1115.90/291.59	WORST_CASE(Omega(n^1), ?)
1126.86/294.37	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1126.86/294.37	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1126.86/294.37	
1126.86/294.37	
1126.86/294.37	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.86/294.37	
1126.86/294.37	(0) CpxTRS
1126.86/294.37	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1126.86/294.37	(2) CpxTRS
1126.86/294.37	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1126.86/294.37	(4) typed CpxTrs
1126.86/294.37	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1126.86/294.37	(6) typed CpxTrs
1126.86/294.37	(7) RewriteLemmaProof [LOWER BOUND(ID), 569 ms]
1126.86/294.37	(8) BEST
1126.86/294.37	    (9) proven lower bound
1126.86/294.37	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1126.86/294.37	        (11) BOUNDS(n^1, INF)
1126.86/294.37	    (12) typed CpxTrs
1126.86/294.37	        (13) RewriteLemmaProof [LOWER BOUND(ID), 133 ms]
1126.86/294.37	        (14) typed CpxTrs
1126.86/294.37	        (15) RewriteLemmaProof [LOWER BOUND(ID), 75 ms]
1126.86/294.37	        (16) typed CpxTrs
1126.86/294.37	        (17) RewriteLemmaProof [LOWER BOUND(ID), 63 ms]
1126.86/294.37	        (18) typed CpxTrs
1126.86/294.37	        (19) RewriteLemmaProof [LOWER BOUND(ID), 37 ms]
1126.86/294.37	        (20) typed CpxTrs
1126.86/294.37	        (21) RewriteLemmaProof [LOWER BOUND(ID), 141 ms]
1126.86/294.37	        (22) typed CpxTrs
1126.86/294.37	        (23) RewriteLemmaProof [LOWER BOUND(ID), 115 ms]
1126.86/294.37	        (24) typed CpxTrs
1126.86/294.37	        (25) RewriteLemmaProof [LOWER BOUND(ID), 113 ms]
1126.86/294.37	        (26) typed CpxTrs
1126.86/294.37	        (27) RewriteLemmaProof [LOWER BOUND(ID), 206 ms]
1126.86/294.37	        (28) typed CpxTrs
1126.86/294.37	
1126.86/294.37	
1126.86/294.37	----------------------------------------
1126.86/294.37	
1126.86/294.37	(0)
1126.86/294.37	Obligation:
1126.86/294.37	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.86/294.37	
1126.86/294.37	
1126.86/294.37	The TRS R consists of the following rules:
1126.86/294.37	
1126.86/294.37	   active(zeros) -> mark(cons(0, zeros))
1126.86/294.37	   active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.37	   active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.37	   active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.37	   active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.37	   active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.37	   active(length(nil)) -> mark(0)
1126.86/294.37	   active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.37	   active(take(0, IL)) -> mark(nil)
1126.86/294.37	   active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.37	   active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.37	   active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.37	   active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.37	   active(s(X)) -> s(active(X))
1126.86/294.37	   active(length(X)) -> length(active(X))
1126.86/294.37	   active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.37	   active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.37	   active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.37	   active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.37	   active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.37	   cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.37	   U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.37	   U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.37	   s(mark(X)) -> mark(s(X))
1126.86/294.37	   length(mark(X)) -> mark(length(X))
1126.86/294.37	   U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.37	   U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.37	   U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.37	   take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.37	   take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.37	   proper(zeros) -> ok(zeros)
1126.86/294.37	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.37	   proper(0) -> ok(0)
1126.86/294.37	   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.37	   proper(tt) -> ok(tt)
1126.86/294.37	   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.37	   proper(s(X)) -> s(proper(X))
1126.86/294.37	   proper(length(X)) -> length(proper(X))
1126.86/294.37	   proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	   proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	   proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.37	   proper(nil) -> ok(nil)
1126.86/294.37	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.37	   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.37	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.37	   s(ok(X)) -> ok(s(X))
1126.86/294.37	   length(ok(X)) -> ok(length(X))
1126.86/294.37	   U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.37	   U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.37	   U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.37	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.37	   top(mark(X)) -> top(proper(X))
1126.86/294.37	   top(ok(X)) -> top(active(X))
1126.86/294.37	
1126.86/294.37	S is empty.
1126.86/294.37	Rewrite Strategy: FULL
1126.86/294.37	----------------------------------------
1126.86/294.37	
1126.86/294.37	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1126.86/294.37	Renamed function symbols to avoid clashes with predefined symbol.
1126.86/294.37	----------------------------------------
1126.86/294.37	
1126.86/294.37	(2)
1126.86/294.37	Obligation:
1126.86/294.37	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1126.86/294.37	
1126.86/294.37	
1126.86/294.37	The TRS R consists of the following rules:
1126.86/294.37	
1126.86/294.37	   active(zeros) -> mark(cons(0', zeros))
1126.86/294.37	   active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.37	   active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.37	   active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.37	   active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.37	   active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.37	   active(length(nil)) -> mark(0')
1126.86/294.37	   active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.37	   active(take(0', IL)) -> mark(nil)
1126.86/294.37	   active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.37	   active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.37	   active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.37	   active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.37	   active(s(X)) -> s(active(X))
1126.86/294.37	   active(length(X)) -> length(active(X))
1126.86/294.37	   active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.37	   active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.37	   active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.37	   active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.37	   active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.37	   cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.37	   U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.37	   U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.37	   s(mark(X)) -> mark(s(X))
1126.86/294.37	   length(mark(X)) -> mark(length(X))
1126.86/294.37	   U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.37	   U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.37	   U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.37	   take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.37	   take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.37	   proper(zeros) -> ok(zeros)
1126.86/294.37	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.37	   proper(0') -> ok(0')
1126.86/294.37	   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.37	   proper(tt) -> ok(tt)
1126.86/294.37	   proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.37	   proper(s(X)) -> s(proper(X))
1126.86/294.37	   proper(length(X)) -> length(proper(X))
1126.86/294.37	   proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	   proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	   proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.37	   proper(nil) -> ok(nil)
1126.86/294.37	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.37	   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.37	   U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.37	   s(ok(X)) -> ok(s(X))
1126.86/294.37	   length(ok(X)) -> ok(length(X))
1126.86/294.37	   U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.37	   U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.37	   U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.37	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.37	   top(mark(X)) -> top(proper(X))
1126.86/294.37	   top(ok(X)) -> top(active(X))
1126.86/294.37	
1126.86/294.37	S is empty.
1126.86/294.37	Rewrite Strategy: FULL
1126.86/294.37	----------------------------------------
1126.86/294.37	
1126.86/294.37	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1126.86/294.37	Infered types.
1126.86/294.37	----------------------------------------
1126.86/294.37	
1126.86/294.37	(4)
1126.86/294.37	Obligation:
1126.86/294.37	TRS:
1126.86/294.37	Rules:
1126.86/294.37	active(zeros) -> mark(cons(0', zeros))
1126.86/294.37	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.37	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.37	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.37	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.37	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.37	active(length(nil)) -> mark(0')
1126.86/294.37	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.37	active(take(0', IL)) -> mark(nil)
1126.86/294.37	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.37	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.37	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.37	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.37	active(s(X)) -> s(active(X))
1126.86/294.37	active(length(X)) -> length(active(X))
1126.86/294.37	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.37	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.37	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.37	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.37	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.37	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.37	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.37	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.37	s(mark(X)) -> mark(s(X))
1126.86/294.37	length(mark(X)) -> mark(length(X))
1126.86/294.37	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.37	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.37	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.37	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.37	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.37	proper(zeros) -> ok(zeros)
1126.86/294.37	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.37	proper(0') -> ok(0')
1126.86/294.37	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.37	proper(tt) -> ok(tt)
1126.86/294.37	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.37	proper(s(X)) -> s(proper(X))
1126.86/294.37	proper(length(X)) -> length(proper(X))
1126.86/294.37	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.37	proper(nil) -> ok(nil)
1126.86/294.37	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.37	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.37	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.37	s(ok(X)) -> ok(s(X))
1126.86/294.37	length(ok(X)) -> ok(length(X))
1126.86/294.37	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.37	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.37	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.37	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.37	top(mark(X)) -> top(proper(X))
1126.86/294.37	top(ok(X)) -> top(active(X))
1126.86/294.37	
1126.86/294.37	Types:
1126.86/294.37	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.37	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.37	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.37	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	U21 :: 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
1126.86/294.37	U22 :: 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
1126.86/294.37	U23 :: 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
1126.86/294.37	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.37	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.37	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.37	hole_top2_0 :: top
1126.86/294.37	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.37	
1126.86/294.37	----------------------------------------
1126.86/294.37	
1126.86/294.37	(5) OrderProof (LOWER BOUND(ID))
1126.86/294.37	Heuristically decided to analyse the following defined symbols:
1126.86/294.37	active, cons, U12, s, length, U22, U23, take, U11, U21, proper, top
1126.86/294.37	
1126.86/294.37	They will be analysed ascendingly in the following order:
1126.86/294.37	cons < active
1126.86/294.37	U12 < active
1126.86/294.37	s < active
1126.86/294.37	length < active
1126.86/294.37	U22 < active
1126.86/294.37	U23 < active
1126.86/294.37	take < active
1126.86/294.37	U11 < active
1126.86/294.37	U21 < active
1126.86/294.37	active < top
1126.86/294.37	cons < proper
1126.86/294.37	U12 < proper
1126.86/294.37	s < proper
1126.86/294.37	length < proper
1126.86/294.37	U22 < proper
1126.86/294.37	U23 < proper
1126.86/294.37	take < proper
1126.86/294.37	U11 < proper
1126.86/294.37	U21 < proper
1126.86/294.37	proper < top
1126.86/294.37	
1126.86/294.37	----------------------------------------
1126.86/294.37	
1126.86/294.37	(6)
1126.86/294.37	Obligation:
1126.86/294.37	TRS:
1126.86/294.37	Rules:
1126.86/294.37	active(zeros) -> mark(cons(0', zeros))
1126.86/294.37	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.37	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.37	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.37	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.37	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.37	active(length(nil)) -> mark(0')
1126.86/294.37	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.37	active(take(0', IL)) -> mark(nil)
1126.86/294.37	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.37	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.37	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.37	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.37	active(s(X)) -> s(active(X))
1126.86/294.37	active(length(X)) -> length(active(X))
1126.86/294.37	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.37	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.37	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.37	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.37	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.37	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.37	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.37	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.37	s(mark(X)) -> mark(s(X))
1126.86/294.37	length(mark(X)) -> mark(length(X))
1126.86/294.37	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.37	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.37	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.37	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.37	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.37	proper(zeros) -> ok(zeros)
1126.86/294.37	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.37	proper(0') -> ok(0')
1126.86/294.37	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.37	proper(tt) -> ok(tt)
1126.86/294.37	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.37	proper(s(X)) -> s(proper(X))
1126.86/294.37	proper(length(X)) -> length(proper(X))
1126.86/294.37	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.37	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.37	proper(nil) -> ok(nil)
1126.86/294.37	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.37	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.37	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.37	s(ok(X)) -> ok(s(X))
1126.86/294.37	length(ok(X)) -> ok(length(X))
1126.86/294.37	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.37	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.37	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.37	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.37	top(mark(X)) -> top(proper(X))
1126.86/294.37	top(ok(X)) -> top(active(X))
1126.86/294.37	
1126.86/294.37	Types:
1126.86/294.37	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.37	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.37	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.37	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	U21 :: 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
1126.86/294.37	U22 :: 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
1126.86/294.37	U23 :: 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
1126.86/294.37	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.37	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.37	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.37	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.37	hole_top2_0 :: top
1126.86/294.37	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.37	
1126.86/294.37	
1126.86/294.37	Generator Equations:
1126.86/294.37	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1126.86/294.37	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1126.86/294.37	
1126.86/294.37	
1126.86/294.37	The following defined symbols remain to be analysed:
1126.86/294.37	cons, active, U12, s, length, U22, U23, take, U11, U21, proper, top
1126.86/294.37	
1126.86/294.37	They will be analysed ascendingly in the following order:
1126.86/294.37	cons < active
1126.86/294.37	U12 < active
1126.86/294.37	s < active
1126.86/294.37	length < active
1126.86/294.37	U22 < active
1126.86/294.37	U23 < active
1126.86/294.37	take < active
1126.86/294.37	U11 < active
1126.86/294.37	U21 < active
1126.86/294.37	active < top
1126.86/294.37	cons < proper
1126.86/294.37	U12 < proper
1126.86/294.37	s < proper
1126.86/294.37	length < proper
1126.86/294.37	U22 < proper
1126.86/294.37	U23 < proper
1126.86/294.37	take < proper
1126.86/294.37	U11 < proper
1126.86/294.37	U21 < proper
1126.86/294.37	proper < top
1126.86/294.37	
1126.86/294.37	----------------------------------------
1126.86/294.37	
1126.86/294.37	(7) RewriteLemmaProof (LOWER BOUND(ID))
1126.86/294.37	Proved the following rewrite lemma:
1126.86/294.37	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)
1126.86/294.37	
1126.86/294.37	Induction Base:
1126.86/294.38	cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))
1126.86/294.38	
1126.86/294.38	Induction Step:
1126.86/294.38	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)
1126.86/294.38	mark(cons(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n5_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
1126.86/294.38	mark(*4_0)
1126.86/294.38	
1126.86/294.38	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(8)
1126.86/294.38	Complex Obligation (BEST)
1126.86/294.38	
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(9)
1126.86/294.38	Obligation:
1126.86/294.38	Proved the lower bound n^1 for the following obligation:
1126.86/294.38	
1126.86/294.38	TRS:
1126.86/294.38	Rules:
1126.86/294.38	active(zeros) -> mark(cons(0', zeros))
1126.86/294.38	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.38	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.38	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.38	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.38	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.38	active(length(nil)) -> mark(0')
1126.86/294.38	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.38	active(take(0', IL)) -> mark(nil)
1126.86/294.38	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.38	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.38	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.38	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.38	active(s(X)) -> s(active(X))
1126.86/294.38	active(length(X)) -> length(active(X))
1126.86/294.38	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.38	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.38	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.38	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.38	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.38	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.38	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.38	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.38	s(mark(X)) -> mark(s(X))
1126.86/294.38	length(mark(X)) -> mark(length(X))
1126.86/294.38	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.38	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.38	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.38	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.38	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.38	proper(zeros) -> ok(zeros)
1126.86/294.38	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.38	proper(0') -> ok(0')
1126.86/294.38	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.38	proper(tt) -> ok(tt)
1126.86/294.38	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.38	proper(s(X)) -> s(proper(X))
1126.86/294.38	proper(length(X)) -> length(proper(X))
1126.86/294.38	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.38	proper(nil) -> ok(nil)
1126.86/294.38	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.38	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.38	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.38	s(ok(X)) -> ok(s(X))
1126.86/294.38	length(ok(X)) -> ok(length(X))
1126.86/294.38	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.38	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.38	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.38	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.38	top(mark(X)) -> top(proper(X))
1126.86/294.38	top(ok(X)) -> top(active(X))
1126.86/294.38	
1126.86/294.38	Types:
1126.86/294.38	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.38	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	U21 :: 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
1126.86/294.38	U22 :: 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
1126.86/294.38	U23 :: 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
1126.86/294.38	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.38	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.38	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.38	hole_top2_0 :: top
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Generator Equations:
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	The following defined symbols remain to be analysed:
1126.86/294.38	cons, active, U12, s, length, U22, U23, take, U11, U21, proper, top
1126.86/294.38	
1126.86/294.38	They will be analysed ascendingly in the following order:
1126.86/294.38	cons < active
1126.86/294.38	U12 < active
1126.86/294.38	s < active
1126.86/294.38	length < active
1126.86/294.38	U22 < active
1126.86/294.38	U23 < active
1126.86/294.38	take < active
1126.86/294.38	U11 < active
1126.86/294.38	U21 < active
1126.86/294.38	active < top
1126.86/294.38	cons < proper
1126.86/294.38	U12 < proper
1126.86/294.38	s < proper
1126.86/294.38	length < proper
1126.86/294.38	U22 < proper
1126.86/294.38	U23 < proper
1126.86/294.38	take < proper
1126.86/294.38	U11 < proper
1126.86/294.38	U21 < proper
1126.86/294.38	proper < top
1126.86/294.38	
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(10) LowerBoundPropagationProof (FINISHED)
1126.86/294.38	Propagated lower bound.
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(11)
1126.86/294.38	BOUNDS(n^1, INF)
1126.86/294.38	
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(12)
1126.86/294.38	Obligation:
1126.86/294.38	TRS:
1126.86/294.38	Rules:
1126.86/294.38	active(zeros) -> mark(cons(0', zeros))
1126.86/294.38	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.38	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.38	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.38	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.38	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.38	active(length(nil)) -> mark(0')
1126.86/294.38	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.38	active(take(0', IL)) -> mark(nil)
1126.86/294.38	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.38	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.38	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.38	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.38	active(s(X)) -> s(active(X))
1126.86/294.38	active(length(X)) -> length(active(X))
1126.86/294.38	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.38	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.38	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.38	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.38	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.38	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.38	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.38	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.38	s(mark(X)) -> mark(s(X))
1126.86/294.38	length(mark(X)) -> mark(length(X))
1126.86/294.38	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.38	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.38	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.38	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.38	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.38	proper(zeros) -> ok(zeros)
1126.86/294.38	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.38	proper(0') -> ok(0')
1126.86/294.38	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.38	proper(tt) -> ok(tt)
1126.86/294.38	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.38	proper(s(X)) -> s(proper(X))
1126.86/294.38	proper(length(X)) -> length(proper(X))
1126.86/294.38	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.38	proper(nil) -> ok(nil)
1126.86/294.38	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.38	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.38	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.38	s(ok(X)) -> ok(s(X))
1126.86/294.38	length(ok(X)) -> ok(length(X))
1126.86/294.38	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.38	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.38	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.38	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.38	top(mark(X)) -> top(proper(X))
1126.86/294.38	top(ok(X)) -> top(active(X))
1126.86/294.38	
1126.86/294.38	Types:
1126.86/294.38	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.38	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	U21 :: 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
1126.86/294.38	U22 :: 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
1126.86/294.38	U23 :: 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
1126.86/294.38	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.38	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.38	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.38	hole_top2_0 :: top
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Lemmas:
1126.86/294.38	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)
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Generator Equations:
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	The following defined symbols remain to be analysed:
1126.86/294.38	U12, active, s, length, U22, U23, take, U11, U21, proper, top
1126.86/294.38	
1126.86/294.38	They will be analysed ascendingly in the following order:
1126.86/294.38	U12 < active
1126.86/294.38	s < active
1126.86/294.38	length < active
1126.86/294.38	U22 < active
1126.86/294.38	U23 < active
1126.86/294.38	take < active
1126.86/294.38	U11 < active
1126.86/294.38	U21 < active
1126.86/294.38	active < top
1126.86/294.38	U12 < proper
1126.86/294.38	s < proper
1126.86/294.38	length < proper
1126.86/294.38	U22 < proper
1126.86/294.38	U23 < proper
1126.86/294.38	take < proper
1126.86/294.38	U11 < proper
1126.86/294.38	U21 < proper
1126.86/294.38	proper < top
1126.86/294.38	
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(13) RewriteLemmaProof (LOWER BOUND(ID))
1126.86/294.38	Proved the following rewrite lemma:
1126.86/294.38	U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
1126.86/294.38	
1126.86/294.38	Induction Base:
1126.86/294.38	U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))
1126.86/294.38	
1126.86/294.38	Induction Step:
1126.86/294.38	U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n1196_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
1126.86/294.38	mark(U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
1126.86/294.38	mark(*4_0)
1126.86/294.38	
1126.86/294.38	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(14)
1126.86/294.38	Obligation:
1126.86/294.38	TRS:
1126.86/294.38	Rules:
1126.86/294.38	active(zeros) -> mark(cons(0', zeros))
1126.86/294.38	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.38	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.38	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.38	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.38	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.38	active(length(nil)) -> mark(0')
1126.86/294.38	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.38	active(take(0', IL)) -> mark(nil)
1126.86/294.38	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.38	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.38	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.38	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.38	active(s(X)) -> s(active(X))
1126.86/294.38	active(length(X)) -> length(active(X))
1126.86/294.38	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.38	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.38	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.38	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.38	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.38	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.38	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.38	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.38	s(mark(X)) -> mark(s(X))
1126.86/294.38	length(mark(X)) -> mark(length(X))
1126.86/294.38	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.38	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.38	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.38	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.38	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.38	proper(zeros) -> ok(zeros)
1126.86/294.38	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.38	proper(0') -> ok(0')
1126.86/294.38	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.38	proper(tt) -> ok(tt)
1126.86/294.38	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.38	proper(s(X)) -> s(proper(X))
1126.86/294.38	proper(length(X)) -> length(proper(X))
1126.86/294.38	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.38	proper(nil) -> ok(nil)
1126.86/294.38	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.38	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.38	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.38	s(ok(X)) -> ok(s(X))
1126.86/294.38	length(ok(X)) -> ok(length(X))
1126.86/294.38	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.38	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.38	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.38	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.38	top(mark(X)) -> top(proper(X))
1126.86/294.38	top(ok(X)) -> top(active(X))
1126.86/294.38	
1126.86/294.38	Types:
1126.86/294.38	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.38	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	U21 :: 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
1126.86/294.38	U22 :: 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
1126.86/294.38	U23 :: 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
1126.86/294.38	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.38	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.38	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.38	hole_top2_0 :: top
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Lemmas:
1126.86/294.38	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)
1126.86/294.38	U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Generator Equations:
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	The following defined symbols remain to be analysed:
1126.86/294.38	s, active, length, U22, U23, take, U11, U21, proper, top
1126.86/294.38	
1126.86/294.38	They will be analysed ascendingly in the following order:
1126.86/294.38	s < active
1126.86/294.38	length < active
1126.86/294.38	U22 < active
1126.86/294.38	U23 < active
1126.86/294.38	take < active
1126.86/294.38	U11 < active
1126.86/294.38	U21 < active
1126.86/294.38	active < top
1126.86/294.38	s < proper
1126.86/294.38	length < proper
1126.86/294.38	U22 < proper
1126.86/294.38	U23 < proper
1126.86/294.38	take < proper
1126.86/294.38	U11 < proper
1126.86/294.38	U21 < proper
1126.86/294.38	proper < top
1126.86/294.38	
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(15) RewriteLemmaProof (LOWER BOUND(ID))
1126.86/294.38	Proved the following rewrite lemma:
1126.86/294.38	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
1126.86/294.38	
1126.86/294.38	Induction Base:
1126.86/294.38	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))
1126.86/294.38	
1126.86/294.38	Induction Step:
1126.86/294.38	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n2693_0, 1)))) ->_R^Omega(1)
1126.86/294.38	mark(s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0)))) ->_IH
1126.86/294.38	mark(*4_0)
1126.86/294.38	
1126.86/294.38	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(16)
1126.86/294.38	Obligation:
1126.86/294.38	TRS:
1126.86/294.38	Rules:
1126.86/294.38	active(zeros) -> mark(cons(0', zeros))
1126.86/294.38	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.38	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.38	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.38	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.38	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.38	active(length(nil)) -> mark(0')
1126.86/294.38	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.38	active(take(0', IL)) -> mark(nil)
1126.86/294.38	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.38	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.38	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.38	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.38	active(s(X)) -> s(active(X))
1126.86/294.38	active(length(X)) -> length(active(X))
1126.86/294.38	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.38	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.38	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.38	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.38	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.38	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.38	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.38	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.38	s(mark(X)) -> mark(s(X))
1126.86/294.38	length(mark(X)) -> mark(length(X))
1126.86/294.38	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.38	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.38	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.38	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.38	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.38	proper(zeros) -> ok(zeros)
1126.86/294.38	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.38	proper(0') -> ok(0')
1126.86/294.38	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.38	proper(tt) -> ok(tt)
1126.86/294.38	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.38	proper(s(X)) -> s(proper(X))
1126.86/294.38	proper(length(X)) -> length(proper(X))
1126.86/294.38	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.38	proper(nil) -> ok(nil)
1126.86/294.38	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.38	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.38	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.38	s(ok(X)) -> ok(s(X))
1126.86/294.38	length(ok(X)) -> ok(length(X))
1126.86/294.38	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.38	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.38	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.38	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.38	top(mark(X)) -> top(proper(X))
1126.86/294.38	top(ok(X)) -> top(active(X))
1126.86/294.38	
1126.86/294.38	Types:
1126.86/294.38	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.38	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	U21 :: 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
1126.86/294.38	U22 :: 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
1126.86/294.38	U23 :: 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
1126.86/294.38	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.38	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.38	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.38	hole_top2_0 :: top
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Lemmas:
1126.86/294.38	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)
1126.86/294.38	U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
1126.86/294.38	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Generator Equations:
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	The following defined symbols remain to be analysed:
1126.86/294.38	length, active, U22, U23, take, U11, U21, proper, top
1126.86/294.38	
1126.86/294.38	They will be analysed ascendingly in the following order:
1126.86/294.38	length < active
1126.86/294.38	U22 < active
1126.86/294.38	U23 < active
1126.86/294.38	take < active
1126.86/294.38	U11 < active
1126.86/294.38	U21 < active
1126.86/294.38	active < top
1126.86/294.38	length < proper
1126.86/294.38	U22 < proper
1126.86/294.38	U23 < proper
1126.86/294.38	take < proper
1126.86/294.38	U11 < proper
1126.86/294.38	U21 < proper
1126.86/294.38	proper < top
1126.86/294.38	
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(17) RewriteLemmaProof (LOWER BOUND(ID))
1126.86/294.38	Proved the following rewrite lemma:
1126.86/294.38	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
1126.86/294.38	
1126.86/294.38	Induction Base:
1126.86/294.38	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)))
1126.86/294.38	
1126.86/294.38	Induction Step:
1126.86/294.38	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n3422_0, 1)))) ->_R^Omega(1)
1126.86/294.38	mark(length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0)))) ->_IH
1126.86/294.38	mark(*4_0)
1126.86/294.38	
1126.86/294.38	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(18)
1126.86/294.38	Obligation:
1126.86/294.38	TRS:
1126.86/294.38	Rules:
1126.86/294.38	active(zeros) -> mark(cons(0', zeros))
1126.86/294.38	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.38	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.38	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.38	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.38	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.38	active(length(nil)) -> mark(0')
1126.86/294.38	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.38	active(take(0', IL)) -> mark(nil)
1126.86/294.38	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.38	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.38	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.38	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.38	active(s(X)) -> s(active(X))
1126.86/294.38	active(length(X)) -> length(active(X))
1126.86/294.38	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.38	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.38	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.38	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.38	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.38	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.38	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.38	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.38	s(mark(X)) -> mark(s(X))
1126.86/294.38	length(mark(X)) -> mark(length(X))
1126.86/294.38	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.38	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.38	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.38	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.38	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.38	proper(zeros) -> ok(zeros)
1126.86/294.38	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.38	proper(0') -> ok(0')
1126.86/294.38	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.38	proper(tt) -> ok(tt)
1126.86/294.38	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.38	proper(s(X)) -> s(proper(X))
1126.86/294.38	proper(length(X)) -> length(proper(X))
1126.86/294.38	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.38	proper(nil) -> ok(nil)
1126.86/294.38	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.38	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.38	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.38	s(ok(X)) -> ok(s(X))
1126.86/294.38	length(ok(X)) -> ok(length(X))
1126.86/294.38	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.38	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.38	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.38	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.38	top(mark(X)) -> top(proper(X))
1126.86/294.38	top(ok(X)) -> top(active(X))
1126.86/294.38	
1126.86/294.38	Types:
1126.86/294.38	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.38	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	U21 :: 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
1126.86/294.38	U22 :: 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
1126.86/294.38	U23 :: 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
1126.86/294.38	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.38	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.38	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.38	hole_top2_0 :: top
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Lemmas:
1126.86/294.38	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)
1126.86/294.38	U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
1126.86/294.38	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
1126.86/294.38	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Generator Equations:
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	The following defined symbols remain to be analysed:
1126.86/294.38	U22, active, U23, take, U11, U21, proper, top
1126.86/294.38	
1126.86/294.38	They will be analysed ascendingly in the following order:
1126.86/294.38	U22 < active
1126.86/294.38	U23 < active
1126.86/294.38	take < active
1126.86/294.38	U11 < active
1126.86/294.38	U21 < active
1126.86/294.38	active < top
1126.86/294.38	U22 < proper
1126.86/294.38	U23 < proper
1126.86/294.38	take < proper
1126.86/294.38	U11 < proper
1126.86/294.38	U21 < proper
1126.86/294.38	proper < top
1126.86/294.38	
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(19) RewriteLemmaProof (LOWER BOUND(ID))
1126.86/294.38	Proved the following rewrite lemma:
1126.86/294.38	U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_0)
1126.86/294.38	
1126.86/294.38	Induction Base:
1126.86/294.38	U22(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))
1126.86/294.38	
1126.86/294.38	Induction Step:
1126.86/294.38	U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n4252_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)
1126.86/294.38	mark(U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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
1126.86/294.38	mark(*4_0)
1126.86/294.38	
1126.86/294.38	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(20)
1126.86/294.38	Obligation:
1126.86/294.38	TRS:
1126.86/294.38	Rules:
1126.86/294.38	active(zeros) -> mark(cons(0', zeros))
1126.86/294.38	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.38	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.38	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.38	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.38	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.38	active(length(nil)) -> mark(0')
1126.86/294.38	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.38	active(take(0', IL)) -> mark(nil)
1126.86/294.38	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.38	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.38	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.38	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.38	active(s(X)) -> s(active(X))
1126.86/294.38	active(length(X)) -> length(active(X))
1126.86/294.38	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.38	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.38	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.38	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.38	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.38	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.38	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.38	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.38	s(mark(X)) -> mark(s(X))
1126.86/294.38	length(mark(X)) -> mark(length(X))
1126.86/294.38	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.38	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.38	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.38	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.38	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.38	proper(zeros) -> ok(zeros)
1126.86/294.38	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.38	proper(0') -> ok(0')
1126.86/294.38	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.38	proper(tt) -> ok(tt)
1126.86/294.38	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.38	proper(s(X)) -> s(proper(X))
1126.86/294.38	proper(length(X)) -> length(proper(X))
1126.86/294.38	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.38	proper(nil) -> ok(nil)
1126.86/294.38	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.38	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.38	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.38	s(ok(X)) -> ok(s(X))
1126.86/294.38	length(ok(X)) -> ok(length(X))
1126.86/294.38	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.38	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.38	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.38	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.38	top(mark(X)) -> top(proper(X))
1126.86/294.38	top(ok(X)) -> top(active(X))
1126.86/294.38	
1126.86/294.38	Types:
1126.86/294.38	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.38	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	U21 :: 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
1126.86/294.38	U22 :: 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
1126.86/294.38	U23 :: 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
1126.86/294.38	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.38	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.38	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.38	hole_top2_0 :: top
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Lemmas:
1126.86/294.38	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)
1126.86/294.38	U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
1126.86/294.38	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
1126.86/294.38	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
1126.86/294.38	U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_0)
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Generator Equations:
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	The following defined symbols remain to be analysed:
1126.86/294.38	U23, active, take, U11, U21, proper, top
1126.86/294.38	
1126.86/294.38	They will be analysed ascendingly in the following order:
1126.86/294.38	U23 < active
1126.86/294.38	take < active
1126.86/294.38	U11 < active
1126.86/294.38	U21 < active
1126.86/294.38	active < top
1126.86/294.38	U23 < proper
1126.86/294.38	take < proper
1126.86/294.38	U11 < proper
1126.86/294.38	U21 < proper
1126.86/294.38	proper < top
1126.86/294.38	
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(21) RewriteLemmaProof (LOWER BOUND(ID))
1126.86/294.38	Proved the following rewrite lemma:
1126.86/294.38	U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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(n9047_0)
1126.86/294.38	
1126.86/294.38	Induction Base:
1126.86/294.38	U23(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))
1126.86/294.38	
1126.86/294.38	Induction Step:
1126.86/294.38	U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n9047_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)
1126.86/294.38	mark(U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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
1126.86/294.38	mark(*4_0)
1126.86/294.38	
1126.86/294.38	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(22)
1126.86/294.38	Obligation:
1126.86/294.38	TRS:
1126.86/294.38	Rules:
1126.86/294.38	active(zeros) -> mark(cons(0', zeros))
1126.86/294.38	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.38	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.38	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.38	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.38	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.38	active(length(nil)) -> mark(0')
1126.86/294.38	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.38	active(take(0', IL)) -> mark(nil)
1126.86/294.38	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.38	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.38	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.38	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.38	active(s(X)) -> s(active(X))
1126.86/294.38	active(length(X)) -> length(active(X))
1126.86/294.38	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.38	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.38	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.38	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.38	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.38	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.38	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.38	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.38	s(mark(X)) -> mark(s(X))
1126.86/294.38	length(mark(X)) -> mark(length(X))
1126.86/294.38	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.38	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.38	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.38	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.38	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.38	proper(zeros) -> ok(zeros)
1126.86/294.38	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.38	proper(0') -> ok(0')
1126.86/294.38	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.38	proper(tt) -> ok(tt)
1126.86/294.38	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.38	proper(s(X)) -> s(proper(X))
1126.86/294.38	proper(length(X)) -> length(proper(X))
1126.86/294.38	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.38	proper(nil) -> ok(nil)
1126.86/294.38	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.38	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.38	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.38	s(ok(X)) -> ok(s(X))
1126.86/294.38	length(ok(X)) -> ok(length(X))
1126.86/294.38	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.38	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.38	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.38	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.38	top(mark(X)) -> top(proper(X))
1126.86/294.38	top(ok(X)) -> top(active(X))
1126.86/294.38	
1126.86/294.38	Types:
1126.86/294.38	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.38	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	U21 :: 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
1126.86/294.38	U22 :: 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
1126.86/294.38	U23 :: 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
1126.86/294.38	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.38	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.38	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.38	hole_top2_0 :: top
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Lemmas:
1126.86/294.38	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)
1126.86/294.38	U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
1126.86/294.38	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
1126.86/294.38	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
1126.86/294.38	U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_0)
1126.86/294.38	U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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(n9047_0)
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Generator Equations:
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	The following defined symbols remain to be analysed:
1126.86/294.38	take, active, U11, U21, proper, top
1126.86/294.38	
1126.86/294.38	They will be analysed ascendingly in the following order:
1126.86/294.38	take < active
1126.86/294.38	U11 < active
1126.86/294.38	U21 < active
1126.86/294.38	active < top
1126.86/294.38	take < proper
1126.86/294.38	U11 < proper
1126.86/294.38	U21 < proper
1126.86/294.38	proper < top
1126.86/294.38	
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(23) RewriteLemmaProof (LOWER BOUND(ID))
1126.86/294.38	Proved the following rewrite lemma:
1126.86/294.38	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14852_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n14852_0)
1126.86/294.38	
1126.86/294.38	Induction Base:
1126.86/294.38	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))
1126.86/294.38	
1126.86/294.38	Induction Step:
1126.86/294.38	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n14852_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
1126.86/294.38	mark(take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14852_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
1126.86/294.38	mark(*4_0)
1126.86/294.38	
1126.86/294.38	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(24)
1126.86/294.38	Obligation:
1126.86/294.38	TRS:
1126.86/294.38	Rules:
1126.86/294.38	active(zeros) -> mark(cons(0', zeros))
1126.86/294.38	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.38	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.38	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.38	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.38	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.38	active(length(nil)) -> mark(0')
1126.86/294.38	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.38	active(take(0', IL)) -> mark(nil)
1126.86/294.38	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.38	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.38	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.38	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.38	active(s(X)) -> s(active(X))
1126.86/294.38	active(length(X)) -> length(active(X))
1126.86/294.38	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.38	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.38	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.38	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.38	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.38	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.38	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.38	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.38	s(mark(X)) -> mark(s(X))
1126.86/294.38	length(mark(X)) -> mark(length(X))
1126.86/294.38	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.38	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.38	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.38	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.38	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.38	proper(zeros) -> ok(zeros)
1126.86/294.38	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.38	proper(0') -> ok(0')
1126.86/294.38	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.38	proper(tt) -> ok(tt)
1126.86/294.38	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.38	proper(s(X)) -> s(proper(X))
1126.86/294.38	proper(length(X)) -> length(proper(X))
1126.86/294.38	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.38	proper(nil) -> ok(nil)
1126.86/294.38	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.38	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.38	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.38	s(ok(X)) -> ok(s(X))
1126.86/294.38	length(ok(X)) -> ok(length(X))
1126.86/294.38	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.38	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.38	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.38	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.38	top(mark(X)) -> top(proper(X))
1126.86/294.38	top(ok(X)) -> top(active(X))
1126.86/294.38	
1126.86/294.38	Types:
1126.86/294.38	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.38	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	U21 :: 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
1126.86/294.38	U22 :: 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
1126.86/294.38	U23 :: 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
1126.86/294.38	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.38	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.38	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.38	hole_top2_0 :: top
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Lemmas:
1126.86/294.38	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)
1126.86/294.38	U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
1126.86/294.38	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
1126.86/294.38	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
1126.86/294.38	U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_0)
1126.86/294.38	U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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(n9047_0)
1126.86/294.38	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14852_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n14852_0)
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Generator Equations:
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	The following defined symbols remain to be analysed:
1126.86/294.38	U11, active, U21, proper, top
1126.86/294.38	
1126.86/294.38	They will be analysed ascendingly in the following order:
1126.86/294.38	U11 < active
1126.86/294.38	U21 < active
1126.86/294.38	active < top
1126.86/294.38	U11 < proper
1126.86/294.38	U21 < proper
1126.86/294.38	proper < top
1126.86/294.38	
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(25) RewriteLemmaProof (LOWER BOUND(ID))
1126.86/294.38	Proved the following rewrite lemma:
1126.86/294.38	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18274_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n18274_0)
1126.86/294.38	
1126.86/294.38	Induction Base:
1126.86/294.38	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, 0)), gen_zeros:0':mark:tt:nil:ok3_0(b))
1126.86/294.38	
1126.86/294.38	Induction Step:
1126.86/294.38	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n18274_0, 1))), gen_zeros:0':mark:tt:nil:ok3_0(b)) ->_R^Omega(1)
1126.86/294.38	mark(U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18274_0)), gen_zeros:0':mark:tt:nil:ok3_0(b))) ->_IH
1126.86/294.38	mark(*4_0)
1126.86/294.38	
1126.86/294.38	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.86/294.38	----------------------------------------
1126.86/294.38	
1126.86/294.38	(26)
1126.86/294.38	Obligation:
1126.86/294.38	TRS:
1126.86/294.38	Rules:
1126.86/294.38	active(zeros) -> mark(cons(0', zeros))
1126.86/294.38	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.38	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.38	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.38	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.38	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.38	active(length(nil)) -> mark(0')
1126.86/294.38	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.38	active(take(0', IL)) -> mark(nil)
1126.86/294.38	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.38	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.38	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.38	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.38	active(s(X)) -> s(active(X))
1126.86/294.38	active(length(X)) -> length(active(X))
1126.86/294.38	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.38	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.38	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.38	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.38	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.38	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.38	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.38	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.38	s(mark(X)) -> mark(s(X))
1126.86/294.38	length(mark(X)) -> mark(length(X))
1126.86/294.38	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.38	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.38	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.38	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.38	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.38	proper(zeros) -> ok(zeros)
1126.86/294.38	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.38	proper(0') -> ok(0')
1126.86/294.38	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.38	proper(tt) -> ok(tt)
1126.86/294.38	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.38	proper(s(X)) -> s(proper(X))
1126.86/294.38	proper(length(X)) -> length(proper(X))
1126.86/294.38	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.38	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.38	proper(nil) -> ok(nil)
1126.86/294.38	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.38	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.38	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.38	s(ok(X)) -> ok(s(X))
1126.86/294.38	length(ok(X)) -> ok(length(X))
1126.86/294.38	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.38	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.38	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.38	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.38	top(mark(X)) -> top(proper(X))
1126.86/294.38	top(ok(X)) -> top(active(X))
1126.86/294.38	
1126.86/294.38	Types:
1126.86/294.38	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.38	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.38	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	U21 :: 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
1126.86/294.38	U22 :: 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
1126.86/294.38	U23 :: 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
1126.86/294.38	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.38	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.38	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.38	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.38	hole_top2_0 :: top
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Lemmas:
1126.86/294.38	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)
1126.86/294.38	U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
1126.86/294.38	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
1126.86/294.38	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
1126.86/294.38	U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_0)
1126.86/294.38	U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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(n9047_0)
1126.86/294.38	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14852_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n14852_0)
1126.86/294.38	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18274_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n18274_0)
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	Generator Equations:
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1126.86/294.38	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1126.86/294.38	
1126.86/294.38	
1126.86/294.38	The following defined symbols remain to be analysed:
1126.86/294.39	U21, active, proper, top
1126.86/294.39	
1126.86/294.39	They will be analysed ascendingly in the following order:
1126.86/294.39	U21 < active
1126.86/294.39	active < top
1126.86/294.39	U21 < proper
1126.86/294.39	proper < top
1126.86/294.39	
1126.86/294.39	----------------------------------------
1126.86/294.39	
1126.86/294.39	(27) RewriteLemmaProof (LOWER BOUND(ID))
1126.86/294.39	Proved the following rewrite lemma:
1126.86/294.39	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21803_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(n21803_0)
1126.86/294.39	
1126.86/294.39	Induction Base:
1126.86/294.39	U21(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))
1126.86/294.39	
1126.86/294.39	Induction Step:
1126.86/294.39	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, +(n21803_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)
1126.86/294.39	mark(U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21803_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
1126.86/294.39	mark(*4_0)
1126.86/294.39	
1126.86/294.39	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.86/294.39	----------------------------------------
1126.86/294.39	
1126.86/294.39	(28)
1126.86/294.39	Obligation:
1126.86/294.39	TRS:
1126.86/294.39	Rules:
1126.86/294.39	active(zeros) -> mark(cons(0', zeros))
1126.86/294.39	active(U11(tt, L)) -> mark(U12(tt, L))
1126.86/294.39	active(U12(tt, L)) -> mark(s(length(L)))
1126.86/294.39	active(U21(tt, IL, M, N)) -> mark(U22(tt, IL, M, N))
1126.86/294.39	active(U22(tt, IL, M, N)) -> mark(U23(tt, IL, M, N))
1126.86/294.39	active(U23(tt, IL, M, N)) -> mark(cons(N, take(M, IL)))
1126.86/294.39	active(length(nil)) -> mark(0')
1126.86/294.39	active(length(cons(N, L))) -> mark(U11(tt, L))
1126.86/294.39	active(take(0', IL)) -> mark(nil)
1126.86/294.39	active(take(s(M), cons(N, IL))) -> mark(U21(tt, IL, M, N))
1126.86/294.39	active(cons(X1, X2)) -> cons(active(X1), X2)
1126.86/294.39	active(U11(X1, X2)) -> U11(active(X1), X2)
1126.86/294.39	active(U12(X1, X2)) -> U12(active(X1), X2)
1126.86/294.39	active(s(X)) -> s(active(X))
1126.86/294.39	active(length(X)) -> length(active(X))
1126.86/294.39	active(U21(X1, X2, X3, X4)) -> U21(active(X1), X2, X3, X4)
1126.86/294.39	active(U22(X1, X2, X3, X4)) -> U22(active(X1), X2, X3, X4)
1126.86/294.39	active(U23(X1, X2, X3, X4)) -> U23(active(X1), X2, X3, X4)
1126.86/294.39	active(take(X1, X2)) -> take(active(X1), X2)
1126.86/294.39	active(take(X1, X2)) -> take(X1, active(X2))
1126.86/294.39	cons(mark(X1), X2) -> mark(cons(X1, X2))
1126.86/294.39	U11(mark(X1), X2) -> mark(U11(X1, X2))
1126.86/294.39	U12(mark(X1), X2) -> mark(U12(X1, X2))
1126.86/294.39	s(mark(X)) -> mark(s(X))
1126.86/294.39	length(mark(X)) -> mark(length(X))
1126.86/294.39	U21(mark(X1), X2, X3, X4) -> mark(U21(X1, X2, X3, X4))
1126.86/294.39	U22(mark(X1), X2, X3, X4) -> mark(U22(X1, X2, X3, X4))
1126.86/294.39	U23(mark(X1), X2, X3, X4) -> mark(U23(X1, X2, X3, X4))
1126.86/294.39	take(mark(X1), X2) -> mark(take(X1, X2))
1126.86/294.39	take(X1, mark(X2)) -> mark(take(X1, X2))
1126.86/294.39	proper(zeros) -> ok(zeros)
1126.86/294.39	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
1126.86/294.39	proper(0') -> ok(0')
1126.86/294.39	proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
1126.86/294.39	proper(tt) -> ok(tt)
1126.86/294.39	proper(U12(X1, X2)) -> U12(proper(X1), proper(X2))
1126.86/294.39	proper(s(X)) -> s(proper(X))
1126.86/294.39	proper(length(X)) -> length(proper(X))
1126.86/294.39	proper(U21(X1, X2, X3, X4)) -> U21(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.39	proper(U22(X1, X2, X3, X4)) -> U22(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.39	proper(U23(X1, X2, X3, X4)) -> U23(proper(X1), proper(X2), proper(X3), proper(X4))
1126.86/294.39	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
1126.86/294.39	proper(nil) -> ok(nil)
1126.86/294.39	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
1126.86/294.39	U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
1126.86/294.39	U12(ok(X1), ok(X2)) -> ok(U12(X1, X2))
1126.86/294.39	s(ok(X)) -> ok(s(X))
1126.86/294.39	length(ok(X)) -> ok(length(X))
1126.86/294.39	U21(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U21(X1, X2, X3, X4))
1126.86/294.39	U22(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U22(X1, X2, X3, X4))
1126.86/294.39	U23(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(U23(X1, X2, X3, X4))
1126.86/294.39	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
1126.86/294.39	top(mark(X)) -> top(proper(X))
1126.86/294.39	top(ok(X)) -> top(active(X))
1126.86/294.39	
1126.86/294.39	Types:
1126.86/294.39	active :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.39	zeros :: zeros:0':mark:tt:nil:ok
1126.86/294.39	mark :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.39	cons :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.39	0' :: zeros:0':mark:tt:nil:ok
1126.86/294.39	U11 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.39	tt :: zeros:0':mark:tt:nil:ok
1126.86/294.39	U12 :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.39	s :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.39	length :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.39	U21 :: 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
1126.86/294.39	U22 :: 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
1126.86/294.39	U23 :: 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
1126.86/294.39	take :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.39	nil :: zeros:0':mark:tt:nil:ok
1126.86/294.39	proper :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.39	ok :: zeros:0':mark:tt:nil:ok -> zeros:0':mark:tt:nil:ok
1126.86/294.39	top :: zeros:0':mark:tt:nil:ok -> top
1126.86/294.39	hole_zeros:0':mark:tt:nil:ok1_0 :: zeros:0':mark:tt:nil:ok
1126.86/294.39	hole_top2_0 :: top
1126.86/294.39	gen_zeros:0':mark:tt:nil:ok3_0 :: Nat -> zeros:0':mark:tt:nil:ok
1126.86/294.39	
1126.86/294.39	
1126.86/294.39	Lemmas:
1126.86/294.39	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)
1126.86/294.39	U12(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n1196_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n1196_0)
1126.86/294.39	s(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n2693_0))) -> *4_0, rt in Omega(n2693_0)
1126.86/294.39	length(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n3422_0))) -> *4_0, rt in Omega(n3422_0)
1126.86/294.39	U22(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n4252_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(n4252_0)
1126.86/294.39	U23(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n9047_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(n9047_0)
1126.86/294.39	take(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n14852_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n14852_0)
1126.86/294.39	U11(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n18274_0)), gen_zeros:0':mark:tt:nil:ok3_0(b)) -> *4_0, rt in Omega(n18274_0)
1126.86/294.39	U21(gen_zeros:0':mark:tt:nil:ok3_0(+(1, n21803_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(n21803_0)
1126.86/294.39	
1126.86/294.39	
1126.86/294.39	Generator Equations:
1126.86/294.39	gen_zeros:0':mark:tt:nil:ok3_0(0) <=> zeros
1126.86/294.39	gen_zeros:0':mark:tt:nil:ok3_0(+(x, 1)) <=> mark(gen_zeros:0':mark:tt:nil:ok3_0(x))
1126.86/294.39	
1126.86/294.39	
1126.86/294.39	The following defined symbols remain to be analysed:
1126.86/294.39	active, proper, top
1126.86/294.39	
1126.86/294.39	They will be analysed ascendingly in the following order:
1126.86/294.39	active < top
1126.86/294.39	proper < top
1127.18/294.49	EOF