52(active(X))
1151.74/293.08	   active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.08	   active(U62(X)) -> U62(active(X))
1151.74/293.08	   active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.08	   active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.08	   active(s(X)) -> s(active(X))
1151.74/293.08	   active(length(X)) -> length(active(X))
1151.74/293.08	   active(U81(X)) -> U81(active(X))
1151.74/293.08	   active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.08	   active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.08	   active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.08	   active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.08	   active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.08	   cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.08	   U11(mark(X)) -> mark(U11(X))
1151.74/293.08	   U21(mark(X)) -> mark(U21(X))
1151.74/293.08	   U31(mark(X)) -> mark(U31(X))
1151.74/293.08	   U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.08	   U42(mark(X)) -> mark(U42(X))
1151.74/293.08	   U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.08	   U52(mark(X)) -> mark(U52(X))
1151.74/293.08	   U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.08	   U62(mark(X)) -> mark(U62(X))
1151.74/293.08	   U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.08	   U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.08	   s(mark(X)) -> mark(s(X))
1151.74/293.08	   length(mark(X)) -> mark(length(X))
1151.74/293.08	   U81(mark(X)) -> mark(U81(X))
1151.74/293.08	   U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.08	   U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.08	   U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.08	   take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.08	   take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.08	   proper(zeros) -> ok(zeros)
1151.74/293.08	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.08	   proper(0') -> ok(0')
1151.74/293.08	   proper(U11(X)) -> U11(proper(X))
1151.74/293.08	   proper(tt) -> ok(tt)
1151.74/293.08	   proper(U21(X)) -> U21(proper(X))
1151.74/293.08	   proper(U31(X)) -> U31(proper(X))
1151.74/293.08	   proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.08	   proper(U42(X)) -> U42(proper(X))
1151.74/293.08	   proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.08	   proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.08	   proper(U52(X)) -> U52(proper(X))
1151.74/293.08	   proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.08	   proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.08	   proper(U62(X)) -> U62(proper(X))
1151.74/293.08	   proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.08	   proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.08	   proper(isNat(X)) -> isNat(proper(X))
1151.74/293.08	   proper(s(X)) -> s(proper(X))
1151.74/293.08	   proper(length(X)) -> length(proper(X))
1151.74/293.08	   proper(U81(X)) -> U81(proper(X))
1151.74/293.08	   proper(nil) -> ok(nil)
1151.74/293.08	   proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	   proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	   proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.08	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.08	   U11(ok(X)) -> ok(U11(X))
1151.74/293.08	   U21(ok(X)) -> ok(U21(X))
1151.74/293.08	   U31(ok(X)) -> ok(U31(X))
1151.74/293.08	   U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.08	   U42(ok(X)) -> ok(U42(X))
1151.74/293.08	   isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.08	   U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.08	   U52(ok(X)) -> ok(U52(X))
1151.74/293.08	   isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.08	   U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.08	   U62(ok(X)) -> ok(U62(X))
1151.74/293.08	   U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.08	   U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.08	   isNat(ok(X)) -> ok(isNat(X))
1151.74/293.08	   s(ok(X)) -> ok(s(X))
1151.74/293.08	   length(ok(X)) -> ok(length(X))
1151.74/293.08	   U81(ok(X)) -> ok(U81(X))
1151.74/293.08	   U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.08	   U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.08	   U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.08	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.08	   top(mark(X)) -> top(proper(X))
1151.74/293.08	   top(ok(X)) -> top(active(X))
1151.74/293.08	
1151.74/293.08	S is empty.
1151.74/293.08	Rewrite Strategy: FULL
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1151.74/293.08	Infered types.
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(4)
1151.74/293.08	Obligation:
1151.74/293.08	TRS:
1151.74/293.08	Rules:
1151.74/293.08	active(zeros) -> mark(cons(0', zeros))
1151.74/293.08	active(U11(tt)) -> mark(tt)
1151.74/293.08	active(U21(tt)) -> mark(tt)
1151.74/293.08	active(U31(tt)) -> mark(tt)
1151.74/293.08	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.08	active(U42(tt)) -> mark(tt)
1151.74/293.08	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.08	active(U52(tt)) -> mark(tt)
1151.74/293.08	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.08	active(U62(tt)) -> mark(tt)
1151.74/293.08	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.08	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.08	active(U81(tt)) -> mark(nil)
1151.74/293.08	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.08	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.08	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.08	active(isNat(0')) -> mark(tt)
1151.74/293.08	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.08	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.08	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.08	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.08	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.08	active(isNatList(nil)) -> mark(tt)
1151.74/293.08	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.08	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.08	active(length(nil)) -> mark(0')
1151.74/293.08	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.08	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.08	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.08	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.08	active(U11(X)) -> U11(active(X))
1151.74/293.08	active(U21(X)) -> U21(active(X))
1151.74/293.08	active(U31(X)) -> U31(active(X))
1151.74/293.08	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.08	active(U42(X)) -> U42(active(X))
1151.74/293.08	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.08	active(U52(X)) -> U52(active(X))
1151.74/293.08	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.08	active(U62(X)) -> U62(active(X))
1151.74/293.08	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.08	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.08	active(s(X)) -> s(active(X))
1151.74/293.08	active(length(X)) -> length(active(X))
1151.74/293.08	active(U81(X)) -> U81(active(X))
1151.74/293.08	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.08	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.08	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.08	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.08	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.08	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.08	U11(mark(X)) -> mark(U11(X))
1151.74/293.08	U21(mark(X)) -> mark(U21(X))
1151.74/293.08	U31(mark(X)) -> mark(U31(X))
1151.74/293.08	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.08	U42(mark(X)) -> mark(U42(X))
1151.74/293.08	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.08	U52(mark(X)) -> mark(U52(X))
1151.74/293.08	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.08	U62(mark(X)) -> mark(U62(X))
1151.74/293.08	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.08	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.08	s(mark(X)) -> mark(s(X))
1151.74/293.08	length(mark(X)) -> mark(length(X))
1151.74/293.08	U81(mark(X)) -> mark(U81(X))
1151.74/293.08	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.08	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.08	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.08	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.08	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.08	proper(zeros) -> ok(zeros)
1151.74/293.08	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.08	proper(0') -> ok(0')
1151.74/293.08	proper(U11(X)) -> U11(proper(X))
1151.74/293.08	proper(tt) -> ok(tt)
1151.74/293.08	proper(U21(X)) -> U21(proper(X))
1151.74/293.08	proper(U31(X)) -> U31(proper(X))
1151.74/293.08	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.08	proper(U42(X)) -> U42(proper(X))
1151.74/293.08	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.08	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.08	proper(U52(X)) -> U52(proper(X))
1151.74/293.08	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.08	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.08	proper(U62(X)) -> U62(proper(X))
1151.74/293.08	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.08	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.08	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.08	proper(s(X)) -> s(proper(X))
1151.74/293.08	proper(length(X)) -> length(proper(X))
1151.74/293.08	proper(U81(X)) -> U81(proper(X))
1151.74/293.08	proper(nil) -> ok(nil)
1151.74/293.08	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.08	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.08	U11(ok(X)) -> ok(U11(X))
1151.74/293.08	U21(ok(X)) -> ok(U21(X))
1151.74/293.08	U31(ok(X)) -> ok(U31(X))
1151.74/293.08	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.08	U42(ok(X)) -> ok(U42(X))
1151.74/293.08	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.08	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.08	U52(ok(X)) -> ok(U52(X))
1151.74/293.08	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.08	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.08	U62(ok(X)) -> ok(U62(X))
1151.74/293.08	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.08	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.08	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.08	s(ok(X)) -> ok(s(X))
1151.74/293.08	length(ok(X)) -> ok(length(X))
1151.74/293.08	U81(ok(X)) -> ok(U81(X))
1151.74/293.08	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.08	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.08	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.08	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.08	top(mark(X)) -> top(proper(X))
1151.74/293.08	top(ok(X)) -> top(active(X))
1151.74/293.08	
1151.74/293.08	Types:
1151.74/293.08	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.08	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.08	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.08	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.08	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.08	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.08	hole_top2_0 :: top
1151.74/293.08	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.08	
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(5) OrderProof (LOWER BOUND(ID))
1151.74/293.08	Heuristically decided to analyse the following defined symbols:
1151.74/293.08	active, cons, U42, isNatIList, U52, isNatList, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.08	
1151.74/293.08	They will be analysed ascendingly in the following order:
1151.74/293.08	cons < active
1151.74/293.08	U42 < active
1151.74/293.08	isNatIList < active
1151.74/293.08	U52 < active
1151.74/293.08	isNatList < active
1151.74/293.08	U62 < active
1151.74/293.08	U72 < active
1151.74/293.08	isNat < active
1151.74/293.08	s < active
1151.74/293.08	length < active
1151.74/293.08	U92 < active
1151.74/293.08	U93 < active
1151.74/293.08	take < active
1151.74/293.08	U11 < active
1151.74/293.08	U21 < active
1151.74/293.08	U31 < active
1151.74/293.08	U41 < active
1151.74/293.08	U51 < active
1151.74/293.08	U61 < active
1151.74/293.08	U71 < active
1151.74/293.08	U81 < active
1151.74/293.08	U91 < active
1151.74/293.08	active < top
1151.74/293.08	cons < proper
1151.74/293.08	U42 < proper
1151.74/293.08	isNatIList < proper
1151.74/293.08	U52 < proper
1151.74/293.08	isNatList < proper
1151.74/293.08	U62 < proper
1151.74/293.08	U72 < proper
1151.74/293.08	isNat < proper
1151.74/293.08	s < proper
1151.74/293.08	length < proper
1151.74/293.08	U92 < proper
1151.74/293.08	U93 < proper
1151.74/293.08	take < proper
1151.74/293.08	U11 < proper
1151.74/293.08	U21 < proper
1151.74/293.08	U31 < proper
1151.74/293.08	U41 < proper
1151.74/293.08	U51 < proper
1151.74/293.08	U61 < proper
1151.74/293.08	U71 < proper
1151.74/293.08	U81 < proper
1151.74/293.08	U91 < proper
1151.74/293.08	proper < top
1151.74/293.08	
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(6)
1151.74/293.08	Obligation:
1151.74/293.08	TRS:
1151.74/293.08	Rules:
1151.74/293.08	active(zeros) -> mark(cons(0', zeros))
1151.74/293.08	active(U11(tt)) -> mark(tt)
1151.74/293.08	active(U21(tt)) -> mark(tt)
1151.74/293.08	active(U31(tt)) -> mark(tt)
1151.74/293.08	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.08	active(U42(tt)) -> mark(tt)
1151.74/293.08	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.08	active(U52(tt)) -> mark(tt)
1151.74/293.08	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.08	active(U62(tt)) -> mark(tt)
1151.74/293.08	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.08	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.08	active(U81(tt)) -> mark(nil)
1151.74/293.08	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.08	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.08	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.08	active(isNat(0')) -> mark(tt)
1151.74/293.08	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.08	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.08	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.08	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.08	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.08	active(isNatList(nil)) -> mark(tt)
1151.74/293.08	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.08	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.08	active(length(nil)) -> mark(0')
1151.74/293.08	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.08	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.08	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.08	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.08	active(U11(X)) -> U11(active(X))
1151.74/293.08	active(U21(X)) -> U21(active(X))
1151.74/293.08	active(U31(X)) -> U31(active(X))
1151.74/293.08	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.08	active(U42(X)) -> U42(active(X))
1151.74/293.08	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.08	active(U52(X)) -> U52(active(X))
1151.74/293.08	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.08	active(U62(X)) -> U62(active(X))
1151.74/293.08	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.08	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.08	active(s(X)) -> s(active(X))
1151.74/293.08	active(length(X)) -> length(active(X))
1151.74/293.08	active(U81(X)) -> U81(active(X))
1151.74/293.08	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.08	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.08	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.08	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.08	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.08	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.08	U11(mark(X)) -> mark(U11(X))
1151.74/293.08	U21(mark(X)) -> mark(U21(X))
1151.74/293.08	U31(mark(X)) -> mark(U31(X))
1151.74/293.08	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.08	U42(mark(X)) -> mark(U42(X))
1151.74/293.08	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.08	U52(mark(X)) -> mark(U52(X))
1151.74/293.08	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.08	U62(mark(X)) -> mark(U62(X))
1151.74/293.08	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.08	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.08	s(mark(X)) -> mark(s(X))
1151.74/293.08	length(mark(X)) -> mark(length(X))
1151.74/293.08	U81(mark(X)) -> mark(U81(X))
1151.74/293.08	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.08	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.08	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.08	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.08	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.08	proper(zeros) -> ok(zeros)
1151.74/293.08	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.08	proper(0') -> ok(0')
1151.74/293.08	proper(U11(X)) -> U11(proper(X))
1151.74/293.08	proper(tt) -> ok(tt)
1151.74/293.08	proper(U21(X)) -> U21(proper(X))
1151.74/293.08	proper(U31(X)) -> U31(proper(X))
1151.74/293.08	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.08	proper(U42(X)) -> U42(proper(X))
1151.74/293.08	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.08	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.08	proper(U52(X)) -> U52(proper(X))
1151.74/293.08	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.08	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.08	proper(U62(X)) -> U62(proper(X))
1151.74/293.08	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.08	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.08	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.08	proper(s(X)) -> s(proper(X))
1151.74/293.08	proper(length(X)) -> length(proper(X))
1151.74/293.08	proper(U81(X)) -> U81(proper(X))
1151.74/293.08	proper(nil) -> ok(nil)
1151.74/293.08	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.08	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.08	U11(ok(X)) -> ok(U11(X))
1151.74/293.08	U21(ok(X)) -> ok(U21(X))
1151.74/293.08	U31(ok(X)) -> ok(U31(X))
1151.74/293.08	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.08	U42(ok(X)) -> ok(U42(X))
1151.74/293.08	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.08	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.08	U52(ok(X)) -> ok(U52(X))
1151.74/293.08	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.08	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.08	U62(ok(X)) -> ok(U62(X))
1151.74/293.08	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.08	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.08	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.08	s(ok(X)) -> ok(s(X))
1151.74/293.08	length(ok(X)) -> ok(length(X))
1151.74/293.08	U81(ok(X)) -> ok(U81(X))
1151.74/293.08	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.08	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.08	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.08	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.08	top(mark(X)) -> top(proper(X))
1151.74/293.08	top(ok(X)) -> top(active(X))
1151.74/293.08	
1151.74/293.08	Types:
1151.74/293.08	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.08	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.08	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.08	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.08	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.08	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.08	hole_top2_0 :: top
1151.74/293.08	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.08	
1151.74/293.08	
1151.74/293.08	Generator Equations:
1151.74/293.08	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.08	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.08	
1151.74/293.08	
1151.74/293.08	The following defined symbols remain to be analysed:
1151.74/293.08	cons, active, U42, isNatIList, U52, isNatList, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.08	
1151.74/293.08	They will be analysed ascendingly in the following order:
1151.74/293.08	cons < active
1151.74/293.08	U42 < active
1151.74/293.08	isNatIList < active
1151.74/293.08	U52 < active
1151.74/293.08	isNatList < active
1151.74/293.08	U62 < active
1151.74/293.08	U72 < active
1151.74/293.08	isNat < active
1151.74/293.08	s < active
1151.74/293.08	length < active
1151.74/293.08	U92 < active
1151.74/293.08	U93 < active
1151.74/293.08	take < active
1151.74/293.08	U11 < active
1151.74/293.08	U21 < active
1151.74/293.08	U31 < active
1151.74/293.08	U41 < active
1151.74/293.08	U51 < active
1151.74/293.08	U61 < active
1151.74/293.08	U71 < active
1151.74/293.08	U81 < active
1151.74/293.08	U91 < active
1151.74/293.08	active < top
1151.74/293.08	cons < proper
1151.74/293.08	U42 < proper
1151.74/293.08	isNatIList < proper
1151.74/293.08	U52 < proper
1151.74/293.08	isNatList < proper
1151.74/293.08	U62 < proper
1151.74/293.08	U72 < proper
1151.74/293.08	isNat < proper
1151.74/293.08	s < proper
1151.74/293.08	length < proper
1151.74/293.08	U92 < proper
1151.74/293.08	U93 < proper
1151.74/293.08	take < proper
1151.74/293.08	U11 < proper
1151.74/293.08	U21 < proper
1151.74/293.08	U31 < proper
1151.74/293.08	U41 < proper
1151.74/293.08	U51 < proper
1151.74/293.08	U61 < proper
1151.74/293.08	U71 < proper
1151.74/293.08	U81 < proper
1151.74/293.08	U91 < proper
1151.74/293.08	proper < top
1151.74/293.08	
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(7) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.08	Proved the following rewrite lemma:
1151.74/293.08	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.08	
1151.74/293.08	Induction Base:
1151.74/293.08	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))
1151.74/293.08	
1151.74/293.08	Induction Step:
1151.74/293.08	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n5_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
1151.74/293.08	mark(cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
1151.74/293.08	mark(*4_0)
1151.74/293.08	
1151.74/293.08	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(8)
1151.74/293.08	Complex Obligation (BEST)
1151.74/293.08	
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(9)
1151.74/293.08	Obligation:
1151.74/293.08	Proved the lower bound n^1 for the following obligation:
1151.74/293.08	
1151.74/293.08	TRS:
1151.74/293.08	Rules:
1151.74/293.08	active(zeros) -> mark(cons(0', zeros))
1151.74/293.08	active(U11(tt)) -> mark(tt)
1151.74/293.08	active(U21(tt)) -> mark(tt)
1151.74/293.08	active(U31(tt)) -> mark(tt)
1151.74/293.08	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.08	active(U42(tt)) -> mark(tt)
1151.74/293.08	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.08	active(U52(tt)) -> mark(tt)
1151.74/293.08	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.08	active(U62(tt)) -> mark(tt)
1151.74/293.08	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.08	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.08	active(U81(tt)) -> mark(nil)
1151.74/293.08	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.08	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.08	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.08	active(isNat(0')) -> mark(tt)
1151.74/293.08	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.08	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.08	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.08	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.08	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.08	active(isNatList(nil)) -> mark(tt)
1151.74/293.08	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.08	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.08	active(length(nil)) -> mark(0')
1151.74/293.08	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.08	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.08	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.08	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.08	active(U11(X)) -> U11(active(X))
1151.74/293.08	active(U21(X)) -> U21(active(X))
1151.74/293.08	active(U31(X)) -> U31(active(X))
1151.74/293.08	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.08	active(U42(X)) -> U42(active(X))
1151.74/293.08	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.08	active(U52(X)) -> U52(active(X))
1151.74/293.08	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.08	active(U62(X)) -> U62(active(X))
1151.74/293.08	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.08	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.08	active(s(X)) -> s(active(X))
1151.74/293.08	active(length(X)) -> length(active(X))
1151.74/293.08	active(U81(X)) -> U81(active(X))
1151.74/293.08	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.08	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.08	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.08	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.08	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.08	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.08	U11(mark(X)) -> mark(U11(X))
1151.74/293.08	U21(mark(X)) -> mark(U21(X))
1151.74/293.08	U31(mark(X)) -> mark(U31(X))
1151.74/293.08	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.08	U42(mark(X)) -> mark(U42(X))
1151.74/293.08	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.08	U52(mark(X)) -> mark(U52(X))
1151.74/293.08	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.08	U62(mark(X)) -> mark(U62(X))
1151.74/293.08	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.08	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.08	s(mark(X)) -> mark(s(X))
1151.74/293.08	length(mark(X)) -> mark(length(X))
1151.74/293.08	U81(mark(X)) -> mark(U81(X))
1151.74/293.08	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.08	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.08	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.08	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.08	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.08	proper(zeros) -> ok(zeros)
1151.74/293.08	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.08	proper(0') -> ok(0')
1151.74/293.08	proper(U11(X)) -> U11(proper(X))
1151.74/293.08	proper(tt) -> ok(tt)
1151.74/293.08	proper(U21(X)) -> U21(proper(X))
1151.74/293.08	proper(U31(X)) -> U31(proper(X))
1151.74/293.08	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.08	proper(U42(X)) -> U42(proper(X))
1151.74/293.08	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.08	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.08	proper(U52(X)) -> U52(proper(X))
1151.74/293.08	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.08	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.08	proper(U62(X)) -> U62(proper(X))
1151.74/293.08	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.08	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.08	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.08	proper(s(X)) -> s(proper(X))
1151.74/293.08	proper(length(X)) -> length(proper(X))
1151.74/293.08	proper(U81(X)) -> U81(proper(X))
1151.74/293.08	proper(nil) -> ok(nil)
1151.74/293.08	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.08	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.08	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.08	U11(ok(X)) -> ok(U11(X))
1151.74/293.08	U21(ok(X)) -> ok(U21(X))
1151.74/293.08	U31(ok(X)) -> ok(U31(X))
1151.74/293.08	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.08	U42(ok(X)) -> ok(U42(X))
1151.74/293.08	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.08	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.08	U52(ok(X)) -> ok(U52(X))
1151.74/293.08	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.08	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.08	U62(ok(X)) -> ok(U62(X))
1151.74/293.08	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.08	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.08	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.08	s(ok(X)) -> ok(s(X))
1151.74/293.08	length(ok(X)) -> ok(length(X))
1151.74/293.08	U81(ok(X)) -> ok(U81(X))
1151.74/293.08	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.08	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.08	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.08	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.08	top(mark(X)) -> top(proper(X))
1151.74/293.08	top(ok(X)) -> top(active(X))
1151.74/293.08	
1151.74/293.08	Types:
1151.74/293.08	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.08	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.08	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.08	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.08	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.08	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.08	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.08	hole_top2_0 :: top
1151.74/293.08	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.08	
1151.74/293.08	
1151.74/293.08	Generator Equations:
1151.74/293.08	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.08	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.08	
1151.74/293.08	
1151.74/293.08	The following defined symbols remain to be analysed:
1151.74/293.08	cons, active, U42, isNatIList, U52, isNatList, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.08	
1151.74/293.08	They will be analysed ascendingly in the following order:
1151.74/293.08	cons < active
1151.74/293.08	U42 < active
1151.74/293.08	isNatIList < active
1151.74/293.08	U52 < active
1151.74/293.08	isNatList < active
1151.74/293.08	U62 < active
1151.74/293.08	U72 < active
1151.74/293.08	isNat < active
1151.74/293.08	s < active
1151.74/293.08	length < active
1151.74/293.08	U92 < active
1151.74/293.08	U93 < active
1151.74/293.08	take < active
1151.74/293.08	U11 < active
1151.74/293.08	U21 < active
1151.74/293.08	U31 < active
1151.74/293.08	U41 < active
1151.74/293.08	U51 < active
1151.74/293.08	U61 < active
1151.74/293.08	U71 < active
1151.74/293.08	U81 < active
1151.74/293.08	U91 < active
1151.74/293.08	active < top
1151.74/293.08	cons < proper
1151.74/293.08	U42 < proper
1151.74/293.08	isNatIList < proper
1151.74/293.08	U52 < proper
1151.74/293.08	isNatList < proper
1151.74/293.08	U62 < proper
1151.74/293.08	U72 < proper
1151.74/293.08	isNat < proper
1151.74/293.08	s < proper
1151.74/293.08	length < proper
1151.74/293.08	U92 < proper
1151.74/293.08	U93 < proper
1151.74/293.08	take < proper
1151.74/293.08	U11 < proper
1151.74/293.08	U21 < proper
1151.74/293.08	U31 < proper
1151.74/293.08	U41 < proper
1151.74/293.08	U51 < proper
1151.74/293.08	U61 < proper
1151.74/293.08	U71 < proper
1151.74/293.08	U81 < proper
1151.74/293.08	U91 < proper
1151.74/293.08	proper < top
1151.74/293.08	
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(10) LowerBoundPropagationProof (FINISHED)
1151.74/293.08	Propagated lower bound.
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(11)
1151.74/293.08	BOUNDS(n^1, INF)
1151.74/293.08	
1151.74/293.08	----------------------------------------
1151.74/293.08	
1151.74/293.08	(12)
1151.74/293.08	Obligation:
1151.74/293.08	TRS:
1151.74/293.08	Rules:
1151.74/293.08	active(zeros) -> mark(cons(0', zeros))
1151.74/293.08	active(U11(tt)) -> mark(tt)
1151.74/293.08	active(U21(tt)) -> mark(tt)
1151.74/293.08	active(U31(tt)) -> mark(tt)
1151.74/293.08	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.08	active(U42(tt)) -> mark(tt)
1151.74/293.08	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.08	active(U52(tt)) -> mark(tt)
1151.74/293.08	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.08	active(U62(tt)) -> mark(tt)
1151.74/293.08	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.08	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.08	active(U81(tt)) -> mark(nil)
1151.74/293.08	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.08	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.08	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.08	active(isNat(0')) -> mark(tt)
1151.74/293.08	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.08	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.08	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.08	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.08	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.08	active(isNatList(nil)) -> mark(tt)
1151.74/293.08	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.08	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.08	active(length(nil)) -> mark(0')
1151.74/293.08	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.08	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.08	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.08	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.08	active(U11(X)) -> U11(active(X))
1151.74/293.08	active(U21(X)) -> U21(active(X))
1151.74/293.08	active(U31(X)) -> U31(active(X))
1151.74/293.08	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.08	active(U42(X)) -> U42(active(X))
1151.74/293.08	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.08	active(U52(X)) -> U52(active(X))
1151.74/293.08	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.08	active(U62(X)) -> U62(active(X))
1151.74/293.08	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.08	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.09	active(s(X)) -> s(active(X))
1151.74/293.09	active(length(X)) -> length(active(X))
1151.74/293.09	active(U81(X)) -> U81(active(X))
1151.74/293.09	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.09	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.09	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.09	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.09	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.09	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.09	U11(mark(X)) -> mark(U11(X))
1151.74/293.09	U21(mark(X)) -> mark(U21(X))
1151.74/293.09	U31(mark(X)) -> mark(U31(X))
1151.74/293.09	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.09	U42(mark(X)) -> mark(U42(X))
1151.74/293.09	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.09	U52(mark(X)) -> mark(U52(X))
1151.74/293.09	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.09	U62(mark(X)) -> mark(U62(X))
1151.74/293.09	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.09	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.09	s(mark(X)) -> mark(s(X))
1151.74/293.09	length(mark(X)) -> mark(length(X))
1151.74/293.09	U81(mark(X)) -> mark(U81(X))
1151.74/293.09	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.09	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.09	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.09	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.09	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.09	proper(zeros) -> ok(zeros)
1151.74/293.09	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.09	proper(0') -> ok(0')
1151.74/293.09	proper(U11(X)) -> U11(proper(X))
1151.74/293.09	proper(tt) -> ok(tt)
1151.74/293.09	proper(U21(X)) -> U21(proper(X))
1151.74/293.09	proper(U31(X)) -> U31(proper(X))
1151.74/293.09	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.09	proper(U42(X)) -> U42(proper(X))
1151.74/293.09	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.09	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.09	proper(U52(X)) -> U52(proper(X))
1151.74/293.09	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.09	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.09	proper(U62(X)) -> U62(proper(X))
1151.74/293.09	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.09	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.09	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.09	proper(s(X)) -> s(proper(X))
1151.74/293.09	proper(length(X)) -> length(proper(X))
1151.74/293.09	proper(U81(X)) -> U81(proper(X))
1151.74/293.09	proper(nil) -> ok(nil)
1151.74/293.09	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.09	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.09	U11(ok(X)) -> ok(U11(X))
1151.74/293.09	U21(ok(X)) -> ok(U21(X))
1151.74/293.09	U31(ok(X)) -> ok(U31(X))
1151.74/293.09	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.09	U42(ok(X)) -> ok(U42(X))
1151.74/293.09	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.09	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.09	U52(ok(X)) -> ok(U52(X))
1151.74/293.09	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.09	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.09	U62(ok(X)) -> ok(U62(X))
1151.74/293.09	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.09	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.09	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.09	s(ok(X)) -> ok(s(X))
1151.74/293.09	length(ok(X)) -> ok(length(X))
1151.74/293.09	U81(ok(X)) -> ok(U81(X))
1151.74/293.09	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.09	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.09	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.09	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.09	top(mark(X)) -> top(proper(X))
1151.74/293.09	top(ok(X)) -> top(active(X))
1151.74/293.09	
1151.74/293.09	Types:
1151.74/293.09	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.09	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.09	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.09	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.09	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.09	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.09	hole_top2_0 :: top
1151.74/293.09	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.09	
1151.74/293.09	
1151.74/293.09	Lemmas:
1151.74/293.09	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.09	
1151.74/293.09	
1151.74/293.09	Generator Equations:
1151.74/293.09	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.09	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.09	
1151.74/293.09	
1151.74/293.09	The following defined symbols remain to be analysed:
1151.74/293.09	U42, active, isNatIList, U52, isNatList, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.09	
1151.74/293.09	They will be analysed ascendingly in the following order:
1151.74/293.09	U42 < active
1151.74/293.09	isNatIList < active
1151.74/293.09	U52 < active
1151.74/293.09	isNatList < active
1151.74/293.09	U62 < active
1151.74/293.09	U72 < active
1151.74/293.09	isNat < active
1151.74/293.09	s < active
1151.74/293.09	length < active
1151.74/293.09	U92 < active
1151.74/293.09	U93 < active
1151.74/293.09	take < active
1151.74/293.09	U11 < active
1151.74/293.09	U21 < active
1151.74/293.09	U31 < active
1151.74/293.09	U41 < active
1151.74/293.09	U51 < active
1151.74/293.09	U61 < active
1151.74/293.09	U71 < active
1151.74/293.09	U81 < active
1151.74/293.09	U91 < active
1151.74/293.09	active < top
1151.74/293.09	U42 < proper
1151.74/293.09	isNatIList < proper
1151.74/293.09	U52 < proper
1151.74/293.09	isNatList < proper
1151.74/293.09	U62 < proper
1151.74/293.09	U72 < proper
1151.74/293.09	isNat < proper
1151.74/293.09	s < proper
1151.74/293.09	length < proper
1151.74/293.09	U92 < proper
1151.74/293.09	U93 < proper
1151.74/293.09	take < proper
1151.74/293.09	U11 < proper
1151.74/293.09	U21 < proper
1151.74/293.09	U31 < proper
1151.74/293.09	U41 < proper
1151.74/293.09	U51 < proper
1151.74/293.09	U61 < proper
1151.74/293.09	U71 < proper
1151.74/293.09	U81 < proper
1151.74/293.09	U91 < proper
1151.74/293.09	proper < top
1151.74/293.09	
1151.74/293.09	----------------------------------------
1151.74/293.09	
1151.74/293.09	(13) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.09	Proved the following rewrite lemma:
1151.74/293.09	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.09	
1151.74/293.09	Induction Base:
1151.74/293.09	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))
1151.74/293.09	
1151.74/293.09	Induction Step:
1151.74/293.09	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n2106_0, 1)))) ->_R^Omega(1)
1151.74/293.09	mark(U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0)))) ->_IH
1151.74/293.09	mark(*4_0)
1151.74/293.09	
1151.74/293.09	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.09	----------------------------------------
1151.74/293.09	
1151.74/293.09	(14)
1151.74/293.09	Obligation:
1151.74/293.09	TRS:
1151.74/293.09	Rules:
1151.74/293.09	active(zeros) -> mark(cons(0', zeros))
1151.74/293.09	active(U11(tt)) -> mark(tt)
1151.74/293.09	active(U21(tt)) -> mark(tt)
1151.74/293.09	active(U31(tt)) -> mark(tt)
1151.74/293.09	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.09	active(U42(tt)) -> mark(tt)
1151.74/293.09	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.09	active(U52(tt)) -> mark(tt)
1151.74/293.09	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.09	active(U62(tt)) -> mark(tt)
1151.74/293.09	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.09	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.09	active(U81(tt)) -> mark(nil)
1151.74/293.09	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.09	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.09	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.09	active(isNat(0')) -> mark(tt)
1151.74/293.09	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.09	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.09	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.09	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.09	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.09	active(isNatList(nil)) -> mark(tt)
1151.74/293.09	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.09	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.09	active(length(nil)) -> mark(0')
1151.74/293.09	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.09	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.09	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.09	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.09	active(U11(X)) -> U11(active(X))
1151.74/293.09	active(U21(X)) -> U21(active(X))
1151.74/293.09	active(U31(X)) -> U31(active(X))
1151.74/293.09	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.09	active(U42(X)) -> U42(active(X))
1151.74/293.09	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.09	active(U52(X)) -> U52(active(X))
1151.74/293.09	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.09	active(U62(X)) -> U62(active(X))
1151.74/293.09	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.09	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.09	active(s(X)) -> s(active(X))
1151.74/293.09	active(length(X)) -> length(active(X))
1151.74/293.09	active(U81(X)) -> U81(active(X))
1151.74/293.09	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.09	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.09	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.09	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.09	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.09	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.09	U11(mark(X)) -> mark(U11(X))
1151.74/293.09	U21(mark(X)) -> mark(U21(X))
1151.74/293.09	U31(mark(X)) -> mark(U31(X))
1151.74/293.09	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.09	U42(mark(X)) -> mark(U42(X))
1151.74/293.09	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.09	U52(mark(X)) -> mark(U52(X))
1151.74/293.09	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.09	U62(mark(X)) -> mark(U62(X))
1151.74/293.09	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.09	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.09	s(mark(X)) -> mark(s(X))
1151.74/293.09	length(mark(X)) -> mark(length(X))
1151.74/293.09	U81(mark(X)) -> mark(U81(X))
1151.74/293.09	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.09	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.09	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.09	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.09	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.09	proper(zeros) -> ok(zeros)
1151.74/293.09	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.09	proper(0') -> ok(0')
1151.74/293.09	proper(U11(X)) -> U11(proper(X))
1151.74/293.09	proper(tt) -> ok(tt)
1151.74/293.09	proper(U21(X)) -> U21(proper(X))
1151.74/293.09	proper(U31(X)) -> U31(proper(X))
1151.74/293.09	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.09	proper(U42(X)) -> U42(proper(X))
1151.74/293.09	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.09	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.09	proper(U52(X)) -> U52(proper(X))
1151.74/293.09	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.09	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.09	proper(U62(X)) -> U62(proper(X))
1151.74/293.09	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.09	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.09	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.09	proper(s(X)) -> s(proper(X))
1151.74/293.09	proper(length(X)) -> length(proper(X))
1151.74/293.09	proper(U81(X)) -> U81(proper(X))
1151.74/293.09	proper(nil) -> ok(nil)
1151.74/293.09	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.09	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.09	U11(ok(X)) -> ok(U11(X))
1151.74/293.09	U21(ok(X)) -> ok(U21(X))
1151.74/293.09	U31(ok(X)) -> ok(U31(X))
1151.74/293.09	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.09	U42(ok(X)) -> ok(U42(X))
1151.74/293.09	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.09	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.09	U52(ok(X)) -> ok(U52(X))
1151.74/293.09	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.09	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.09	U62(ok(X)) -> ok(U62(X))
1151.74/293.09	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.09	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.09	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.09	s(ok(X)) -> ok(s(X))
1151.74/293.09	length(ok(X)) -> ok(length(X))
1151.74/293.09	U81(ok(X)) -> ok(U81(X))
1151.74/293.09	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.09	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.09	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.09	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.09	top(mark(X)) -> top(proper(X))
1151.74/293.09	top(ok(X)) -> top(active(X))
1151.74/293.09	
1151.74/293.09	Types:
1151.74/293.09	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.09	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.09	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.09	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.09	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.09	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.09	hole_top2_0 :: top
1151.74/293.09	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.09	
1151.74/293.09	
1151.74/293.09	Lemmas:
1151.74/293.09	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.09	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.09	
1151.74/293.09	
1151.74/293.09	Generator Equations:
1151.74/293.09	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.09	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.09	
1151.74/293.09	
1151.74/293.09	The following defined symbols remain to be analysed:
1151.74/293.09	isNatIList, active, U52, isNatList, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.09	
1151.74/293.09	They will be analysed ascendingly in the following order:
1151.74/293.09	isNatIList < active
1151.74/293.09	U52 < active
1151.74/293.09	isNatList < active
1151.74/293.09	U62 < active
1151.74/293.09	U72 < active
1151.74/293.09	isNat < active
1151.74/293.09	s < active
1151.74/293.09	length < active
1151.74/293.09	U92 < active
1151.74/293.09	U93 < active
1151.74/293.09	take < active
1151.74/293.09	U11 < active
1151.74/293.09	U21 < active
1151.74/293.09	U31 < active
1151.74/293.09	U41 < active
1151.74/293.09	U51 < active
1151.74/293.09	U61 < active
1151.74/293.09	U71 < active
1151.74/293.09	U81 < active
1151.74/293.09	U91 < active
1151.74/293.09	active < top
1151.74/293.09	isNatIList < proper
1151.74/293.09	U52 < proper
1151.74/293.09	isNatList < proper
1151.74/293.09	U62 < proper
1151.74/293.09	U72 < proper
1151.74/293.09	isNat < proper
1151.74/293.09	s < proper
1151.74/293.09	length < proper
1151.74/293.09	U92 < proper
1151.74/293.09	U93 < proper
1151.74/293.09	take < proper
1151.74/293.09	U11 < proper
1151.74/293.09	U21 < proper
1151.74/293.09	U31 < proper
1151.74/293.09	U41 < proper
1151.74/293.09	U51 < proper
1151.74/293.09	U61 < proper
1151.74/293.09	U71 < proper
1151.74/293.09	U81 < proper
1151.74/293.09	U91 < proper
1151.74/293.09	proper < top
1151.74/293.09	
1151.74/293.09	----------------------------------------
1151.74/293.09	
1151.74/293.09	(15) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.09	Proved the following rewrite lemma:
1151.74/293.09	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.09	
1151.74/293.09	Induction Base:
1151.74/293.09	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))
1151.74/293.09	
1151.74/293.09	Induction Step:
1151.74/293.09	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n2892_0, 1)))) ->_R^Omega(1)
1151.74/293.09	mark(U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0)))) ->_IH
1151.74/293.09	mark(*4_0)
1151.74/293.09	
1151.74/293.09	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.09	----------------------------------------
1151.74/293.09	
1151.74/293.09	(16)
1151.74/293.09	Obligation:
1151.74/293.09	TRS:
1151.74/293.09	Rules:
1151.74/293.09	active(zeros) -> mark(cons(0', zeros))
1151.74/293.09	active(U11(tt)) -> mark(tt)
1151.74/293.09	active(U21(tt)) -> mark(tt)
1151.74/293.09	active(U31(tt)) -> mark(tt)
1151.74/293.09	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.09	active(U42(tt)) -> mark(tt)
1151.74/293.09	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.09	active(U52(tt)) -> mark(tt)
1151.74/293.09	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.09	active(U62(tt)) -> mark(tt)
1151.74/293.09	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.09	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.09	active(U81(tt)) -> mark(nil)
1151.74/293.09	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.09	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.09	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.09	active(isNat(0')) -> mark(tt)
1151.74/293.09	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.09	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.09	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.09	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.09	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.09	active(isNatList(nil)) -> mark(tt)
1151.74/293.09	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.09	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.09	active(length(nil)) -> mark(0')
1151.74/293.09	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.09	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.09	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.09	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.09	active(U11(X)) -> U11(active(X))
1151.74/293.09	active(U21(X)) -> U21(active(X))
1151.74/293.09	active(U31(X)) -> U31(active(X))
1151.74/293.09	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.09	active(U42(X)) -> U42(active(X))
1151.74/293.09	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.09	active(U52(X)) -> U52(active(X))
1151.74/293.09	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.09	active(U62(X)) -> U62(active(X))
1151.74/293.09	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.09	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.09	active(s(X)) -> s(active(X))
1151.74/293.09	active(length(X)) -> length(active(X))
1151.74/293.09	active(U81(X)) -> U81(active(X))
1151.74/293.09	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.09	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.09	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.09	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.09	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.09	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.09	U11(mark(X)) -> mark(U11(X))
1151.74/293.09	U21(mark(X)) -> mark(U21(X))
1151.74/293.09	U31(mark(X)) -> mark(U31(X))
1151.74/293.09	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.09	U42(mark(X)) -> mark(U42(X))
1151.74/293.09	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.09	U52(mark(X)) -> mark(U52(X))
1151.74/293.09	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.09	U62(mark(X)) -> mark(U62(X))
1151.74/293.09	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.09	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.09	s(mark(X)) -> mark(s(X))
1151.74/293.09	length(mark(X)) -> mark(length(X))
1151.74/293.09	U81(mark(X)) -> mark(U81(X))
1151.74/293.09	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.09	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.09	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.09	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.09	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.09	proper(zeros) -> ok(zeros)
1151.74/293.09	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.09	proper(0') -> ok(0')
1151.74/293.09	proper(U11(X)) -> U11(proper(X))
1151.74/293.09	proper(tt) -> ok(tt)
1151.74/293.09	proper(U21(X)) -> U21(proper(X))
1151.74/293.09	proper(U31(X)) -> U31(proper(X))
1151.74/293.09	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.09	proper(U42(X)) -> U42(proper(X))
1151.74/293.09	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.09	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.09	proper(U52(X)) -> U52(proper(X))
1151.74/293.09	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.09	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.09	proper(U62(X)) -> U62(proper(X))
1151.74/293.09	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.09	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.09	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.09	proper(s(X)) -> s(proper(X))
1151.74/293.09	proper(length(X)) -> length(proper(X))
1151.74/293.09	proper(U81(X)) -> U81(proper(X))
1151.74/293.09	proper(nil) -> ok(nil)
1151.74/293.09	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.09	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.09	U11(ok(X)) -> ok(U11(X))
1151.74/293.09	U21(ok(X)) -> ok(U21(X))
1151.74/293.09	U31(ok(X)) -> ok(U31(X))
1151.74/293.09	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.09	U42(ok(X)) -> ok(U42(X))
1151.74/293.09	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.09	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.09	U52(ok(X)) -> ok(U52(X))
1151.74/293.09	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.09	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.09	U62(ok(X)) -> ok(U62(X))
1151.74/293.09	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.09	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.09	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.09	s(ok(X)) -> ok(s(X))
1151.74/293.09	length(ok(X)) -> ok(length(X))
1151.74/293.09	U81(ok(X)) -> ok(U81(X))
1151.74/293.09	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.09	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.09	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.09	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.09	top(mark(X)) -> top(proper(X))
1151.74/293.09	top(ok(X)) -> top(active(X))
1151.74/293.09	
1151.74/293.09	Types:
1151.74/293.09	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.09	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.09	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.09	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.09	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.09	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.09	hole_top2_0 :: top
1151.74/293.09	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.09	
1151.74/293.09	
1151.74/293.09	Lemmas:
1151.74/293.09	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.09	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.09	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.09	
1151.74/293.09	
1151.74/293.09	Generator Equations:
1151.74/293.09	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.09	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.09	
1151.74/293.09	
1151.74/293.09	The following defined symbols remain to be analysed:
1151.74/293.09	isNatList, active, U62, U72, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.09	
1151.74/293.09	They will be analysed ascendingly in the following order:
1151.74/293.09	isNatList < active
1151.74/293.09	U62 < active
1151.74/293.09	U72 < active
1151.74/293.09	isNat < active
1151.74/293.09	s < active
1151.74/293.09	length < active
1151.74/293.09	U92 < active
1151.74/293.09	U93 < active
1151.74/293.09	take < active
1151.74/293.09	U11 < active
1151.74/293.09	U21 < active
1151.74/293.09	U31 < active
1151.74/293.09	U41 < active
1151.74/293.09	U51 < active
1151.74/293.09	U61 < active
1151.74/293.09	U71 < active
1151.74/293.09	U81 < active
1151.74/293.09	U91 < active
1151.74/293.09	active < top
1151.74/293.09	isNatList < proper
1151.74/293.09	U62 < proper
1151.74/293.09	U72 < proper
1151.74/293.09	isNat < proper
1151.74/293.09	s < proper
1151.74/293.09	length < proper
1151.74/293.09	U92 < proper
1151.74/293.09	U93 < proper
1151.74/293.09	take < proper
1151.74/293.09	U11 < proper
1151.74/293.09	U21 < proper
1151.74/293.09	U31 < proper
1151.74/293.09	U41 < proper
1151.74/293.09	U51 < proper
1151.74/293.09	U61 < proper
1151.74/293.09	U71 < proper
1151.74/293.09	U81 < proper
1151.74/293.09	U91 < proper
1151.74/293.09	proper < top
1151.74/293.09	
1151.74/293.09	----------------------------------------
1151.74/293.09	
1151.74/293.09	(17) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.09	Proved the following rewrite lemma:
1151.74/293.09	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.74/293.09	
1151.74/293.09	Induction Base:
1151.74/293.09	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))
1151.74/293.09	
1151.74/293.09	Induction Step:
1151.74/293.09	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n3782_0, 1)))) ->_R^Omega(1)
1151.74/293.09	mark(U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0)))) ->_IH
1151.74/293.09	mark(*4_0)
1151.74/293.09	
1151.74/293.09	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.09	----------------------------------------
1151.74/293.09	
1151.74/293.09	(18)
1151.74/293.09	Obligation:
1151.74/293.09	TRS:
1151.74/293.09	Rules:
1151.74/293.09	active(zeros) -> mark(cons(0', zeros))
1151.74/293.09	active(U11(tt)) -> mark(tt)
1151.74/293.09	active(U21(tt)) -> mark(tt)
1151.74/293.09	active(U31(tt)) -> mark(tt)
1151.74/293.09	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.09	active(U42(tt)) -> mark(tt)
1151.74/293.09	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.09	active(U52(tt)) -> mark(tt)
1151.74/293.09	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.09	active(U62(tt)) -> mark(tt)
1151.74/293.09	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.09	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.09	active(U81(tt)) -> mark(nil)
1151.74/293.09	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.09	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.09	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.09	active(isNat(0')) -> mark(tt)
1151.74/293.09	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.09	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.09	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.09	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.09	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.09	active(isNatList(nil)) -> mark(tt)
1151.74/293.09	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.09	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.09	active(length(nil)) -> mark(0')
1151.74/293.09	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.09	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.09	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.09	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.09	active(U11(X)) -> U11(active(X))
1151.74/293.09	active(U21(X)) -> U21(active(X))
1151.74/293.09	active(U31(X)) -> U31(active(X))
1151.74/293.09	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.09	active(U42(X)) -> U42(active(X))
1151.74/293.09	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.09	active(U52(X)) -> U52(active(X))
1151.74/293.09	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.09	active(U62(X)) -> U62(active(X))
1151.74/293.09	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.09	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.09	active(s(X)) -> s(active(X))
1151.74/293.09	active(length(X)) -> length(active(X))
1151.74/293.09	active(U81(X)) -> U81(active(X))
1151.74/293.09	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.09	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.09	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.09	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.09	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.09	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.09	U11(mark(X)) -> mark(U11(X))
1151.74/293.09	U21(mark(X)) -> mark(U21(X))
1151.74/293.09	U31(mark(X)) -> mark(U31(X))
1151.74/293.09	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.09	U42(mark(X)) -> mark(U42(X))
1151.74/293.09	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.09	U52(mark(X)) -> mark(U52(X))
1151.74/293.09	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.09	U62(mark(X)) -> mark(U62(X))
1151.74/293.09	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.09	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.09	s(mark(X)) -> mark(s(X))
1151.74/293.09	length(mark(X)) -> mark(length(X))
1151.74/293.09	U81(mark(X)) -> mark(U81(X))
1151.74/293.09	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.09	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.09	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.09	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.09	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.09	proper(zeros) -> ok(zeros)
1151.74/293.09	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.09	proper(0') -> ok(0')
1151.74/293.09	proper(U11(X)) -> U11(proper(X))
1151.74/293.09	proper(tt) -> ok(tt)
1151.74/293.09	proper(U21(X)) -> U21(proper(X))
1151.74/293.09	proper(U31(X)) -> U31(proper(X))
1151.74/293.09	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.09	proper(U42(X)) -> U42(proper(X))
1151.74/293.09	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.09	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.09	proper(U52(X)) -> U52(proper(X))
1151.74/293.09	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.09	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.09	proper(U62(X)) -> U62(proper(X))
1151.74/293.09	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.09	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.09	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.09	proper(s(X)) -> s(proper(X))
1151.74/293.09	proper(length(X)) -> length(proper(X))
1151.74/293.09	proper(U81(X)) -> U81(proper(X))
1151.74/293.09	proper(nil) -> ok(nil)
1151.74/293.09	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.09	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.09	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.09	U11(ok(X)) -> ok(U11(X))
1151.74/293.09	U21(ok(X)) -> ok(U21(X))
1151.74/293.09	U31(ok(X)) -> ok(U31(X))
1151.74/293.09	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.09	U42(ok(X)) -> ok(U42(X))
1151.74/293.09	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.09	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.09	U52(ok(X)) -> ok(U52(X))
1151.74/293.09	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.09	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.09	U62(ok(X)) -> ok(U62(X))
1151.74/293.09	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.09	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.09	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.09	s(ok(X)) -> ok(s(X))
1151.74/293.09	length(ok(X)) -> ok(length(X))
1151.74/293.09	U81(ok(X)) -> ok(U81(X))
1151.74/293.09	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.09	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.09	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.09	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.09	top(mark(X)) -> top(proper(X))
1151.74/293.09	top(ok(X)) -> top(active(X))
1151.74/293.09	
1151.74/293.09	Types:
1151.74/293.09	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.09	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.09	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.09	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.09	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.10	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.10	hole_top2_0 :: top
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	Lemmas:
1151.74/293.10	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.10	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.10	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.10	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	Generator Equations:
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	The following defined symbols remain to be analysed:
1151.74/293.10	U72, active, isNat, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.10	
1151.74/293.10	They will be analysed ascendingly in the following order:
1151.74/293.10	U72 < active
1151.74/293.10	isNat < active
1151.74/293.10	s < active
1151.74/293.10	length < active
1151.74/293.10	U92 < active
1151.74/293.10	U93 < active
1151.74/293.10	take < active
1151.74/293.10	U11 < active
1151.74/293.10	U21 < active
1151.74/293.10	U31 < active
1151.74/293.10	U41 < active
1151.74/293.10	U51 < active
1151.74/293.10	U61 < active
1151.74/293.10	U71 < active
1151.74/293.10	U81 < active
1151.74/293.10	U91 < active
1151.74/293.10	active < top
1151.74/293.10	U72 < proper
1151.74/293.10	isNat < proper
1151.74/293.10	s < proper
1151.74/293.10	length < proper
1151.74/293.10	U92 < proper
1151.74/293.10	U93 < proper
1151.74/293.10	take < proper
1151.74/293.10	U11 < proper
1151.74/293.10	U21 < proper
1151.74/293.10	U31 < proper
1151.74/293.10	U41 < proper
1151.74/293.10	U51 < proper
1151.74/293.10	U61 < proper
1151.74/293.10	U71 < proper
1151.74/293.10	U81 < proper
1151.74/293.10	U91 < proper
1151.74/293.10	proper < top
1151.74/293.10	
1151.74/293.10	----------------------------------------
1151.74/293.10	
1151.74/293.10	(19) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.10	Proved the following rewrite lemma:
1151.74/293.10	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.74/293.10	
1151.74/293.10	Induction Base:
1151.74/293.10	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))
1151.74/293.10	
1151.74/293.10	Induction Step:
1151.74/293.10	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n4758_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
1151.74/293.10	mark(U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
1151.74/293.10	mark(*4_0)
1151.74/293.10	
1151.74/293.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.10	----------------------------------------
1151.74/293.10	
1151.74/293.10	(20)
1151.74/293.10	Obligation:
1151.74/293.10	TRS:
1151.74/293.10	Rules:
1151.74/293.10	active(zeros) -> mark(cons(0', zeros))
1151.74/293.10	active(U11(tt)) -> mark(tt)
1151.74/293.10	active(U21(tt)) -> mark(tt)
1151.74/293.10	active(U31(tt)) -> mark(tt)
1151.74/293.10	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.10	active(U42(tt)) -> mark(tt)
1151.74/293.10	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.10	active(U52(tt)) -> mark(tt)
1151.74/293.10	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.10	active(U62(tt)) -> mark(tt)
1151.74/293.10	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.10	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.10	active(U81(tt)) -> mark(nil)
1151.74/293.10	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.10	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.10	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.10	active(isNat(0')) -> mark(tt)
1151.74/293.10	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.10	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.10	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.10	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.10	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.10	active(isNatList(nil)) -> mark(tt)
1151.74/293.10	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.10	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.10	active(length(nil)) -> mark(0')
1151.74/293.10	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.10	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.10	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.10	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.10	active(U11(X)) -> U11(active(X))
1151.74/293.10	active(U21(X)) -> U21(active(X))
1151.74/293.10	active(U31(X)) -> U31(active(X))
1151.74/293.10	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.10	active(U42(X)) -> U42(active(X))
1151.74/293.10	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.10	active(U52(X)) -> U52(active(X))
1151.74/293.10	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.10	active(U62(X)) -> U62(active(X))
1151.74/293.10	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.10	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.10	active(s(X)) -> s(active(X))
1151.74/293.10	active(length(X)) -> length(active(X))
1151.74/293.10	active(U81(X)) -> U81(active(X))
1151.74/293.10	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.10	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.10	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.10	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.10	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.10	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.10	U11(mark(X)) -> mark(U11(X))
1151.74/293.10	U21(mark(X)) -> mark(U21(X))
1151.74/293.10	U31(mark(X)) -> mark(U31(X))
1151.74/293.10	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.10	U42(mark(X)) -> mark(U42(X))
1151.74/293.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.10	U52(mark(X)) -> mark(U52(X))
1151.74/293.10	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.10	U62(mark(X)) -> mark(U62(X))
1151.74/293.10	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.10	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.10	s(mark(X)) -> mark(s(X))
1151.74/293.10	length(mark(X)) -> mark(length(X))
1151.74/293.10	U81(mark(X)) -> mark(U81(X))
1151.74/293.10	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.10	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.10	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.10	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.10	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.10	proper(zeros) -> ok(zeros)
1151.74/293.10	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.10	proper(0') -> ok(0')
1151.74/293.10	proper(U11(X)) -> U11(proper(X))
1151.74/293.10	proper(tt) -> ok(tt)
1151.74/293.10	proper(U21(X)) -> U21(proper(X))
1151.74/293.10	proper(U31(X)) -> U31(proper(X))
1151.74/293.10	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.10	proper(U42(X)) -> U42(proper(X))
1151.74/293.10	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.10	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.10	proper(U52(X)) -> U52(proper(X))
1151.74/293.10	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.10	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.10	proper(U62(X)) -> U62(proper(X))
1151.74/293.10	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.10	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.10	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.10	proper(s(X)) -> s(proper(X))
1151.74/293.10	proper(length(X)) -> length(proper(X))
1151.74/293.10	proper(U81(X)) -> U81(proper(X))
1151.74/293.10	proper(nil) -> ok(nil)
1151.74/293.10	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.10	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.10	U11(ok(X)) -> ok(U11(X))
1151.74/293.10	U21(ok(X)) -> ok(U21(X))
1151.74/293.10	U31(ok(X)) -> ok(U31(X))
1151.74/293.10	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.10	U42(ok(X)) -> ok(U42(X))
1151.74/293.10	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.10	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.10	U52(ok(X)) -> ok(U52(X))
1151.74/293.10	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.10	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.10	U62(ok(X)) -> ok(U62(X))
1151.74/293.10	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.10	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.10	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.10	s(ok(X)) -> ok(s(X))
1151.74/293.10	length(ok(X)) -> ok(length(X))
1151.74/293.10	U81(ok(X)) -> ok(U81(X))
1151.74/293.10	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.10	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.10	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.10	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.10	top(mark(X)) -> top(proper(X))
1151.74/293.10	top(ok(X)) -> top(active(X))
1151.74/293.10	
1151.74/293.10	Types:
1151.74/293.10	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.10	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.10	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.10	hole_top2_0 :: top
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	Lemmas:
1151.74/293.10	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.10	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.10	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.10	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.74/293.10	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	Generator Equations:
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	The following defined symbols remain to be analysed:
1151.74/293.10	isNat, active, s, length, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.10	
1151.74/293.10	They will be analysed ascendingly in the following order:
1151.74/293.10	isNat < active
1151.74/293.10	s < active
1151.74/293.10	length < active
1151.74/293.10	U92 < active
1151.74/293.10	U93 < active
1151.74/293.10	take < active
1151.74/293.10	U11 < active
1151.74/293.10	U21 < active
1151.74/293.10	U31 < active
1151.74/293.10	U41 < active
1151.74/293.10	U51 < active
1151.74/293.10	U61 < active
1151.74/293.10	U71 < active
1151.74/293.10	U81 < active
1151.74/293.10	U91 < active
1151.74/293.10	active < top
1151.74/293.10	isNat < proper
1151.74/293.10	s < proper
1151.74/293.10	length < proper
1151.74/293.10	U92 < proper
1151.74/293.10	U93 < proper
1151.74/293.10	take < proper
1151.74/293.10	U11 < proper
1151.74/293.10	U21 < proper
1151.74/293.10	U31 < proper
1151.74/293.10	U41 < proper
1151.74/293.10	U51 < proper
1151.74/293.10	U61 < proper
1151.74/293.10	U71 < proper
1151.74/293.10	U81 < proper
1151.74/293.10	U91 < proper
1151.74/293.10	proper < top
1151.74/293.10	
1151.74/293.10	----------------------------------------
1151.74/293.10	
1151.74/293.10	(21) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.10	Proved the following rewrite lemma:
1151.74/293.10	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.74/293.10	
1151.74/293.10	Induction Base:
1151.74/293.10	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))
1151.74/293.10	
1151.74/293.10	Induction Step:
1151.74/293.10	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n7812_0, 1)))) ->_R^Omega(1)
1151.74/293.10	mark(s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0)))) ->_IH
1151.74/293.10	mark(*4_0)
1151.74/293.10	
1151.74/293.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.10	----------------------------------------
1151.74/293.10	
1151.74/293.10	(22)
1151.74/293.10	Obligation:
1151.74/293.10	TRS:
1151.74/293.10	Rules:
1151.74/293.10	active(zeros) -> mark(cons(0', zeros))
1151.74/293.10	active(U11(tt)) -> mark(tt)
1151.74/293.10	active(U21(tt)) -> mark(tt)
1151.74/293.10	active(U31(tt)) -> mark(tt)
1151.74/293.10	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.10	active(U42(tt)) -> mark(tt)
1151.74/293.10	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.10	active(U52(tt)) -> mark(tt)
1151.74/293.10	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.10	active(U62(tt)) -> mark(tt)
1151.74/293.10	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.10	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.10	active(U81(tt)) -> mark(nil)
1151.74/293.10	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.10	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.10	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.10	active(isNat(0')) -> mark(tt)
1151.74/293.10	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.10	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.10	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.10	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.10	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.10	active(isNatList(nil)) -> mark(tt)
1151.74/293.10	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.10	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.10	active(length(nil)) -> mark(0')
1151.74/293.10	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.10	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.10	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.10	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.10	active(U11(X)) -> U11(active(X))
1151.74/293.10	active(U21(X)) -> U21(active(X))
1151.74/293.10	active(U31(X)) -> U31(active(X))
1151.74/293.10	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.10	active(U42(X)) -> U42(active(X))
1151.74/293.10	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.10	active(U52(X)) -> U52(active(X))
1151.74/293.10	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.10	active(U62(X)) -> U62(active(X))
1151.74/293.10	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.10	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.10	active(s(X)) -> s(active(X))
1151.74/293.10	active(length(X)) -> length(active(X))
1151.74/293.10	active(U81(X)) -> U81(active(X))
1151.74/293.10	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.10	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.10	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.10	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.10	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.10	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.10	U11(mark(X)) -> mark(U11(X))
1151.74/293.10	U21(mark(X)) -> mark(U21(X))
1151.74/293.10	U31(mark(X)) -> mark(U31(X))
1151.74/293.10	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.10	U42(mark(X)) -> mark(U42(X))
1151.74/293.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.10	U52(mark(X)) -> mark(U52(X))
1151.74/293.10	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.10	U62(mark(X)) -> mark(U62(X))
1151.74/293.10	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.10	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.10	s(mark(X)) -> mark(s(X))
1151.74/293.10	length(mark(X)) -> mark(length(X))
1151.74/293.10	U81(mark(X)) -> mark(U81(X))
1151.74/293.10	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.10	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.10	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.10	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.10	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.10	proper(zeros) -> ok(zeros)
1151.74/293.10	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.10	proper(0') -> ok(0')
1151.74/293.10	proper(U11(X)) -> U11(proper(X))
1151.74/293.10	proper(tt) -> ok(tt)
1151.74/293.10	proper(U21(X)) -> U21(proper(X))
1151.74/293.10	proper(U31(X)) -> U31(proper(X))
1151.74/293.10	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.10	proper(U42(X)) -> U42(proper(X))
1151.74/293.10	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.10	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.10	proper(U52(X)) -> U52(proper(X))
1151.74/293.10	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.10	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.10	proper(U62(X)) -> U62(proper(X))
1151.74/293.10	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.10	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.10	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.10	proper(s(X)) -> s(proper(X))
1151.74/293.10	proper(length(X)) -> length(proper(X))
1151.74/293.10	proper(U81(X)) -> U81(proper(X))
1151.74/293.10	proper(nil) -> ok(nil)
1151.74/293.10	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.10	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.10	U11(ok(X)) -> ok(U11(X))
1151.74/293.10	U21(ok(X)) -> ok(U21(X))
1151.74/293.10	U31(ok(X)) -> ok(U31(X))
1151.74/293.10	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.10	U42(ok(X)) -> ok(U42(X))
1151.74/293.10	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.10	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.10	U52(ok(X)) -> ok(U52(X))
1151.74/293.10	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.10	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.10	U62(ok(X)) -> ok(U62(X))
1151.74/293.10	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.10	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.10	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.10	s(ok(X)) -> ok(s(X))
1151.74/293.10	length(ok(X)) -> ok(length(X))
1151.74/293.10	U81(ok(X)) -> ok(U81(X))
1151.74/293.10	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.10	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.10	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.10	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.10	top(mark(X)) -> top(proper(X))
1151.74/293.10	top(ok(X)) -> top(active(X))
1151.74/293.10	
1151.74/293.10	Types:
1151.74/293.10	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.10	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.10	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.10	hole_top2_0 :: top
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	Lemmas:
1151.74/293.10	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.10	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.10	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.10	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.74/293.10	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.74/293.10	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	Generator Equations:
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	The following defined symbols remain to be analysed:
1151.74/293.10	length, active, U92, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.10	
1151.74/293.10	They will be analysed ascendingly in the following order:
1151.74/293.10	length < active
1151.74/293.10	U92 < active
1151.74/293.10	U93 < active
1151.74/293.10	take < active
1151.74/293.10	U11 < active
1151.74/293.10	U21 < active
1151.74/293.10	U31 < active
1151.74/293.10	U41 < active
1151.74/293.10	U51 < active
1151.74/293.10	U61 < active
1151.74/293.10	U71 < active
1151.74/293.10	U81 < active
1151.74/293.10	U91 < active
1151.74/293.10	active < top
1151.74/293.10	length < proper
1151.74/293.10	U92 < proper
1151.74/293.10	U93 < proper
1151.74/293.10	take < proper
1151.74/293.10	U11 < proper
1151.74/293.10	U21 < proper
1151.74/293.10	U31 < proper
1151.74/293.10	U41 < proper
1151.74/293.10	U51 < proper
1151.74/293.10	U61 < proper
1151.74/293.10	U71 < proper
1151.74/293.10	U81 < proper
1151.74/293.10	U91 < proper
1151.74/293.10	proper < top
1151.74/293.10	
1151.74/293.10	----------------------------------------
1151.74/293.10	
1151.74/293.10	(23) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.10	Proved the following rewrite lemma:
1151.74/293.10	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.74/293.10	
1151.74/293.10	Induction Base:
1151.74/293.10	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))
1151.74/293.10	
1151.74/293.10	Induction Step:
1151.74/293.10	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n9039_0, 1)))) ->_R^Omega(1)
1151.74/293.10	mark(length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0)))) ->_IH
1151.74/293.10	mark(*4_0)
1151.74/293.10	
1151.74/293.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.10	----------------------------------------
1151.74/293.10	
1151.74/293.10	(24)
1151.74/293.10	Obligation:
1151.74/293.10	TRS:
1151.74/293.10	Rules:
1151.74/293.10	active(zeros) -> mark(cons(0', zeros))
1151.74/293.10	active(U11(tt)) -> mark(tt)
1151.74/293.10	active(U21(tt)) -> mark(tt)
1151.74/293.10	active(U31(tt)) -> mark(tt)
1151.74/293.10	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.10	active(U42(tt)) -> mark(tt)
1151.74/293.10	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.10	active(U52(tt)) -> mark(tt)
1151.74/293.10	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.10	active(U62(tt)) -> mark(tt)
1151.74/293.10	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.10	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.10	active(U81(tt)) -> mark(nil)
1151.74/293.10	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.10	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.10	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.10	active(isNat(0')) -> mark(tt)
1151.74/293.10	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.10	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.10	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.10	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.10	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.10	active(isNatList(nil)) -> mark(tt)
1151.74/293.10	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.10	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.10	active(length(nil)) -> mark(0')
1151.74/293.10	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.10	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.10	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.10	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.10	active(U11(X)) -> U11(active(X))
1151.74/293.10	active(U21(X)) -> U21(active(X))
1151.74/293.10	active(U31(X)) -> U31(active(X))
1151.74/293.10	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.10	active(U42(X)) -> U42(active(X))
1151.74/293.10	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.10	active(U52(X)) -> U52(active(X))
1151.74/293.10	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.10	active(U62(X)) -> U62(active(X))
1151.74/293.10	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.10	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.10	active(s(X)) -> s(active(X))
1151.74/293.10	active(length(X)) -> length(active(X))
1151.74/293.10	active(U81(X)) -> U81(active(X))
1151.74/293.10	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.10	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.10	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.10	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.10	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.10	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.10	U11(mark(X)) -> mark(U11(X))
1151.74/293.10	U21(mark(X)) -> mark(U21(X))
1151.74/293.10	U31(mark(X)) -> mark(U31(X))
1151.74/293.10	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.10	U42(mark(X)) -> mark(U42(X))
1151.74/293.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.10	U52(mark(X)) -> mark(U52(X))
1151.74/293.10	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.10	U62(mark(X)) -> mark(U62(X))
1151.74/293.10	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.10	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.10	s(mark(X)) -> mark(s(X))
1151.74/293.10	length(mark(X)) -> mark(length(X))
1151.74/293.10	U81(mark(X)) -> mark(U81(X))
1151.74/293.10	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.10	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.10	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.10	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.10	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.10	proper(zeros) -> ok(zeros)
1151.74/293.10	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.10	proper(0') -> ok(0')
1151.74/293.10	proper(U11(X)) -> U11(proper(X))
1151.74/293.10	proper(tt) -> ok(tt)
1151.74/293.10	proper(U21(X)) -> U21(proper(X))
1151.74/293.10	proper(U31(X)) -> U31(proper(X))
1151.74/293.10	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.10	proper(U42(X)) -> U42(proper(X))
1151.74/293.10	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.10	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.10	proper(U52(X)) -> U52(proper(X))
1151.74/293.10	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.10	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.10	proper(U62(X)) -> U62(proper(X))
1151.74/293.10	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.10	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.10	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.10	proper(s(X)) -> s(proper(X))
1151.74/293.10	proper(length(X)) -> length(proper(X))
1151.74/293.10	proper(U81(X)) -> U81(proper(X))
1151.74/293.10	proper(nil) -> ok(nil)
1151.74/293.10	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.10	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.10	U11(ok(X)) -> ok(U11(X))
1151.74/293.10	U21(ok(X)) -> ok(U21(X))
1151.74/293.10	U31(ok(X)) -> ok(U31(X))
1151.74/293.10	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.10	U42(ok(X)) -> ok(U42(X))
1151.74/293.10	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.10	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.10	U52(ok(X)) -> ok(U52(X))
1151.74/293.10	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.10	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.10	U62(ok(X)) -> ok(U62(X))
1151.74/293.10	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.10	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.10	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.10	s(ok(X)) -> ok(s(X))
1151.74/293.10	length(ok(X)) -> ok(length(X))
1151.74/293.10	U81(ok(X)) -> ok(U81(X))
1151.74/293.10	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.10	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.10	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.10	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.10	top(mark(X)) -> top(proper(X))
1151.74/293.10	top(ok(X)) -> top(active(X))
1151.74/293.10	
1151.74/293.10	Types:
1151.74/293.10	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.10	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.10	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.10	hole_top2_0 :: top
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	Lemmas:
1151.74/293.10	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.10	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.10	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.10	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.74/293.10	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.74/293.10	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.74/293.10	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	Generator Equations:
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	The following defined symbols remain to be analysed:
1151.74/293.10	U92, active, U93, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.10	
1151.74/293.10	They will be analysed ascendingly in the following order:
1151.74/293.10	U92 < active
1151.74/293.10	U93 < active
1151.74/293.10	take < active
1151.74/293.10	U11 < active
1151.74/293.10	U21 < active
1151.74/293.10	U31 < active
1151.74/293.10	U41 < active
1151.74/293.10	U51 < active
1151.74/293.10	U61 < active
1151.74/293.10	U71 < active
1151.74/293.10	U81 < active
1151.74/293.10	U91 < active
1151.74/293.10	active < top
1151.74/293.10	U92 < proper
1151.74/293.10	U93 < proper
1151.74/293.10	take < proper
1151.74/293.10	U11 < proper
1151.74/293.10	U21 < proper
1151.74/293.10	U31 < proper
1151.74/293.10	U41 < proper
1151.74/293.10	U51 < proper
1151.74/293.10	U61 < proper
1151.74/293.10	U71 < proper
1151.74/293.10	U81 < proper
1151.74/293.10	U91 < proper
1151.74/293.10	proper < top
1151.74/293.10	
1151.74/293.10	----------------------------------------
1151.74/293.10	
1151.74/293.10	(25) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.10	Proved the following rewrite lemma:
1151.74/293.10	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.74/293.10	
1151.74/293.10	Induction Base:
1151.74/293.10	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))
1151.74/293.10	
1151.74/293.10	Induction Step:
1151.74/293.10	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n10367_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) ->_R^Omega(1)
1151.74/293.10	mark(U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))) ->_IH
1151.74/293.10	mark(*4_0)
1151.74/293.10	
1151.74/293.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.10	----------------------------------------
1151.74/293.10	
1151.74/293.10	(26)
1151.74/293.10	Obligation:
1151.74/293.10	TRS:
1151.74/293.10	Rules:
1151.74/293.10	active(zeros) -> mark(cons(0', zeros))
1151.74/293.10	active(U11(tt)) -> mark(tt)
1151.74/293.10	active(U21(tt)) -> mark(tt)
1151.74/293.10	active(U31(tt)) -> mark(tt)
1151.74/293.10	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.10	active(U42(tt)) -> mark(tt)
1151.74/293.10	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.10	active(U52(tt)) -> mark(tt)
1151.74/293.10	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.10	active(U62(tt)) -> mark(tt)
1151.74/293.10	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.10	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.10	active(U81(tt)) -> mark(nil)
1151.74/293.10	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.10	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.10	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.10	active(isNat(0')) -> mark(tt)
1151.74/293.10	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.10	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.10	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.10	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.10	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.10	active(isNatList(nil)) -> mark(tt)
1151.74/293.10	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.10	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.10	active(length(nil)) -> mark(0')
1151.74/293.10	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.10	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.10	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.10	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.10	active(U11(X)) -> U11(active(X))
1151.74/293.10	active(U21(X)) -> U21(active(X))
1151.74/293.10	active(U31(X)) -> U31(active(X))
1151.74/293.10	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.10	active(U42(X)) -> U42(active(X))
1151.74/293.10	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.10	active(U52(X)) -> U52(active(X))
1151.74/293.10	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.10	active(U62(X)) -> U62(active(X))
1151.74/293.10	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.10	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.10	active(s(X)) -> s(active(X))
1151.74/293.10	active(length(X)) -> length(active(X))
1151.74/293.10	active(U81(X)) -> U81(active(X))
1151.74/293.10	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.10	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.10	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.10	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.10	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.10	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.10	U11(mark(X)) -> mark(U11(X))
1151.74/293.10	U21(mark(X)) -> mark(U21(X))
1151.74/293.10	U31(mark(X)) -> mark(U31(X))
1151.74/293.10	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.10	U42(mark(X)) -> mark(U42(X))
1151.74/293.10	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.10	U52(mark(X)) -> mark(U52(X))
1151.74/293.10	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.10	U62(mark(X)) -> mark(U62(X))
1151.74/293.10	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.10	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.10	s(mark(X)) -> mark(s(X))
1151.74/293.10	length(mark(X)) -> mark(length(X))
1151.74/293.10	U81(mark(X)) -> mark(U81(X))
1151.74/293.10	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.10	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.10	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.10	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.10	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.10	proper(zeros) -> ok(zeros)
1151.74/293.10	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.10	proper(0') -> ok(0')
1151.74/293.10	proper(U11(X)) -> U11(proper(X))
1151.74/293.10	proper(tt) -> ok(tt)
1151.74/293.10	proper(U21(X)) -> U21(proper(X))
1151.74/293.10	proper(U31(X)) -> U31(proper(X))
1151.74/293.10	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.10	proper(U42(X)) -> U42(proper(X))
1151.74/293.10	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.10	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.10	proper(U52(X)) -> U52(proper(X))
1151.74/293.10	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.10	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.10	proper(U62(X)) -> U62(proper(X))
1151.74/293.10	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.10	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.10	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.10	proper(s(X)) -> s(proper(X))
1151.74/293.10	proper(length(X)) -> length(proper(X))
1151.74/293.10	proper(U81(X)) -> U81(proper(X))
1151.74/293.10	proper(nil) -> ok(nil)
1151.74/293.10	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.10	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.10	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.10	U11(ok(X)) -> ok(U11(X))
1151.74/293.10	U21(ok(X)) -> ok(U21(X))
1151.74/293.10	U31(ok(X)) -> ok(U31(X))
1151.74/293.10	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.10	U42(ok(X)) -> ok(U42(X))
1151.74/293.10	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.10	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.10	U52(ok(X)) -> ok(U52(X))
1151.74/293.10	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.10	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.10	U62(ok(X)) -> ok(U62(X))
1151.74/293.10	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.10	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.10	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.10	s(ok(X)) -> ok(s(X))
1151.74/293.10	length(ok(X)) -> ok(length(X))
1151.74/293.10	U81(ok(X)) -> ok(U81(X))
1151.74/293.10	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.10	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.10	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.10	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.10	top(mark(X)) -> top(proper(X))
1151.74/293.10	top(ok(X)) -> top(active(X))
1151.74/293.10	
1151.74/293.10	Types:
1151.74/293.10	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.10	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.10	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.10	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.10	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.10	hole_top2_0 :: top
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	Lemmas:
1151.74/293.10	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.10	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.10	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.10	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.74/293.10	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.74/293.10	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.74/293.10	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.74/293.10	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	Generator Equations:
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.10	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.10	
1151.74/293.10	
1151.74/293.10	The following defined symbols remain to be analysed:
1151.74/293.10	U93, active, take, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.10	
1151.74/293.10	They will be analysed ascendingly in the following order:
1151.74/293.10	U93 < active
1151.74/293.10	take < active
1151.74/293.10	U11 < active
1151.74/293.10	U21 < active
1151.74/293.10	U31 < active
1151.74/293.10	U41 < active
1151.74/293.10	U51 < active
1151.74/293.10	U61 < active
1151.74/293.10	U71 < active
1151.74/293.10	U81 < active
1151.74/293.10	U91 < active
1151.74/293.10	active < top
1151.74/293.10	U93 < proper
1151.74/293.10	take < proper
1151.74/293.10	U11 < proper
1151.74/293.10	U21 < proper
1151.74/293.10	U31 < proper
1151.74/293.10	U41 < proper
1151.74/293.10	U51 < proper
1151.74/293.10	U61 < proper
1151.74/293.10	U71 < proper
1151.74/293.10	U81 < proper
1151.74/293.10	U91 < proper
1151.74/293.10	proper < top
1151.74/293.10	
1151.74/293.10	----------------------------------------
1151.74/293.10	
1151.74/293.10	(27) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.10	Proved the following rewrite lemma:
1151.74/293.10	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.74/293.10	
1151.74/293.10	Induction Base:
1151.74/293.10	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))
1151.74/293.10	
1151.74/293.10	Induction Step:
1151.74/293.10	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n18768_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) ->_R^Omega(1)
1151.74/293.10	mark(U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))) ->_IH
1151.74/293.10	mark(*4_0)
1151.74/293.10	
1151.74/293.10	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.10	----------------------------------------
1151.74/293.10	
1151.74/293.10	(28)
1151.74/293.10	Obligation:
1151.74/293.10	TRS:
1151.74/293.10	Rules:
1151.74/293.10	active(zeros) -> mark(cons(0', zeros))
1151.74/293.10	active(U11(tt)) -> mark(tt)
1151.74/293.10	active(U21(tt)) -> mark(tt)
1151.74/293.10	active(U31(tt)) -> mark(tt)
1151.74/293.10	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.10	active(U42(tt)) -> mark(tt)
1151.74/293.10	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.11	active(U52(tt)) -> mark(tt)
1151.74/293.11	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.11	active(U62(tt)) -> mark(tt)
1151.74/293.11	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.11	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.11	active(U81(tt)) -> mark(nil)
1151.74/293.11	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.11	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.11	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.11	active(isNat(0')) -> mark(tt)
1151.74/293.11	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.11	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.11	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.11	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.11	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.11	active(isNatList(nil)) -> mark(tt)
1151.74/293.11	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.11	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.11	active(length(nil)) -> mark(0')
1151.74/293.11	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.11	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.11	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.11	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.11	active(U11(X)) -> U11(active(X))
1151.74/293.11	active(U21(X)) -> U21(active(X))
1151.74/293.11	active(U31(X)) -> U31(active(X))
1151.74/293.11	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.11	active(U42(X)) -> U42(active(X))
1151.74/293.11	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.11	active(U52(X)) -> U52(active(X))
1151.74/293.11	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.11	active(U62(X)) -> U62(active(X))
1151.74/293.11	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.11	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.11	active(s(X)) -> s(active(X))
1151.74/293.11	active(length(X)) -> length(active(X))
1151.74/293.11	active(U81(X)) -> U81(active(X))
1151.74/293.11	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.11	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.11	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.11	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.11	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.11	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.11	U11(mark(X)) -> mark(U11(X))
1151.74/293.11	U21(mark(X)) -> mark(U21(X))
1151.74/293.11	U31(mark(X)) -> mark(U31(X))
1151.74/293.11	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.11	U42(mark(X)) -> mark(U42(X))
1151.74/293.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.11	U52(mark(X)) -> mark(U52(X))
1151.74/293.11	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.11	U62(mark(X)) -> mark(U62(X))
1151.74/293.11	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.11	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.11	s(mark(X)) -> mark(s(X))
1151.74/293.11	length(mark(X)) -> mark(length(X))
1151.74/293.11	U81(mark(X)) -> mark(U81(X))
1151.74/293.11	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.11	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.11	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.11	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.11	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.11	proper(zeros) -> ok(zeros)
1151.74/293.11	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.11	proper(0') -> ok(0')
1151.74/293.11	proper(U11(X)) -> U11(proper(X))
1151.74/293.11	proper(tt) -> ok(tt)
1151.74/293.11	proper(U21(X)) -> U21(proper(X))
1151.74/293.11	proper(U31(X)) -> U31(proper(X))
1151.74/293.11	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.11	proper(U42(X)) -> U42(proper(X))
1151.74/293.11	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.11	proper(U52(X)) -> U52(proper(X))
1151.74/293.11	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.11	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.11	proper(U62(X)) -> U62(proper(X))
1151.74/293.11	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.11	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.11	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.11	proper(s(X)) -> s(proper(X))
1151.74/293.11	proper(length(X)) -> length(proper(X))
1151.74/293.11	proper(U81(X)) -> U81(proper(X))
1151.74/293.11	proper(nil) -> ok(nil)
1151.74/293.11	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.11	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.11	U11(ok(X)) -> ok(U11(X))
1151.74/293.11	U21(ok(X)) -> ok(U21(X))
1151.74/293.11	U31(ok(X)) -> ok(U31(X))
1151.74/293.11	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.11	U42(ok(X)) -> ok(U42(X))
1151.74/293.11	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.11	U52(ok(X)) -> ok(U52(X))
1151.74/293.11	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.11	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.11	U62(ok(X)) -> ok(U62(X))
1151.74/293.11	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.11	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.11	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.11	s(ok(X)) -> ok(s(X))
1151.74/293.11	length(ok(X)) -> ok(length(X))
1151.74/293.11	U81(ok(X)) -> ok(U81(X))
1151.74/293.11	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.11	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.11	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.11	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.11	top(mark(X)) -> top(proper(X))
1151.74/293.11	top(ok(X)) -> top(active(X))
1151.74/293.11	
1151.74/293.11	Types:
1151.74/293.11	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.11	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.11	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.11	hole_top2_0 :: top
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	Lemmas:
1151.74/293.11	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.11	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.11	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.11	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.74/293.11	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.74/293.11	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.74/293.11	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.74/293.11	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.74/293.11	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	Generator Equations:
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	The following defined symbols remain to be analysed:
1151.74/293.11	take, active, U11, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.11	
1151.74/293.11	They will be analysed ascendingly in the following order:
1151.74/293.11	take < active
1151.74/293.11	U11 < active
1151.74/293.11	U21 < active
1151.74/293.11	U31 < active
1151.74/293.11	U41 < active
1151.74/293.11	U51 < active
1151.74/293.11	U61 < active
1151.74/293.11	U71 < active
1151.74/293.11	U81 < active
1151.74/293.11	U91 < active
1151.74/293.11	active < top
1151.74/293.11	take < proper
1151.74/293.11	U11 < proper
1151.74/293.11	U21 < proper
1151.74/293.11	U31 < proper
1151.74/293.11	U41 < proper
1151.74/293.11	U51 < proper
1151.74/293.11	U61 < proper
1151.74/293.11	U71 < proper
1151.74/293.11	U81 < proper
1151.74/293.11	U91 < proper
1151.74/293.11	proper < top
1151.74/293.11	
1151.74/293.11	----------------------------------------
1151.74/293.11	
1151.74/293.11	(29) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.11	Proved the following rewrite lemma:
1151.74/293.11	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0)
1151.74/293.11	
1151.74/293.11	Induction Base:
1151.74/293.11	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))
1151.74/293.11	
1151.74/293.11	Induction Step:
1151.74/293.11	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n28179_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
1151.74/293.11	mark(take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
1151.74/293.11	mark(*4_0)
1151.74/293.11	
1151.74/293.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.11	----------------------------------------
1151.74/293.11	
1151.74/293.11	(30)
1151.74/293.11	Obligation:
1151.74/293.11	TRS:
1151.74/293.11	Rules:
1151.74/293.11	active(zeros) -> mark(cons(0', zeros))
1151.74/293.11	active(U11(tt)) -> mark(tt)
1151.74/293.11	active(U21(tt)) -> mark(tt)
1151.74/293.11	active(U31(tt)) -> mark(tt)
1151.74/293.11	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.11	active(U42(tt)) -> mark(tt)
1151.74/293.11	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.11	active(U52(tt)) -> mark(tt)
1151.74/293.11	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.11	active(U62(tt)) -> mark(tt)
1151.74/293.11	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.11	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.11	active(U81(tt)) -> mark(nil)
1151.74/293.11	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.11	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.11	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.11	active(isNat(0')) -> mark(tt)
1151.74/293.11	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.11	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.11	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.11	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.11	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.11	active(isNatList(nil)) -> mark(tt)
1151.74/293.11	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.11	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.11	active(length(nil)) -> mark(0')
1151.74/293.11	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.11	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.11	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.11	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.11	active(U11(X)) -> U11(active(X))
1151.74/293.11	active(U21(X)) -> U21(active(X))
1151.74/293.11	active(U31(X)) -> U31(active(X))
1151.74/293.11	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.11	active(U42(X)) -> U42(active(X))
1151.74/293.11	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.11	active(U52(X)) -> U52(active(X))
1151.74/293.11	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.11	active(U62(X)) -> U62(active(X))
1151.74/293.11	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.11	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.11	active(s(X)) -> s(active(X))
1151.74/293.11	active(length(X)) -> length(active(X))
1151.74/293.11	active(U81(X)) -> U81(active(X))
1151.74/293.11	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.11	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.11	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.11	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.11	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.11	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.11	U11(mark(X)) -> mark(U11(X))
1151.74/293.11	U21(mark(X)) -> mark(U21(X))
1151.74/293.11	U31(mark(X)) -> mark(U31(X))
1151.74/293.11	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.11	U42(mark(X)) -> mark(U42(X))
1151.74/293.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.11	U52(mark(X)) -> mark(U52(X))
1151.74/293.11	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.11	U62(mark(X)) -> mark(U62(X))
1151.74/293.11	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.11	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.11	s(mark(X)) -> mark(s(X))
1151.74/293.11	length(mark(X)) -> mark(length(X))
1151.74/293.11	U81(mark(X)) -> mark(U81(X))
1151.74/293.11	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.11	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.11	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.11	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.11	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.11	proper(zeros) -> ok(zeros)
1151.74/293.11	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.11	proper(0') -> ok(0')
1151.74/293.11	proper(U11(X)) -> U11(proper(X))
1151.74/293.11	proper(tt) -> ok(tt)
1151.74/293.11	proper(U21(X)) -> U21(proper(X))
1151.74/293.11	proper(U31(X)) -> U31(proper(X))
1151.74/293.11	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.11	proper(U42(X)) -> U42(proper(X))
1151.74/293.11	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.11	proper(U52(X)) -> U52(proper(X))
1151.74/293.11	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.11	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.11	proper(U62(X)) -> U62(proper(X))
1151.74/293.11	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.11	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.11	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.11	proper(s(X)) -> s(proper(X))
1151.74/293.11	proper(length(X)) -> length(proper(X))
1151.74/293.11	proper(U81(X)) -> U81(proper(X))
1151.74/293.11	proper(nil) -> ok(nil)
1151.74/293.11	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.11	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.11	U11(ok(X)) -> ok(U11(X))
1151.74/293.11	U21(ok(X)) -> ok(U21(X))
1151.74/293.11	U31(ok(X)) -> ok(U31(X))
1151.74/293.11	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.11	U42(ok(X)) -> ok(U42(X))
1151.74/293.11	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.11	U52(ok(X)) -> ok(U52(X))
1151.74/293.11	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.11	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.11	U62(ok(X)) -> ok(U62(X))
1151.74/293.11	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.11	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.11	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.11	s(ok(X)) -> ok(s(X))
1151.74/293.11	length(ok(X)) -> ok(length(X))
1151.74/293.11	U81(ok(X)) -> ok(U81(X))
1151.74/293.11	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.11	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.11	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.11	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.11	top(mark(X)) -> top(proper(X))
1151.74/293.11	top(ok(X)) -> top(active(X))
1151.74/293.11	
1151.74/293.11	Types:
1151.74/293.11	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.11	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.11	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.11	hole_top2_0 :: top
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	Lemmas:
1151.74/293.11	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.11	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.11	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.11	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.74/293.11	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.74/293.11	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.74/293.11	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.74/293.11	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.74/293.11	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.74/293.11	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0)
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	Generator Equations:
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	The following defined symbols remain to be analysed:
1151.74/293.11	U11, active, U21, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.11	
1151.74/293.11	They will be analysed ascendingly in the following order:
1151.74/293.11	U11 < active
1151.74/293.11	U21 < active
1151.74/293.11	U31 < active
1151.74/293.11	U41 < active
1151.74/293.11	U51 < active
1151.74/293.11	U61 < active
1151.74/293.11	U71 < active
1151.74/293.11	U81 < active
1151.74/293.11	U91 < active
1151.74/293.11	active < top
1151.74/293.11	U11 < proper
1151.74/293.11	U21 < proper
1151.74/293.11	U31 < proper
1151.74/293.11	U41 < proper
1151.74/293.11	U51 < proper
1151.74/293.11	U61 < proper
1151.74/293.11	U71 < proper
1151.74/293.11	U81 < proper
1151.74/293.11	U91 < proper
1151.74/293.11	proper < top
1151.74/293.11	
1151.74/293.11	----------------------------------------
1151.74/293.11	
1151.74/293.11	(31) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.11	Proved the following rewrite lemma:
1151.74/293.11	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0)
1151.74/293.11	
1151.74/293.11	Induction Base:
1151.74/293.11	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))
1151.74/293.11	
1151.74/293.11	Induction Step:
1151.74/293.11	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n33135_0, 1)))) ->_R^Omega(1)
1151.74/293.11	mark(U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0)))) ->_IH
1151.74/293.11	mark(*4_0)
1151.74/293.11	
1151.74/293.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.11	----------------------------------------
1151.74/293.11	
1151.74/293.11	(32)
1151.74/293.11	Obligation:
1151.74/293.11	TRS:
1151.74/293.11	Rules:
1151.74/293.11	active(zeros) -> mark(cons(0', zeros))
1151.74/293.11	active(U11(tt)) -> mark(tt)
1151.74/293.11	active(U21(tt)) -> mark(tt)
1151.74/293.11	active(U31(tt)) -> mark(tt)
1151.74/293.11	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.11	active(U42(tt)) -> mark(tt)
1151.74/293.11	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.11	active(U52(tt)) -> mark(tt)
1151.74/293.11	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.11	active(U62(tt)) -> mark(tt)
1151.74/293.11	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.11	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.11	active(U81(tt)) -> mark(nil)
1151.74/293.11	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.11	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.11	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.11	active(isNat(0')) -> mark(tt)
1151.74/293.11	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.11	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.11	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.11	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.11	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.11	active(isNatList(nil)) -> mark(tt)
1151.74/293.11	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.11	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.11	active(length(nil)) -> mark(0')
1151.74/293.11	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.11	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.11	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.11	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.11	active(U11(X)) -> U11(active(X))
1151.74/293.11	active(U21(X)) -> U21(active(X))
1151.74/293.11	active(U31(X)) -> U31(active(X))
1151.74/293.11	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.11	active(U42(X)) -> U42(active(X))
1151.74/293.11	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.11	active(U52(X)) -> U52(active(X))
1151.74/293.11	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.11	active(U62(X)) -> U62(active(X))
1151.74/293.11	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.11	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.11	active(s(X)) -> s(active(X))
1151.74/293.11	active(length(X)) -> length(active(X))
1151.74/293.11	active(U81(X)) -> U81(active(X))
1151.74/293.11	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.11	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.11	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.11	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.11	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.11	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.11	U11(mark(X)) -> mark(U11(X))
1151.74/293.11	U21(mark(X)) -> mark(U21(X))
1151.74/293.11	U31(mark(X)) -> mark(U31(X))
1151.74/293.11	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.11	U42(mark(X)) -> mark(U42(X))
1151.74/293.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.11	U52(mark(X)) -> mark(U52(X))
1151.74/293.11	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.11	U62(mark(X)) -> mark(U62(X))
1151.74/293.11	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.11	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.11	s(mark(X)) -> mark(s(X))
1151.74/293.11	length(mark(X)) -> mark(length(X))
1151.74/293.11	U81(mark(X)) -> mark(U81(X))
1151.74/293.11	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.11	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.11	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.11	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.11	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.11	proper(zeros) -> ok(zeros)
1151.74/293.11	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.11	proper(0') -> ok(0')
1151.74/293.11	proper(U11(X)) -> U11(proper(X))
1151.74/293.11	proper(tt) -> ok(tt)
1151.74/293.11	proper(U21(X)) -> U21(proper(X))
1151.74/293.11	proper(U31(X)) -> U31(proper(X))
1151.74/293.11	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.11	proper(U42(X)) -> U42(proper(X))
1151.74/293.11	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.11	proper(U52(X)) -> U52(proper(X))
1151.74/293.11	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.11	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.11	proper(U62(X)) -> U62(proper(X))
1151.74/293.11	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.11	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.11	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.11	proper(s(X)) -> s(proper(X))
1151.74/293.11	proper(length(X)) -> length(proper(X))
1151.74/293.11	proper(U81(X)) -> U81(proper(X))
1151.74/293.11	proper(nil) -> ok(nil)
1151.74/293.11	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.11	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.11	U11(ok(X)) -> ok(U11(X))
1151.74/293.11	U21(ok(X)) -> ok(U21(X))
1151.74/293.11	U31(ok(X)) -> ok(U31(X))
1151.74/293.11	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.11	U42(ok(X)) -> ok(U42(X))
1151.74/293.11	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.11	U52(ok(X)) -> ok(U52(X))
1151.74/293.11	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.11	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.11	U62(ok(X)) -> ok(U62(X))
1151.74/293.11	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.11	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.11	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.11	s(ok(X)) -> ok(s(X))
1151.74/293.11	length(ok(X)) -> ok(length(X))
1151.74/293.11	U81(ok(X)) -> ok(U81(X))
1151.74/293.11	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.11	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.11	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.11	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.11	top(mark(X)) -> top(proper(X))
1151.74/293.11	top(ok(X)) -> top(active(X))
1151.74/293.11	
1151.74/293.11	Types:
1151.74/293.11	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.11	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.11	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.11	hole_top2_0 :: top
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	Lemmas:
1151.74/293.11	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.11	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.11	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.11	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.74/293.11	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.74/293.11	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.74/293.11	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.74/293.11	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.74/293.11	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.74/293.11	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0)
1151.74/293.11	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0)
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	Generator Equations:
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	The following defined symbols remain to be analysed:
1151.74/293.11	U21, active, U31, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.11	
1151.74/293.11	They will be analysed ascendingly in the following order:
1151.74/293.11	U21 < active
1151.74/293.11	U31 < active
1151.74/293.11	U41 < active
1151.74/293.11	U51 < active
1151.74/293.11	U61 < active
1151.74/293.11	U71 < active
1151.74/293.11	U81 < active
1151.74/293.11	U91 < active
1151.74/293.11	active < top
1151.74/293.11	U21 < proper
1151.74/293.11	U31 < proper
1151.74/293.11	U41 < proper
1151.74/293.11	U51 < proper
1151.74/293.11	U61 < proper
1151.74/293.11	U71 < proper
1151.74/293.11	U81 < proper
1151.74/293.11	U91 < proper
1151.74/293.11	proper < top
1151.74/293.11	
1151.74/293.11	----------------------------------------
1151.74/293.11	
1151.74/293.11	(33) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.11	Proved the following rewrite lemma:
1151.74/293.11	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0)
1151.74/293.11	
1151.74/293.11	Induction Base:
1151.74/293.11	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))
1151.74/293.11	
1151.74/293.11	Induction Step:
1151.74/293.11	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n35210_0, 1)))) ->_R^Omega(1)
1151.74/293.11	mark(U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0)))) ->_IH
1151.74/293.11	mark(*4_0)
1151.74/293.11	
1151.74/293.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.11	----------------------------------------
1151.74/293.11	
1151.74/293.11	(34)
1151.74/293.11	Obligation:
1151.74/293.11	TRS:
1151.74/293.11	Rules:
1151.74/293.11	active(zeros) -> mark(cons(0', zeros))
1151.74/293.11	active(U11(tt)) -> mark(tt)
1151.74/293.11	active(U21(tt)) -> mark(tt)
1151.74/293.11	active(U31(tt)) -> mark(tt)
1151.74/293.11	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.11	active(U42(tt)) -> mark(tt)
1151.74/293.11	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.11	active(U52(tt)) -> mark(tt)
1151.74/293.11	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.11	active(U62(tt)) -> mark(tt)
1151.74/293.11	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.11	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.11	active(U81(tt)) -> mark(nil)
1151.74/293.11	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.11	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.11	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.11	active(isNat(0')) -> mark(tt)
1151.74/293.11	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.11	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.11	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.11	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.11	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.11	active(isNatList(nil)) -> mark(tt)
1151.74/293.11	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.11	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.11	active(length(nil)) -> mark(0')
1151.74/293.11	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.11	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.11	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.11	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.11	active(U11(X)) -> U11(active(X))
1151.74/293.11	active(U21(X)) -> U21(active(X))
1151.74/293.11	active(U31(X)) -> U31(active(X))
1151.74/293.11	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.11	active(U42(X)) -> U42(active(X))
1151.74/293.11	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.11	active(U52(X)) -> U52(active(X))
1151.74/293.11	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.11	active(U62(X)) -> U62(active(X))
1151.74/293.11	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.11	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.11	active(s(X)) -> s(active(X))
1151.74/293.11	active(length(X)) -> length(active(X))
1151.74/293.11	active(U81(X)) -> U81(active(X))
1151.74/293.11	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.11	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.11	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.11	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.11	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.11	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.11	U11(mark(X)) -> mark(U11(X))
1151.74/293.11	U21(mark(X)) -> mark(U21(X))
1151.74/293.11	U31(mark(X)) -> mark(U31(X))
1151.74/293.11	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.11	U42(mark(X)) -> mark(U42(X))
1151.74/293.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.11	U52(mark(X)) -> mark(U52(X))
1151.74/293.11	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.11	U62(mark(X)) -> mark(U62(X))
1151.74/293.11	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.11	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.11	s(mark(X)) -> mark(s(X))
1151.74/293.11	length(mark(X)) -> mark(length(X))
1151.74/293.11	U81(mark(X)) -> mark(U81(X))
1151.74/293.11	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.11	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.11	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.11	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.11	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.11	proper(zeros) -> ok(zeros)
1151.74/293.11	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.11	proper(0') -> ok(0')
1151.74/293.11	proper(U11(X)) -> U11(proper(X))
1151.74/293.11	proper(tt) -> ok(tt)
1151.74/293.11	proper(U21(X)) -> U21(proper(X))
1151.74/293.11	proper(U31(X)) -> U31(proper(X))
1151.74/293.11	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.11	proper(U42(X)) -> U42(proper(X))
1151.74/293.11	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.11	proper(U52(X)) -> U52(proper(X))
1151.74/293.11	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.11	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.11	proper(U62(X)) -> U62(proper(X))
1151.74/293.11	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.11	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.11	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.11	proper(s(X)) -> s(proper(X))
1151.74/293.11	proper(length(X)) -> length(proper(X))
1151.74/293.11	proper(U81(X)) -> U81(proper(X))
1151.74/293.11	proper(nil) -> ok(nil)
1151.74/293.11	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.11	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.11	U11(ok(X)) -> ok(U11(X))
1151.74/293.11	U21(ok(X)) -> ok(U21(X))
1151.74/293.11	U31(ok(X)) -> ok(U31(X))
1151.74/293.11	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.11	U42(ok(X)) -> ok(U42(X))
1151.74/293.11	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.11	U52(ok(X)) -> ok(U52(X))
1151.74/293.11	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.11	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.11	U62(ok(X)) -> ok(U62(X))
1151.74/293.11	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.11	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.11	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.11	s(ok(X)) -> ok(s(X))
1151.74/293.11	length(ok(X)) -> ok(length(X))
1151.74/293.11	U81(ok(X)) -> ok(U81(X))
1151.74/293.11	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.11	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.11	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.11	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.11	top(mark(X)) -> top(proper(X))
1151.74/293.11	top(ok(X)) -> top(active(X))
1151.74/293.11	
1151.74/293.11	Types:
1151.74/293.11	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.11	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.11	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.11	hole_top2_0 :: top
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	Lemmas:
1151.74/293.11	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.11	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.11	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.11	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.74/293.11	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.74/293.11	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.74/293.11	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.74/293.11	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.74/293.11	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.74/293.11	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0)
1151.74/293.11	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0)
1151.74/293.11	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0)
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	Generator Equations:
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	The following defined symbols remain to be analysed:
1151.74/293.11	U31, active, U41, U51, U61, U71, U81, U91, proper, top
1151.74/293.11	
1151.74/293.11	They will be analysed ascendingly in the following order:
1151.74/293.11	U31 < active
1151.74/293.11	U41 < active
1151.74/293.11	U51 < active
1151.74/293.11	U61 < active
1151.74/293.11	U71 < active
1151.74/293.11	U81 < active
1151.74/293.11	U91 < active
1151.74/293.11	active < top
1151.74/293.11	U31 < proper
1151.74/293.11	U41 < proper
1151.74/293.11	U51 < proper
1151.74/293.11	U61 < proper
1151.74/293.11	U71 < proper
1151.74/293.11	U81 < proper
1151.74/293.11	U91 < proper
1151.74/293.11	proper < top
1151.74/293.11	
1151.74/293.11	----------------------------------------
1151.74/293.11	
1151.74/293.11	(35) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.11	Proved the following rewrite lemma:
1151.74/293.11	U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0)
1151.74/293.11	
1151.74/293.11	Induction Base:
1151.74/293.11	U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))
1151.74/293.11	
1151.74/293.11	Induction Step:
1151.74/293.11	U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n37386_0, 1)))) ->_R^Omega(1)
1151.74/293.11	mark(U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0)))) ->_IH
1151.74/293.11	mark(*4_0)
1151.74/293.11	
1151.74/293.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.11	----------------------------------------
1151.74/293.11	
1151.74/293.11	(36)
1151.74/293.11	Obligation:
1151.74/293.11	TRS:
1151.74/293.11	Rules:
1151.74/293.11	active(zeros) -> mark(cons(0', zeros))
1151.74/293.11	active(U11(tt)) -> mark(tt)
1151.74/293.11	active(U21(tt)) -> mark(tt)
1151.74/293.11	active(U31(tt)) -> mark(tt)
1151.74/293.11	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.11	active(U42(tt)) -> mark(tt)
1151.74/293.11	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.11	active(U52(tt)) -> mark(tt)
1151.74/293.11	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.11	active(U62(tt)) -> mark(tt)
1151.74/293.11	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.11	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.11	active(U81(tt)) -> mark(nil)
1151.74/293.11	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.11	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.11	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.11	active(isNat(0')) -> mark(tt)
1151.74/293.11	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.11	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.11	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.11	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.11	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.11	active(isNatList(nil)) -> mark(tt)
1151.74/293.11	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.11	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.11	active(length(nil)) -> mark(0')
1151.74/293.11	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.11	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.11	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.11	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.11	active(U11(X)) -> U11(active(X))
1151.74/293.11	active(U21(X)) -> U21(active(X))
1151.74/293.11	active(U31(X)) -> U31(active(X))
1151.74/293.11	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.11	active(U42(X)) -> U42(active(X))
1151.74/293.11	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.11	active(U52(X)) -> U52(active(X))
1151.74/293.11	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.11	active(U62(X)) -> U62(active(X))
1151.74/293.11	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.11	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.11	active(s(X)) -> s(active(X))
1151.74/293.11	active(length(X)) -> length(active(X))
1151.74/293.11	active(U81(X)) -> U81(active(X))
1151.74/293.11	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.11	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.11	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.11	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.11	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.11	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.11	U11(mark(X)) -> mark(U11(X))
1151.74/293.11	U21(mark(X)) -> mark(U21(X))
1151.74/293.11	U31(mark(X)) -> mark(U31(X))
1151.74/293.11	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.11	U42(mark(X)) -> mark(U42(X))
1151.74/293.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.11	U52(mark(X)) -> mark(U52(X))
1151.74/293.11	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.11	U62(mark(X)) -> mark(U62(X))
1151.74/293.11	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.11	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.11	s(mark(X)) -> mark(s(X))
1151.74/293.11	length(mark(X)) -> mark(length(X))
1151.74/293.11	U81(mark(X)) -> mark(U81(X))
1151.74/293.11	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.11	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.11	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.11	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.11	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.11	proper(zeros) -> ok(zeros)
1151.74/293.11	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.11	proper(0') -> ok(0')
1151.74/293.11	proper(U11(X)) -> U11(proper(X))
1151.74/293.11	proper(tt) -> ok(tt)
1151.74/293.11	proper(U21(X)) -> U21(proper(X))
1151.74/293.11	proper(U31(X)) -> U31(proper(X))
1151.74/293.11	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.11	proper(U42(X)) -> U42(proper(X))
1151.74/293.11	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.11	proper(U52(X)) -> U52(proper(X))
1151.74/293.11	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.11	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.74/293.11	proper(U62(X)) -> U62(proper(X))
1151.74/293.11	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.74/293.11	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.74/293.11	proper(isNat(X)) -> isNat(proper(X))
1151.74/293.11	proper(s(X)) -> s(proper(X))
1151.74/293.11	proper(length(X)) -> length(proper(X))
1151.74/293.11	proper(U81(X)) -> U81(proper(X))
1151.74/293.11	proper(nil) -> ok(nil)
1151.74/293.11	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.74/293.11	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.74/293.11	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.74/293.11	U11(ok(X)) -> ok(U11(X))
1151.74/293.11	U21(ok(X)) -> ok(U21(X))
1151.74/293.11	U31(ok(X)) -> ok(U31(X))
1151.74/293.11	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.74/293.11	U42(ok(X)) -> ok(U42(X))
1151.74/293.11	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.74/293.11	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.74/293.11	U52(ok(X)) -> ok(U52(X))
1151.74/293.11	isNatList(ok(X)) -> ok(isNatList(X))
1151.74/293.11	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.74/293.11	U62(ok(X)) -> ok(U62(X))
1151.74/293.11	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.74/293.11	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.74/293.11	isNat(ok(X)) -> ok(isNat(X))
1151.74/293.11	s(ok(X)) -> ok(s(X))
1151.74/293.11	length(ok(X)) -> ok(length(X))
1151.74/293.11	U81(ok(X)) -> ok(U81(X))
1151.74/293.11	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.74/293.11	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.74/293.11	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.74/293.11	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.74/293.11	top(mark(X)) -> top(proper(X))
1151.74/293.11	top(ok(X)) -> top(active(X))
1151.74/293.11	
1151.74/293.11	Types:
1151.74/293.11	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	zeros :: zeros:0':mark:tt:nil:ok
1151.74/293.11	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	0' :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	tt :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	nil :: zeros:0':mark:tt:nil:ok
1151.74/293.11	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.74/293.11	top :: zeros:0':mark:tt:nil:ok -> top
1151.74/293.11	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.74/293.11	hole_top2_0 :: top
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	Lemmas:
1151.74/293.11	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.74/293.11	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.74/293.11	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.74/293.11	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.74/293.11	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.74/293.11	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.74/293.11	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.74/293.11	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.74/293.11	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.74/293.11	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0)
1151.74/293.11	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0)
1151.74/293.11	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0)
1151.74/293.11	U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0)
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	Generator Equations:
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.74/293.11	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.74/293.11	
1151.74/293.11	
1151.74/293.11	The following defined symbols remain to be analysed:
1151.74/293.11	U41, active, U51, U61, U71, U81, U91, proper, top
1151.74/293.11	
1151.74/293.11	They will be analysed ascendingly in the following order:
1151.74/293.11	U41 < active
1151.74/293.11	U51 < active
1151.74/293.11	U61 < active
1151.74/293.11	U71 < active
1151.74/293.11	U81 < active
1151.74/293.11	U91 < active
1151.74/293.11	active < top
1151.74/293.11	U41 < proper
1151.74/293.11	U51 < proper
1151.74/293.11	U61 < proper
1151.74/293.11	U71 < proper
1151.74/293.11	U81 < proper
1151.74/293.11	U91 < proper
1151.74/293.11	proper < top
1151.74/293.11	
1151.74/293.11	----------------------------------------
1151.74/293.11	
1151.74/293.11	(37) RewriteLemmaProof (LOWER BOUND(ID))
1151.74/293.11	Proved the following rewrite lemma:
1151.74/293.11	U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0)
1151.74/293.11	
1151.74/293.11	Induction Base:
1151.74/293.11	U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))
1151.74/293.11	
1151.74/293.11	Induction Step:
1151.74/293.11	U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n39663_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
1151.74/293.11	mark(U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
1151.74/293.11	mark(*4_0)
1151.74/293.11	
1151.74/293.11	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.74/293.11	----------------------------------------
1151.74/293.11	
1151.74/293.11	(38)
1151.74/293.11	Obligation:
1151.74/293.11	TRS:
1151.74/293.11	Rules:
1151.74/293.11	active(zeros) -> mark(cons(0', zeros))
1151.74/293.11	active(U11(tt)) -> mark(tt)
1151.74/293.11	active(U21(tt)) -> mark(tt)
1151.74/293.11	active(U31(tt)) -> mark(tt)
1151.74/293.11	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.74/293.11	active(U42(tt)) -> mark(tt)
1151.74/293.11	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.74/293.11	active(U52(tt)) -> mark(tt)
1151.74/293.11	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.74/293.11	active(U62(tt)) -> mark(tt)
1151.74/293.11	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.74/293.11	active(U72(tt, L)) -> mark(s(length(L)))
1151.74/293.11	active(U81(tt)) -> mark(nil)
1151.74/293.11	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.74/293.11	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.74/293.11	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.74/293.11	active(isNat(0')) -> mark(tt)
1151.74/293.11	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.74/293.11	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.74/293.11	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.74/293.11	active(isNatIList(zeros)) -> mark(tt)
1151.74/293.11	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.74/293.11	active(isNatList(nil)) -> mark(tt)
1151.74/293.11	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.74/293.11	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.74/293.11	active(length(nil)) -> mark(0')
1151.74/293.11	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.74/293.11	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.74/293.11	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.74/293.11	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.74/293.11	active(U11(X)) -> U11(active(X))
1151.74/293.11	active(U21(X)) -> U21(active(X))
1151.74/293.11	active(U31(X)) -> U31(active(X))
1151.74/293.11	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.74/293.11	active(U42(X)) -> U42(active(X))
1151.74/293.11	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.74/293.11	active(U52(X)) -> U52(active(X))
1151.74/293.11	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.74/293.11	active(U62(X)) -> U62(active(X))
1151.74/293.11	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.74/293.11	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.74/293.11	active(s(X)) -> s(active(X))
1151.74/293.11	active(length(X)) -> length(active(X))
1151.74/293.11	active(U81(X)) -> U81(active(X))
1151.74/293.11	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.74/293.11	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.74/293.11	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.74/293.11	active(take(X1, X2)) -> take(active(X1), X2)
1151.74/293.11	active(take(X1, X2)) -> take(X1, active(X2))
1151.74/293.11	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.74/293.11	U11(mark(X)) -> mark(U11(X))
1151.74/293.11	U21(mark(X)) -> mark(U21(X))
1151.74/293.11	U31(mark(X)) -> mark(U31(X))
1151.74/293.11	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.74/293.11	U42(mark(X)) -> mark(U42(X))
1151.74/293.11	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.74/293.11	U52(mark(X)) -> mark(U52(X))
1151.74/293.11	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.74/293.11	U62(mark(X)) -> mark(U62(X))
1151.74/293.11	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.74/293.11	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.74/293.11	s(mark(X)) -> mark(s(X))
1151.74/293.11	length(mark(X)) -> mark(length(X))
1151.74/293.11	U81(mark(X)) -> mark(U81(X))
1151.74/293.11	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.74/293.11	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.74/293.11	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.74/293.11	take(mark(X1), X2) -> mark(take(X1, X2))
1151.74/293.11	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.74/293.11	proper(zeros) -> ok(zeros)
1151.74/293.11	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.74/293.11	proper(0') -> ok(0')
1151.74/293.11	proper(U11(X)) -> U11(proper(X))
1151.74/293.11	proper(tt) -> ok(tt)
1151.74/293.11	proper(U21(X)) -> U21(proper(X))
1151.74/293.11	proper(U31(X)) -> U31(proper(X))
1151.74/293.11	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.74/293.11	proper(U42(X)) -> U42(proper(X))
1151.74/293.11	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.74/293.11	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.74/293.11	proper(U52(X)) -> U52(proper(X))
1151.74/293.11	proper(isNatList(X)) -> isNatList(proper(X))
1151.74/293.11	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.89/293.12	proper(U62(X)) -> U62(proper(X))
1151.89/293.12	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.89/293.12	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.89/293.12	proper(isNat(X)) -> isNat(proper(X))
1151.89/293.12	proper(s(X)) -> s(proper(X))
1151.89/293.12	proper(length(X)) -> length(proper(X))
1151.89/293.12	proper(U81(X)) -> U81(proper(X))
1151.89/293.12	proper(nil) -> ok(nil)
1151.89/293.12	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.89/293.12	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.89/293.12	U11(ok(X)) -> ok(U11(X))
1151.89/293.12	U21(ok(X)) -> ok(U21(X))
1151.89/293.12	U31(ok(X)) -> ok(U31(X))
1151.89/293.12	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.89/293.12	U42(ok(X)) -> ok(U42(X))
1151.89/293.12	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.89/293.12	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.89/293.12	U52(ok(X)) -> ok(U52(X))
1151.89/293.12	isNatList(ok(X)) -> ok(isNatList(X))
1151.89/293.12	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.89/293.12	U62(ok(X)) -> ok(U62(X))
1151.89/293.12	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.89/293.12	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.89/293.12	isNat(ok(X)) -> ok(isNat(X))
1151.89/293.12	s(ok(X)) -> ok(s(X))
1151.89/293.12	length(ok(X)) -> ok(length(X))
1151.89/293.12	U81(ok(X)) -> ok(U81(X))
1151.89/293.12	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.89/293.12	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.89/293.12	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.89/293.12	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.89/293.12	top(mark(X)) -> top(proper(X))
1151.89/293.12	top(ok(X)) -> top(active(X))
1151.89/293.12	
1151.89/293.12	Types:
1151.89/293.12	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	zeros :: zeros:0':mark:tt:nil:ok
1151.89/293.12	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	0' :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	tt :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	nil :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	top :: zeros:0':mark:tt:nil:ok -> top
1151.89/293.12	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.89/293.12	hole_top2_0 :: top
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Lemmas:
1151.89/293.12	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.89/293.12	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.89/293.12	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.89/293.12	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.89/293.12	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.89/293.12	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.89/293.12	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.89/293.12	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.89/293.12	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.89/293.12	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0)
1151.89/293.12	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0)
1151.89/293.12	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0)
1151.89/293.12	U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0)
1151.89/293.12	U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0)
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Generator Equations:
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	The following defined symbols remain to be analysed:
1151.89/293.12	U51, active, U61, U71, U81, U91, proper, top
1151.89/293.12	
1151.89/293.12	They will be analysed ascendingly in the following order:
1151.89/293.12	U51 < active
1151.89/293.12	U61 < active
1151.89/293.12	U71 < active
1151.89/293.12	U81 < active
1151.89/293.12	U91 < active
1151.89/293.12	active < top
1151.89/293.12	U51 < proper
1151.89/293.12	U61 < proper
1151.89/293.12	U71 < proper
1151.89/293.12	U81 < proper
1151.89/293.12	U91 < proper
1151.89/293.12	proper < top
1151.89/293.12	
1151.89/293.12	----------------------------------------
1151.89/293.12	
1151.89/293.12	(39) RewriteLemmaProof (LOWER BOUND(ID))
1151.89/293.12	Proved the following rewrite lemma:
1151.89/293.12	U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0)
1151.89/293.12	
1151.89/293.12	Induction Base:
1151.89/293.12	U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))
1151.89/293.12	
1151.89/293.12	Induction Step:
1151.89/293.12	U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n45350_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
1151.89/293.12	mark(U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
1151.89/293.12	mark(*4_0)
1151.89/293.12	
1151.89/293.12	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.89/293.12	----------------------------------------
1151.89/293.12	
1151.89/293.12	(40)
1151.89/293.12	Obligation:
1151.89/293.12	TRS:
1151.89/293.12	Rules:
1151.89/293.12	active(zeros) -> mark(cons(0', zeros))
1151.89/293.12	active(U11(tt)) -> mark(tt)
1151.89/293.12	active(U21(tt)) -> mark(tt)
1151.89/293.12	active(U31(tt)) -> mark(tt)
1151.89/293.12	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.89/293.12	active(U42(tt)) -> mark(tt)
1151.89/293.12	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.89/293.12	active(U52(tt)) -> mark(tt)
1151.89/293.12	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.89/293.12	active(U62(tt)) -> mark(tt)
1151.89/293.12	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.89/293.12	active(U72(tt, L)) -> mark(s(length(L)))
1151.89/293.12	active(U81(tt)) -> mark(nil)
1151.89/293.12	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.89/293.12	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.89/293.12	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.89/293.12	active(isNat(0')) -> mark(tt)
1151.89/293.12	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.89/293.12	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.89/293.12	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.89/293.12	active(isNatIList(zeros)) -> mark(tt)
1151.89/293.12	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.89/293.12	active(isNatList(nil)) -> mark(tt)
1151.89/293.12	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.89/293.12	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.89/293.12	active(length(nil)) -> mark(0')
1151.89/293.12	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.89/293.12	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.89/293.12	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.89/293.12	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.89/293.12	active(U11(X)) -> U11(active(X))
1151.89/293.12	active(U21(X)) -> U21(active(X))
1151.89/293.12	active(U31(X)) -> U31(active(X))
1151.89/293.12	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.89/293.12	active(U42(X)) -> U42(active(X))
1151.89/293.12	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.89/293.12	active(U52(X)) -> U52(active(X))
1151.89/293.12	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.89/293.12	active(U62(X)) -> U62(active(X))
1151.89/293.12	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.89/293.12	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.89/293.12	active(s(X)) -> s(active(X))
1151.89/293.12	active(length(X)) -> length(active(X))
1151.89/293.12	active(U81(X)) -> U81(active(X))
1151.89/293.12	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.89/293.12	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.89/293.12	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.89/293.12	active(take(X1, X2)) -> take(active(X1), X2)
1151.89/293.12	active(take(X1, X2)) -> take(X1, active(X2))
1151.89/293.12	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.89/293.12	U11(mark(X)) -> mark(U11(X))
1151.89/293.12	U21(mark(X)) -> mark(U21(X))
1151.89/293.12	U31(mark(X)) -> mark(U31(X))
1151.89/293.12	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.89/293.12	U42(mark(X)) -> mark(U42(X))
1151.89/293.12	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.89/293.12	U52(mark(X)) -> mark(U52(X))
1151.89/293.12	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.89/293.12	U62(mark(X)) -> mark(U62(X))
1151.89/293.12	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.89/293.12	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.89/293.12	s(mark(X)) -> mark(s(X))
1151.89/293.12	length(mark(X)) -> mark(length(X))
1151.89/293.12	U81(mark(X)) -> mark(U81(X))
1151.89/293.12	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.89/293.12	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.89/293.12	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.89/293.12	take(mark(X1), X2) -> mark(take(X1, X2))
1151.89/293.12	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.89/293.12	proper(zeros) -> ok(zeros)
1151.89/293.12	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.89/293.12	proper(0') -> ok(0')
1151.89/293.12	proper(U11(X)) -> U11(proper(X))
1151.89/293.12	proper(tt) -> ok(tt)
1151.89/293.12	proper(U21(X)) -> U21(proper(X))
1151.89/293.12	proper(U31(X)) -> U31(proper(X))
1151.89/293.12	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.89/293.12	proper(U42(X)) -> U42(proper(X))
1151.89/293.12	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.89/293.12	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.89/293.12	proper(U52(X)) -> U52(proper(X))
1151.89/293.12	proper(isNatList(X)) -> isNatList(proper(X))
1151.89/293.12	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.89/293.12	proper(U62(X)) -> U62(proper(X))
1151.89/293.12	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.89/293.12	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.89/293.12	proper(isNat(X)) -> isNat(proper(X))
1151.89/293.12	proper(s(X)) -> s(proper(X))
1151.89/293.12	proper(length(X)) -> length(proper(X))
1151.89/293.12	proper(U81(X)) -> U81(proper(X))
1151.89/293.12	proper(nil) -> ok(nil)
1151.89/293.12	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.89/293.12	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.89/293.12	U11(ok(X)) -> ok(U11(X))
1151.89/293.12	U21(ok(X)) -> ok(U21(X))
1151.89/293.12	U31(ok(X)) -> ok(U31(X))
1151.89/293.12	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.89/293.12	U42(ok(X)) -> ok(U42(X))
1151.89/293.12	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.89/293.12	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.89/293.12	U52(ok(X)) -> ok(U52(X))
1151.89/293.12	isNatList(ok(X)) -> ok(isNatList(X))
1151.89/293.12	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.89/293.12	U62(ok(X)) -> ok(U62(X))
1151.89/293.12	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.89/293.12	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.89/293.12	isNat(ok(X)) -> ok(isNat(X))
1151.89/293.12	s(ok(X)) -> ok(s(X))
1151.89/293.12	length(ok(X)) -> ok(length(X))
1151.89/293.12	U81(ok(X)) -> ok(U81(X))
1151.89/293.12	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.89/293.12	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.89/293.12	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.89/293.12	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.89/293.12	top(mark(X)) -> top(proper(X))
1151.89/293.12	top(ok(X)) -> top(active(X))
1151.89/293.12	
1151.89/293.12	Types:
1151.89/293.12	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	zeros :: zeros:0':mark:tt:nil:ok
1151.89/293.12	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	0' :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	tt :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	nil :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	top :: zeros:0':mark:tt:nil:ok -> top
1151.89/293.12	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.89/293.12	hole_top2_0 :: top
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Lemmas:
1151.89/293.12	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.89/293.12	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.89/293.12	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.89/293.12	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.89/293.12	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.89/293.12	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.89/293.12	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.89/293.12	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.89/293.12	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.89/293.12	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0)
1151.89/293.12	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0)
1151.89/293.12	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0)
1151.89/293.12	U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0)
1151.89/293.12	U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0)
1151.89/293.12	U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0)
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Generator Equations:
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	The following defined symbols remain to be analysed:
1151.89/293.12	U61, active, U71, U81, U91, proper, top
1151.89/293.12	
1151.89/293.12	They will be analysed ascendingly in the following order:
1151.89/293.12	U61 < active
1151.89/293.12	U71 < active
1151.89/293.12	U81 < active
1151.89/293.12	U91 < active
1151.89/293.12	active < top
1151.89/293.12	U61 < proper
1151.89/293.12	U71 < proper
1151.89/293.12	U81 < proper
1151.89/293.12	U91 < proper
1151.89/293.12	proper < top
1151.89/293.12	
1151.89/293.12	----------------------------------------
1151.89/293.12	
1151.89/293.12	(41) RewriteLemmaProof (LOWER BOUND(ID))
1151.89/293.12	Proved the following rewrite lemma:
1151.89/293.12	U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n51343_0)
1151.89/293.12	
1151.89/293.12	Induction Base:
1151.89/293.12	U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))
1151.89/293.12	
1151.89/293.12	Induction Step:
1151.89/293.12	U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n51343_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
1151.89/293.12	mark(U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
1151.89/293.12	mark(*4_0)
1151.89/293.12	
1151.89/293.12	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.89/293.12	----------------------------------------
1151.89/293.12	
1151.89/293.12	(42)
1151.89/293.12	Obligation:
1151.89/293.12	TRS:
1151.89/293.12	Rules:
1151.89/293.12	active(zeros) -> mark(cons(0', zeros))
1151.89/293.12	active(U11(tt)) -> mark(tt)
1151.89/293.12	active(U21(tt)) -> mark(tt)
1151.89/293.12	active(U31(tt)) -> mark(tt)
1151.89/293.12	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.89/293.12	active(U42(tt)) -> mark(tt)
1151.89/293.12	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.89/293.12	active(U52(tt)) -> mark(tt)
1151.89/293.12	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.89/293.12	active(U62(tt)) -> mark(tt)
1151.89/293.12	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.89/293.12	active(U72(tt, L)) -> mark(s(length(L)))
1151.89/293.12	active(U81(tt)) -> mark(nil)
1151.89/293.12	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.89/293.12	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.89/293.12	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.89/293.12	active(isNat(0')) -> mark(tt)
1151.89/293.12	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.89/293.12	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.89/293.12	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.89/293.12	active(isNatIList(zeros)) -> mark(tt)
1151.89/293.12	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.89/293.12	active(isNatList(nil)) -> mark(tt)
1151.89/293.12	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.89/293.12	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.89/293.12	active(length(nil)) -> mark(0')
1151.89/293.12	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.89/293.12	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.89/293.12	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.89/293.12	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.89/293.12	active(U11(X)) -> U11(active(X))
1151.89/293.12	active(U21(X)) -> U21(active(X))
1151.89/293.12	active(U31(X)) -> U31(active(X))
1151.89/293.12	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.89/293.12	active(U42(X)) -> U42(active(X))
1151.89/293.12	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.89/293.12	active(U52(X)) -> U52(active(X))
1151.89/293.12	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.89/293.12	active(U62(X)) -> U62(active(X))
1151.89/293.12	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.89/293.12	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.89/293.12	active(s(X)) -> s(active(X))
1151.89/293.12	active(length(X)) -> length(active(X))
1151.89/293.12	active(U81(X)) -> U81(active(X))
1151.89/293.12	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.89/293.12	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.89/293.12	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.89/293.12	active(take(X1, X2)) -> take(active(X1), X2)
1151.89/293.12	active(take(X1, X2)) -> take(X1, active(X2))
1151.89/293.12	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.89/293.12	U11(mark(X)) -> mark(U11(X))
1151.89/293.12	U21(mark(X)) -> mark(U21(X))
1151.89/293.12	U31(mark(X)) -> mark(U31(X))
1151.89/293.12	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.89/293.12	U42(mark(X)) -> mark(U42(X))
1151.89/293.12	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.89/293.12	U52(mark(X)) -> mark(U52(X))
1151.89/293.12	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.89/293.12	U62(mark(X)) -> mark(U62(X))
1151.89/293.12	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.89/293.12	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.89/293.12	s(mark(X)) -> mark(s(X))
1151.89/293.12	length(mark(X)) -> mark(length(X))
1151.89/293.12	U81(mark(X)) -> mark(U81(X))
1151.89/293.12	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.89/293.12	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.89/293.12	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.89/293.12	take(mark(X1), X2) -> mark(take(X1, X2))
1151.89/293.12	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.89/293.12	proper(zeros) -> ok(zeros)
1151.89/293.12	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.89/293.12	proper(0') -> ok(0')
1151.89/293.12	proper(U11(X)) -> U11(proper(X))
1151.89/293.12	proper(tt) -> ok(tt)
1151.89/293.12	proper(U21(X)) -> U21(proper(X))
1151.89/293.12	proper(U31(X)) -> U31(proper(X))
1151.89/293.12	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.89/293.12	proper(U42(X)) -> U42(proper(X))
1151.89/293.12	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.89/293.12	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.89/293.12	proper(U52(X)) -> U52(proper(X))
1151.89/293.12	proper(isNatList(X)) -> isNatList(proper(X))
1151.89/293.12	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.89/293.12	proper(U62(X)) -> U62(proper(X))
1151.89/293.12	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.89/293.12	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.89/293.12	proper(isNat(X)) -> isNat(proper(X))
1151.89/293.12	proper(s(X)) -> s(proper(X))
1151.89/293.12	proper(length(X)) -> length(proper(X))
1151.89/293.12	proper(U81(X)) -> U81(proper(X))
1151.89/293.12	proper(nil) -> ok(nil)
1151.89/293.12	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.89/293.12	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.89/293.12	U11(ok(X)) -> ok(U11(X))
1151.89/293.12	U21(ok(X)) -> ok(U21(X))
1151.89/293.12	U31(ok(X)) -> ok(U31(X))
1151.89/293.12	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.89/293.12	U42(ok(X)) -> ok(U42(X))
1151.89/293.12	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.89/293.12	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.89/293.12	U52(ok(X)) -> ok(U52(X))
1151.89/293.12	isNatList(ok(X)) -> ok(isNatList(X))
1151.89/293.12	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.89/293.12	U62(ok(X)) -> ok(U62(X))
1151.89/293.12	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.89/293.12	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.89/293.12	isNat(ok(X)) -> ok(isNat(X))
1151.89/293.12	s(ok(X)) -> ok(s(X))
1151.89/293.12	length(ok(X)) -> ok(length(X))
1151.89/293.12	U81(ok(X)) -> ok(U81(X))
1151.89/293.12	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.89/293.12	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.89/293.12	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.89/293.12	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.89/293.12	top(mark(X)) -> top(proper(X))
1151.89/293.12	top(ok(X)) -> top(active(X))
1151.89/293.12	
1151.89/293.12	Types:
1151.89/293.12	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	zeros :: zeros:0':mark:tt:nil:ok
1151.89/293.12	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	0' :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	tt :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	nil :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	top :: zeros:0':mark:tt:nil:ok -> top
1151.89/293.12	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.89/293.12	hole_top2_0 :: top
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Lemmas:
1151.89/293.12	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.89/293.12	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.89/293.12	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.89/293.12	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.89/293.12	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.89/293.12	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.89/293.12	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.89/293.12	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.89/293.12	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.89/293.12	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0)
1151.89/293.12	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0)
1151.89/293.12	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0)
1151.89/293.12	U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0)
1151.89/293.12	U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0)
1151.89/293.12	U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0)
1151.89/293.12	U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n51343_0)
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Generator Equations:
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	The following defined symbols remain to be analysed:
1151.89/293.12	U71, active, U81, U91, proper, top
1151.89/293.12	
1151.89/293.12	They will be analysed ascendingly in the following order:
1151.89/293.12	U71 < active
1151.89/293.12	U81 < active
1151.89/293.12	U91 < active
1151.89/293.12	active < top
1151.89/293.12	U71 < proper
1151.89/293.12	U81 < proper
1151.89/293.12	U91 < proper
1151.89/293.12	proper < top
1151.89/293.12	
1151.89/293.12	----------------------------------------
1151.89/293.12	
1151.89/293.12	(43) RewriteLemmaProof (LOWER BOUND(ID))
1151.89/293.12	Proved the following rewrite lemma:
1151.89/293.12	U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n57642_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n57642_0)
1151.89/293.12	
1151.89/293.12	Induction Base:
1151.89/293.12	U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c))
1151.89/293.12	
1151.89/293.12	Induction Step:
1151.89/293.12	U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n57642_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) ->_R^Omega(1)
1151.89/293.12	mark(U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n57642_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c))) ->_IH
1151.89/293.12	mark(*4_0)
1151.89/293.12	
1151.89/293.12	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.89/293.12	----------------------------------------
1151.89/293.12	
1151.89/293.12	(44)
1151.89/293.12	Obligation:
1151.89/293.12	TRS:
1151.89/293.12	Rules:
1151.89/293.12	active(zeros) -> mark(cons(0', zeros))
1151.89/293.12	active(U11(tt)) -> mark(tt)
1151.89/293.12	active(U21(tt)) -> mark(tt)
1151.89/293.12	active(U31(tt)) -> mark(tt)
1151.89/293.12	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.89/293.12	active(U42(tt)) -> mark(tt)
1151.89/293.12	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.89/293.12	active(U52(tt)) -> mark(tt)
1151.89/293.12	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.89/293.12	active(U62(tt)) -> mark(tt)
1151.89/293.12	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.89/293.12	active(U72(tt, L)) -> mark(s(length(L)))
1151.89/293.12	active(U81(tt)) -> mark(nil)
1151.89/293.12	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.89/293.12	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.89/293.12	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.89/293.12	active(isNat(0')) -> mark(tt)
1151.89/293.12	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.89/293.12	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.89/293.12	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.89/293.12	active(isNatIList(zeros)) -> mark(tt)
1151.89/293.12	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.89/293.12	active(isNatList(nil)) -> mark(tt)
1151.89/293.12	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.89/293.12	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.89/293.12	active(length(nil)) -> mark(0')
1151.89/293.12	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.89/293.12	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.89/293.12	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.89/293.12	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.89/293.12	active(U11(X)) -> U11(active(X))
1151.89/293.12	active(U21(X)) -> U21(active(X))
1151.89/293.12	active(U31(X)) -> U31(active(X))
1151.89/293.12	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.89/293.12	active(U42(X)) -> U42(active(X))
1151.89/293.12	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.89/293.12	active(U52(X)) -> U52(active(X))
1151.89/293.12	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.89/293.12	active(U62(X)) -> U62(active(X))
1151.89/293.12	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.89/293.12	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.89/293.12	active(s(X)) -> s(active(X))
1151.89/293.12	active(length(X)) -> length(active(X))
1151.89/293.12	active(U81(X)) -> U81(active(X))
1151.89/293.12	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.89/293.12	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.89/293.12	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.89/293.12	active(take(X1, X2)) -> take(active(X1), X2)
1151.89/293.12	active(take(X1, X2)) -> take(X1, active(X2))
1151.89/293.12	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.89/293.12	U11(mark(X)) -> mark(U11(X))
1151.89/293.12	U21(mark(X)) -> mark(U21(X))
1151.89/293.12	U31(mark(X)) -> mark(U31(X))
1151.89/293.12	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.89/293.12	U42(mark(X)) -> mark(U42(X))
1151.89/293.12	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.89/293.12	U52(mark(X)) -> mark(U52(X))
1151.89/293.12	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.89/293.12	U62(mark(X)) -> mark(U62(X))
1151.89/293.12	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.89/293.12	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.89/293.12	s(mark(X)) -> mark(s(X))
1151.89/293.12	length(mark(X)) -> mark(length(X))
1151.89/293.12	U81(mark(X)) -> mark(U81(X))
1151.89/293.12	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.89/293.12	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.89/293.12	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.89/293.12	take(mark(X1), X2) -> mark(take(X1, X2))
1151.89/293.12	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.89/293.12	proper(zeros) -> ok(zeros)
1151.89/293.12	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.89/293.12	proper(0') -> ok(0')
1151.89/293.12	proper(U11(X)) -> U11(proper(X))
1151.89/293.12	proper(tt) -> ok(tt)
1151.89/293.12	proper(U21(X)) -> U21(proper(X))
1151.89/293.12	proper(U31(X)) -> U31(proper(X))
1151.89/293.12	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.89/293.12	proper(U42(X)) -> U42(proper(X))
1151.89/293.12	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.89/293.12	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.89/293.12	proper(U52(X)) -> U52(proper(X))
1151.89/293.12	proper(isNatList(X)) -> isNatList(proper(X))
1151.89/293.12	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.89/293.12	proper(U62(X)) -> U62(proper(X))
1151.89/293.12	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.89/293.12	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.89/293.12	proper(isNat(X)) -> isNat(proper(X))
1151.89/293.12	proper(s(X)) -> s(proper(X))
1151.89/293.12	proper(length(X)) -> length(proper(X))
1151.89/293.12	proper(U81(X)) -> U81(proper(X))
1151.89/293.12	proper(nil) -> ok(nil)
1151.89/293.12	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.89/293.12	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.89/293.12	U11(ok(X)) -> ok(U11(X))
1151.89/293.12	U21(ok(X)) -> ok(U21(X))
1151.89/293.12	U31(ok(X)) -> ok(U31(X))
1151.89/293.12	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.89/293.12	U42(ok(X)) -> ok(U42(X))
1151.89/293.12	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.89/293.12	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.89/293.12	U52(ok(X)) -> ok(U52(X))
1151.89/293.12	isNatList(ok(X)) -> ok(isNatList(X))
1151.89/293.12	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.89/293.12	U62(ok(X)) -> ok(U62(X))
1151.89/293.12	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.89/293.12	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.89/293.12	isNat(ok(X)) -> ok(isNat(X))
1151.89/293.12	s(ok(X)) -> ok(s(X))
1151.89/293.12	length(ok(X)) -> ok(length(X))
1151.89/293.12	U81(ok(X)) -> ok(U81(X))
1151.89/293.12	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.89/293.12	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.89/293.12	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.89/293.12	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.89/293.12	top(mark(X)) -> top(proper(X))
1151.89/293.12	top(ok(X)) -> top(active(X))
1151.89/293.12	
1151.89/293.12	Types:
1151.89/293.12	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	zeros :: zeros:0':mark:tt:nil:ok
1151.89/293.12	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	0' :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	tt :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	nil :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	top :: zeros:0':mark:tt:nil:ok -> top
1151.89/293.12	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.89/293.12	hole_top2_0 :: top
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Lemmas:
1151.89/293.12	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.89/293.12	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.89/293.12	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.89/293.12	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.89/293.12	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.89/293.12	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.89/293.12	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.89/293.12	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.89/293.12	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.89/293.12	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0)
1151.89/293.12	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0)
1151.89/293.12	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0)
1151.89/293.12	U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0)
1151.89/293.12	U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0)
1151.89/293.12	U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0)
1151.89/293.12	U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n51343_0)
1151.89/293.12	U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n57642_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n57642_0)
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Generator Equations:
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	The following defined symbols remain to be analysed:
1151.89/293.12	U81, active, U91, proper, top
1151.89/293.12	
1151.89/293.12	They will be analysed ascendingly in the following order:
1151.89/293.12	U81 < active
1151.89/293.12	U91 < active
1151.89/293.12	active < top
1151.89/293.12	U81 < proper
1151.89/293.12	U91 < proper
1151.89/293.12	proper < top
1151.89/293.12	
1151.89/293.12	----------------------------------------
1151.89/293.12	
1151.89/293.12	(45) RewriteLemmaProof (LOWER BOUND(ID))
1151.89/293.12	Proved the following rewrite lemma:
1151.89/293.12	U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n68024_0))) -> *4_0, rt in Omega(n68024_0)
1151.89/293.12	
1151.89/293.12	Induction Base:
1151.89/293.12	U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))
1151.89/293.12	
1151.89/293.12	Induction Step:
1151.89/293.12	U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n68024_0, 1)))) ->_R^Omega(1)
1151.89/293.12	mark(U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n68024_0)))) ->_IH
1151.89/293.12	mark(*4_0)
1151.89/293.12	
1151.89/293.12	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.89/293.12	----------------------------------------
1151.89/293.12	
1151.89/293.12	(46)
1151.89/293.12	Obligation:
1151.89/293.12	TRS:
1151.89/293.12	Rules:
1151.89/293.12	active(zeros) -> mark(cons(0', zeros))
1151.89/293.12	active(U11(tt)) -> mark(tt)
1151.89/293.12	active(U21(tt)) -> mark(tt)
1151.89/293.12	active(U31(tt)) -> mark(tt)
1151.89/293.12	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.89/293.12	active(U42(tt)) -> mark(tt)
1151.89/293.12	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.89/293.12	active(U52(tt)) -> mark(tt)
1151.89/293.12	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.89/293.12	active(U62(tt)) -> mark(tt)
1151.89/293.12	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.89/293.12	active(U72(tt, L)) -> mark(s(length(L)))
1151.89/293.12	active(U81(tt)) -> mark(nil)
1151.89/293.12	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.89/293.12	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.89/293.12	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.89/293.12	active(isNat(0')) -> mark(tt)
1151.89/293.12	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.89/293.12	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.89/293.12	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.89/293.12	active(isNatIList(zeros)) -> mark(tt)
1151.89/293.12	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.89/293.12	active(isNatList(nil)) -> mark(tt)
1151.89/293.12	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.89/293.12	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.89/293.12	active(length(nil)) -> mark(0')
1151.89/293.12	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.89/293.12	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.89/293.12	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.89/293.12	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.89/293.12	active(U11(X)) -> U11(active(X))
1151.89/293.12	active(U21(X)) -> U21(active(X))
1151.89/293.12	active(U31(X)) -> U31(active(X))
1151.89/293.12	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.89/293.12	active(U42(X)) -> U42(active(X))
1151.89/293.12	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.89/293.12	active(U52(X)) -> U52(active(X))
1151.89/293.12	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.89/293.12	active(U62(X)) -> U62(active(X))
1151.89/293.12	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.89/293.12	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.89/293.12	active(s(X)) -> s(active(X))
1151.89/293.12	active(length(X)) -> length(active(X))
1151.89/293.12	active(U81(X)) -> U81(active(X))
1151.89/293.12	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.89/293.12	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.89/293.12	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.89/293.12	active(take(X1, X2)) -> take(active(X1), X2)
1151.89/293.12	active(take(X1, X2)) -> take(X1, active(X2))
1151.89/293.12	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.89/293.12	U11(mark(X)) -> mark(U11(X))
1151.89/293.12	U21(mark(X)) -> mark(U21(X))
1151.89/293.12	U31(mark(X)) -> mark(U31(X))
1151.89/293.12	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.89/293.12	U42(mark(X)) -> mark(U42(X))
1151.89/293.12	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.89/293.12	U52(mark(X)) -> mark(U52(X))
1151.89/293.12	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.89/293.12	U62(mark(X)) -> mark(U62(X))
1151.89/293.12	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.89/293.12	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.89/293.12	s(mark(X)) -> mark(s(X))
1151.89/293.12	length(mark(X)) -> mark(length(X))
1151.89/293.12	U81(mark(X)) -> mark(U81(X))
1151.89/293.12	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.89/293.12	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.89/293.12	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.89/293.12	take(mark(X1), X2) -> mark(take(X1, X2))
1151.89/293.12	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.89/293.12	proper(zeros) -> ok(zeros)
1151.89/293.12	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.89/293.12	proper(0') -> ok(0')
1151.89/293.12	proper(U11(X)) -> U11(proper(X))
1151.89/293.12	proper(tt) -> ok(tt)
1151.89/293.12	proper(U21(X)) -> U21(proper(X))
1151.89/293.12	proper(U31(X)) -> U31(proper(X))
1151.89/293.12	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.89/293.12	proper(U42(X)) -> U42(proper(X))
1151.89/293.12	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.89/293.12	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.89/293.12	proper(U52(X)) -> U52(proper(X))
1151.89/293.12	proper(isNatList(X)) -> isNatList(proper(X))
1151.89/293.12	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.89/293.12	proper(U62(X)) -> U62(proper(X))
1151.89/293.12	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.89/293.12	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.89/293.12	proper(isNat(X)) -> isNat(proper(X))
1151.89/293.12	proper(s(X)) -> s(proper(X))
1151.89/293.12	proper(length(X)) -> length(proper(X))
1151.89/293.12	proper(U81(X)) -> U81(proper(X))
1151.89/293.12	proper(nil) -> ok(nil)
1151.89/293.12	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.89/293.12	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.89/293.12	U11(ok(X)) -> ok(U11(X))
1151.89/293.12	U21(ok(X)) -> ok(U21(X))
1151.89/293.12	U31(ok(X)) -> ok(U31(X))
1151.89/293.12	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.89/293.12	U42(ok(X)) -> ok(U42(X))
1151.89/293.12	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.89/293.12	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.89/293.12	U52(ok(X)) -> ok(U52(X))
1151.89/293.12	isNatList(ok(X)) -> ok(isNatList(X))
1151.89/293.12	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.89/293.12	U62(ok(X)) -> ok(U62(X))
1151.89/293.12	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.89/293.12	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.89/293.12	isNat(ok(X)) -> ok(isNat(X))
1151.89/293.12	s(ok(X)) -> ok(s(X))
1151.89/293.12	length(ok(X)) -> ok(length(X))
1151.89/293.12	U81(ok(X)) -> ok(U81(X))
1151.89/293.12	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.89/293.12	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.89/293.12	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.89/293.12	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.89/293.12	top(mark(X)) -> top(proper(X))
1151.89/293.12	top(ok(X)) -> top(active(X))
1151.89/293.12	
1151.89/293.12	Types:
1151.89/293.12	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	zeros :: zeros:0':mark:tt:nil:ok
1151.89/293.12	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	0' :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	tt :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	nil :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	top :: zeros:0':mark:tt:nil:ok -> top
1151.89/293.12	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.89/293.12	hole_top2_0 :: top
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Lemmas:
1151.89/293.12	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.89/293.12	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.89/293.12	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.89/293.12	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.89/293.12	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.89/293.12	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.89/293.12	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.89/293.12	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.89/293.12	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.89/293.12	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0)
1151.89/293.12	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0)
1151.89/293.12	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0)
1151.89/293.12	U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0)
1151.89/293.12	U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0)
1151.89/293.12	U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0)
1151.89/293.12	U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n51343_0)
1151.89/293.12	U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n57642_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n57642_0)
1151.89/293.12	U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n68024_0))) -> *4_0, rt in Omega(n68024_0)
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Generator Equations:
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	The following defined symbols remain to be analysed:
1151.89/293.12	U91, active, proper, top
1151.89/293.12	
1151.89/293.12	They will be analysed ascendingly in the following order:
1151.89/293.12	U91 < active
1151.89/293.12	active < top
1151.89/293.12	U91 < proper
1151.89/293.12	proper < top
1151.89/293.12	
1151.89/293.12	----------------------------------------
1151.89/293.12	
1151.89/293.12	(47) RewriteLemmaProof (LOWER BOUND(ID))
1151.89/293.12	Proved the following rewrite lemma:
1151.89/293.12	U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n71051_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n71051_0)
1151.89/293.12	
1151.89/293.12	Induction Base:
1151.89/293.12	U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))
1151.89/293.12	
1151.89/293.12	Induction Step:
1151.89/293.12	U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n71051_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) ->_R^Omega(1)
1151.89/293.12	mark(U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n71051_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d))) ->_IH
1151.89/293.12	mark(*4_0)
1151.89/293.12	
1151.89/293.12	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1151.89/293.12	----------------------------------------
1151.89/293.12	
1151.89/293.12	(48)
1151.89/293.12	Obligation:
1151.89/293.12	TRS:
1151.89/293.12	Rules:
1151.89/293.12	active(zeros) -> mark(cons(0', zeros))
1151.89/293.12	active(U11(tt)) -> mark(tt)
1151.89/293.12	active(U21(tt)) -> mark(tt)
1151.89/293.12	active(U31(tt)) -> mark(tt)
1151.89/293.12	active(U41(tt, V2)) -> mark(U42(isNatIList(V2)))
1151.89/293.12	active(U42(tt)) -> mark(tt)
1151.89/293.12	active(U51(tt, V2)) -> mark(U52(isNatList(V2)))
1151.89/293.12	active(U52(tt)) -> mark(tt)
1151.89/293.12	active(U61(tt, V2)) -> mark(U62(isNatIList(V2)))
1151.89/293.12	active(U62(tt)) -> mark(tt)
1151.89/293.12	active(U71(tt, L, N)) -> mark(U72(isNat(N), L))
1151.89/293.12	active(U72(tt, L)) -> mark(s(length(L)))
1151.89/293.12	active(U81(tt)) -> mark(nil)
1151.89/293.12	active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N))
1151.89/293.12	active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N))
1151.89/293.12	active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1151.89/293.12	active(isNat(0')) -> mark(tt)
1151.89/293.12	active(isNat(length(V1))) -> mark(U11(isNatList(V1)))
1151.89/293.12	active(isNat(s(V1))) -> mark(U21(isNat(V1)))
1151.89/293.12	active(isNatIList(V)) -> mark(U31(isNatList(V)))
1151.89/293.12	active(isNatIList(zeros)) -> mark(tt)
1151.89/293.12	active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2))
1151.89/293.12	active(isNatList(nil)) -> mark(tt)
1151.89/293.12	active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2))
1151.89/293.12	active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2))
1151.89/293.12	active(length(nil)) -> mark(0')
1151.89/293.12	active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N))
1151.89/293.12	active(take(0', IL)) -> mark(U81(isNatIList(IL)))
1151.89/293.12	active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N))
1151.89/293.12	active(cons(X1, X2)) -> cons(active(X1), X2)
1151.89/293.12	active(U11(X)) -> U11(active(X))
1151.89/293.12	active(U21(X)) -> U21(active(X))
1151.89/293.12	active(U31(X)) -> U31(active(X))
1151.89/293.12	active(U41(X1, X2)) -> U41(active(X1), X2)
1151.89/293.12	active(U42(X)) -> U42(active(X))
1151.89/293.12	active(U51(X1, X2)) -> U51(active(X1), X2)
1151.89/293.12	active(U52(X)) -> U52(active(X))
1151.89/293.12	active(U61(X1, X2)) -> U61(active(X1), X2)
1151.89/293.12	active(U62(X)) -> U62(active(X))
1151.89/293.12	active(U71(X1, X2, X3)) -> U71(active(X1), X2, X3)
1151.89/293.12	active(U72(X1, X2)) -> U72(active(X1), X2)
1151.89/293.12	active(s(X)) -> s(active(X))
1151.89/293.12	active(length(X)) -> length(active(X))
1151.89/293.12	active(U81(X)) -> U81(active(X))
1151.89/293.12	active(U91(X1, X2, X3, X4)) -> U91(active(X1), X2, X3, X4)
1151.89/293.12	active(U92(X1, X2, X3, X4)) -> U92(active(X1), X2, X3, X4)
1151.89/293.12	active(U93(X1, X2, X3, X4)) -> U93(active(X1), X2, X3, X4)
1151.89/293.12	active(take(X1, X2)) -> take(active(X1), X2)
1151.89/293.12	active(take(X1, X2)) -> take(X1, active(X2))
1151.89/293.12	cons(mark(X1), X2) -> mark(cons(X1, X2))
1151.89/293.12	U11(mark(X)) -> mark(U11(X))
1151.89/293.12	U21(mark(X)) -> mark(U21(X))
1151.89/293.12	U31(mark(X)) -> mark(U31(X))
1151.89/293.12	U41(mark(X1), X2) -> mark(U41(X1, X2))
1151.89/293.12	U42(mark(X)) -> mark(U42(X))
1151.89/293.12	U51(mark(X1), X2) -> mark(U51(X1, X2))
1151.89/293.12	U52(mark(X)) -> mark(U52(X))
1151.89/293.12	U61(mark(X1), X2) -> mark(U61(X1, X2))
1151.89/293.12	U62(mark(X)) -> mark(U62(X))
1151.89/293.12	U71(mark(X1), X2, X3) -> mark(U71(X1, X2, X3))
1151.89/293.12	U72(mark(X1), X2) -> mark(U72(X1, X2))
1151.89/293.12	s(mark(X)) -> mark(s(X))
1151.89/293.12	length(mark(X)) -> mark(length(X))
1151.89/293.12	U81(mark(X)) -> mark(U81(X))
1151.89/293.12	U91(mark(X1), X2, X3, X4) -> mark(U91(X1, X2, X3, X4))
1151.89/293.12	U92(mark(X1), X2, X3, X4) -> mark(U92(X1, X2, X3, X4))
1151.89/293.12	U93(mark(X1), X2, X3, X4) -> mark(U93(X1, X2, X3, X4))
1151.89/293.12	take(mark(X1), X2) -> mark(take(X1, X2))
1151.89/293.12	take(X1, mark(X2)) -> mark(take(X1, X2))
1151.89/293.12	proper(zeros) -> ok(zeros)
1151.89/293.12	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1151.89/293.12	proper(0') -> ok(0')
1151.89/293.12	proper(U11(X)) -> U11(proper(X))
1151.89/293.12	proper(tt) -> ok(tt)
1151.89/293.12	proper(U21(X)) -> U21(proper(X))
1151.89/293.12	proper(U31(X)) -> U31(proper(X))
1151.89/293.12	proper(U41(X1, X2)) -> U41(proper(X1), proper(X2))
1151.89/293.12	proper(U42(X)) -> U42(proper(X))
1151.89/293.12	proper(isNatIList(X)) -> isNatIList(proper(X))
1151.89/293.12	proper(U51(X1, X2)) -> U51(proper(X1), proper(X2))
1151.89/293.12	proper(U52(X)) -> U52(proper(X))
1151.89/293.12	proper(isNatList(X)) -> isNatList(proper(X))
1151.89/293.12	proper(U61(X1, X2)) -> U61(proper(X1), proper(X2))
1151.89/293.12	proper(U62(X)) -> U62(proper(X))
1151.89/293.12	proper(U71(X1, X2, X3)) -> U71(proper(X1), proper(X2), proper(X3))
1151.89/293.12	proper(U72(X1, X2)) -> U72(proper(X1), proper(X2))
1151.89/293.12	proper(isNat(X)) -> isNat(proper(X))
1151.89/293.12	proper(s(X)) -> s(proper(X))
1151.89/293.12	proper(length(X)) -> length(proper(X))
1151.89/293.12	proper(U81(X)) -> U81(proper(X))
1151.89/293.12	proper(nil) -> ok(nil)
1151.89/293.12	proper(U91(X1, X2, X3, X4)) -> U91(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U92(X1, X2, X3, X4)) -> U92(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(U93(X1, X2, X3, X4)) -> U93(proper(X1), proper(X2), proper(X3), proper(X4))
1151.89/293.12	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1151.89/293.12	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1151.89/293.12	U11(ok(X)) -> ok(U11(X))
1151.89/293.12	U21(ok(X)) -> ok(U21(X))
1151.89/293.12	U31(ok(X)) -> ok(U31(X))
1151.89/293.12	U41(ok(X1), ok(X2)) -> ok(U41(X1, X2))
1151.89/293.12	U42(ok(X)) -> ok(U42(X))
1151.89/293.12	isNatIList(ok(X)) -> ok(isNatIList(X))
1151.89/293.12	U51(ok(X1), ok(X2)) -> ok(U51(X1, X2))
1151.89/293.12	U52(ok(X)) -> ok(U52(X))
1151.89/293.12	isNatList(ok(X)) -> ok(isNatList(X))
1151.89/293.12	U61(ok(X1), ok(X2)) -> ok(U61(X1, X2))
1151.89/293.12	U62(ok(X)) -> ok(U62(X))
1151.89/293.12	U71(ok(X1), ok(X2), ok(X3)) -> ok(U71(X1, X2, X3))
1151.89/293.12	U72(ok(X1), ok(X2)) -> ok(U72(X1, X2))
1151.89/293.12	isNat(ok(X)) -> ok(isNat(X))
1151.89/293.12	s(ok(X)) -> ok(s(X))
1151.89/293.12	length(ok(X)) -> ok(length(X))
1151.89/293.12	U81(ok(X)) -> ok(U81(X))
1151.89/293.12	U91(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U91(X1, X2, X3, X4))
1151.89/293.12	U92(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U92(X1, X2, X3, X4))
1151.89/293.12	U93(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U93(X1, X2, X3, X4))
1151.89/293.12	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1151.89/293.12	top(mark(X)) -> top(proper(X))
1151.89/293.12	top(ok(X)) -> top(active(X))
1151.89/293.12	
1151.89/293.12	Types:
1151.89/293.12	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	zeros :: zeros:0':mark:tt:nil:ok
1151.89/293.12	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	0' :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	tt :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U21 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U31 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U41 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U42 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatIList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U51 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U52 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNatList :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U61 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U62 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U71 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U72 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	isNat :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U81 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	nil :: zeros:0':mark:tt:nil:ok
1151.89/293.12	U91 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U92 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	U93 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1151.89/293.12	top :: zeros:0':mark:tt:nil:ok -> top
1151.89/293.12	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1151.89/293.12	hole_top2_0 :: top
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Lemmas:
1151.89/293.12	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
1151.89/293.12	U42(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2106_0))) -> *4_0, rt in Omega(n2106_0)
1151.89/293.12	U52(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2892_0))) -> *4_0, rt in Omega(n2892_0)
1151.89/293.12	U62(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3782_0))) -> *4_0, rt in Omega(n3782_0)
1151.89/293.12	U72(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4758_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n4758_0)
1151.89/293.12	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n7812_0))) -> *4_0, rt in Omega(n7812_0)
1151.89/293.12	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9039_0))) -> *4_0, rt in Omega(n9039_0)
1151.89/293.12	U92(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n10367_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n10367_0)
1151.89/293.12	U93(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18768_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n18768_0)
1151.89/293.12	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n28179_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n28179_0)
1151.89/293.12	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n33135_0))) -> *4_0, rt in Omega(n33135_0)
1151.89/293.12	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n35210_0))) -> *4_0, rt in Omega(n35210_0)
1151.89/293.12	U31(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n37386_0))) -> *4_0, rt in Omega(n37386_0)
1151.89/293.12	U41(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n39663_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n39663_0)
1151.89/293.12	U51(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n45350_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n45350_0)
1151.89/293.12	U61(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n51343_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n51343_0)
1151.89/293.12	U71(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n57642_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c)) -> *4_0, rt in Omega(n57642_0)
1151.89/293.12	U81(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n68024_0))) -> *4_0, rt in Omega(n68024_0)
1151.89/293.12	U91(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n71051_0)), gen_zeros:0':mark:tt:nil:ok3_0(b), gen_zeros:0':mark:tt:nil:ok3_0(c), gen_zeros:0':mark:tt:nil:ok3_0(d)) -> *4_0, rt in Omega(n71051_0)
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	Generator Equations:
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1151.89/293.12	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1151.89/293.12	
1151.89/293.12	
1151.89/293.12	The following defined symbols remain to be analysed:
1151.89/293.12	active, proper, top
1151.89/293.12	
1151.89/293.12	They will be analysed ascendingly in the following order:
1151.89/293.12	active < top
1151.89/293.12	proper < top
1152.00/293.20	EOF