38.10/11.00	WORST_CASE(Omega(n^1), O(n^1))
38.34/11.02	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
38.34/11.02	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
38.34/11.02	
38.34/11.02	
38.34/11.02	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
38.34/11.02	
38.34/11.02	(0) CpxTRS
38.34/11.02	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
38.34/11.02	(2) CpxTRS
38.34/11.02	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
38.34/11.02	    (4) CpxTRS
38.34/11.02	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 342 ms]
38.34/11.02	    (6) BOUNDS(1, n^1)
38.34/11.02	(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
38.34/11.02	(8) CpxTRS
38.34/11.02	    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
38.34/11.02	    (10) typed CpxTrs
38.34/11.02	    (11) OrderProof [LOWER BOUND(ID), 0 ms]
38.34/11.02	    (12) typed CpxTrs
38.34/11.02	    (13) RewriteLemmaProof [LOWER BOUND(ID), 488 ms]
38.34/11.02	    (14) BEST
38.34/11.02	        (15) proven lower bound
38.34/11.02	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
38.34/11.02	            (17) BOUNDS(n^1, INF)
38.34/11.02	        (18) typed CpxTrs
38.34/11.02	            (19) RewriteLemmaProof [LOWER BOUND(ID), 118 ms]
38.34/11.02	            (20) typed CpxTrs
38.34/11.02	            (21) RewriteLemmaProof [LOWER BOUND(ID), 65 ms]
38.34/11.02	            (22) typed CpxTrs
38.34/11.02	            (23) RewriteLemmaProof [LOWER BOUND(ID), 62 ms]
38.34/11.02	            (24) typed CpxTrs
38.34/11.02	            (25) RewriteLemmaProof [LOWER BOUND(ID), 125 ms]
38.34/11.02	            (26) typed CpxTrs
38.34/11.02	            (27) RewriteLemmaProof [LOWER BOUND(ID), 94 ms]
38.34/11.02	            (28) typed CpxTrs
38.34/11.02	            (29) RewriteLemmaProof [LOWER BOUND(ID), 92 ms]
38.34/11.02	            (30) typed CpxTrs
38.34/11.02	            (31) RewriteLemmaProof [LOWER BOUND(ID), 70 ms]
38.34/11.02	            (32) typed CpxTrs
38.34/11.02	            (33) RewriteLemmaProof [LOWER BOUND(ID), 114 ms]
38.34/11.02	            (34) typed CpxTrs
38.34/11.02	            (35) RewriteLemmaProof [LOWER BOUND(ID), 123 ms]
38.34/11.02	            (36) typed CpxTrs
38.34/11.02	            (37) RewriteLemmaProof [LOWER BOUND(ID), 135 ms]
38.34/11.02	            (38) typed CpxTrs
38.34/11.02	            (39) RewriteLemmaProof [LOWER BOUND(ID), 134 ms]
38.34/11.02	            (40) typed CpxTrs
38.34/11.02	            (41) RewriteLemmaProof [LOWER BOUND(ID), 159 ms]
38.34/11.02	            (42) typed CpxTrs
38.34/11.02	
38.34/11.02	
38.34/11.02	----------------------------------------
38.34/11.02	
38.34/11.02	(0)
38.34/11.02	Obligation:
38.34/11.02	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
38.34/11.02	
38.34/11.02	
38.34/11.02	The TRS R consists of the following rules:
38.34/11.02	
38.34/11.02	   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.34/11.02	   active(fst(pair(XS, YS))) -> mark(XS)
38.34/11.02	   active(snd(pair(XS, YS))) -> mark(YS)
38.34/11.02	   active(splitAt(0, XS)) -> mark(pair(nil, XS))
38.34/11.02	   active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.34/11.02	   active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.34/11.02	   active(head(cons(N, XS))) -> mark(N)
38.34/11.02	   active(tail(cons(N, XS))) -> mark(XS)
38.34/11.02	   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.34/11.02	   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.34/11.02	   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.34/11.02	   active(natsFrom(X)) -> natsFrom(active(X))
38.34/11.02	   active(cons(X1, X2)) -> cons(active(X1), X2)
38.34/11.02	   active(s(X)) -> s(active(X))
38.34/11.02	   active(fst(X)) -> fst(active(X))
38.34/11.02	   active(pair(X1, X2)) -> pair(active(X1), X2)
38.34/11.02	   active(pair(X1, X2)) -> pair(X1, active(X2))
38.34/11.02	   active(snd(X)) -> snd(active(X))
38.34/11.02	   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.34/11.02	   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.34/11.02	   active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.34/11.02	   active(head(X)) -> head(active(X))
38.34/11.02	   active(tail(X)) -> tail(active(X))
38.34/11.02	   active(sel(X1, X2)) -> sel(active(X1), X2)
38.34/11.02	   active(sel(X1, X2)) -> sel(X1, active(X2))
38.34/11.02	   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.34/11.02	   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.34/11.02	   active(take(X1, X2)) -> take(active(X1), X2)
38.34/11.02	   active(take(X1, X2)) -> take(X1, active(X2))
38.34/11.02	   natsFrom(mark(X)) -> mark(natsFrom(X))
38.34/11.02	   cons(mark(X1), X2) -> mark(cons(X1, X2))
38.34/11.02	   s(mark(X)) -> mark(s(X))
38.34/11.02	   fst(mark(X)) -> mark(fst(X))
38.34/11.02	   pair(mark(X1), X2) -> mark(pair(X1, X2))
38.34/11.02	   pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.34/11.02	   snd(mark(X)) -> mark(snd(X))
38.34/11.02	   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.34/11.02	   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.34/11.02	   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.34/11.02	   head(mark(X)) -> mark(head(X))
38.34/11.02	   tail(mark(X)) -> mark(tail(X))
38.34/11.02	   sel(mark(X1), X2) -> mark(sel(X1, X2))
38.34/11.02	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.34/11.02	   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.34/11.02	   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.34/11.02	   take(mark(X1), X2) -> mark(take(X1, X2))
38.34/11.02	   take(X1, mark(X2)) -> mark(take(X1, X2))
38.34/11.02	   proper(natsFrom(X)) -> natsFrom(proper(X))
38.34/11.02	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.34/11.02	   proper(s(X)) -> s(proper(X))
38.34/11.02	   proper(fst(X)) -> fst(proper(X))
38.34/11.02	   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.34/11.02	   proper(snd(X)) -> snd(proper(X))
38.34/11.02	   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.34/11.02	   proper(0) -> ok(0)
38.34/11.02	   proper(nil) -> ok(nil)
38.34/11.02	   proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.34/11.02	   proper(head(X)) -> head(proper(X))
38.34/11.02	   proper(tail(X)) -> tail(proper(X))
38.34/11.02	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.34/11.02	   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.34/11.02	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.34/11.02	   natsFrom(ok(X)) -> ok(natsFrom(X))
38.34/11.02	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.34/11.02	   s(ok(X)) -> ok(s(X))
38.34/11.02	   fst(ok(X)) -> ok(fst(X))
38.34/11.02	   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.34/11.02	   snd(ok(X)) -> ok(snd(X))
38.34/11.02	   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.34/11.02	   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.34/11.02	   head(ok(X)) -> ok(head(X))
38.34/11.02	   tail(ok(X)) -> ok(tail(X))
38.34/11.02	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.34/11.02	   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.34/11.02	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.34/11.02	   top(mark(X)) -> top(proper(X))
38.34/11.02	   top(ok(X)) -> top(active(X))
38.34/11.02	
38.34/11.02	S is empty.
38.34/11.02	Rewrite Strategy: FULL
38.34/11.02	----------------------------------------
38.34/11.02	
38.34/11.02	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
38.34/11.02	The following defined symbols can occur below the 0th argument of top: proper, active
38.34/11.02	The following defined symbols can occur below the 0th argument of proper: proper, active
38.34/11.02	The following defined symbols can occur below the 0th argument of active: proper, active
38.34/11.02	
38.34/11.02	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
38.34/11.02	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.34/11.02	active(fst(pair(XS, YS))) -> mark(XS)
38.34/11.02	active(snd(pair(XS, YS))) -> mark(YS)
38.34/11.02	active(splitAt(0, XS)) -> mark(pair(nil, XS))
38.34/11.02	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.34/11.02	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.34/11.02	active(head(cons(N, XS))) -> mark(N)
38.34/11.02	active(tail(cons(N, XS))) -> mark(XS)
38.34/11.02	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.34/11.02	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.34/11.02	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.34/11.02	active(natsFrom(X)) -> natsFrom(active(X))
38.34/11.02	active(cons(X1, X2)) -> cons(active(X1), X2)
38.34/11.02	active(s(X)) -> s(active(X))
38.34/11.02	active(fst(X)) -> fst(active(X))
38.34/11.02	active(pair(X1, X2)) -> pair(active(X1), X2)
38.34/11.02	active(pair(X1, X2)) -> pair(X1, active(X2))
38.34/11.02	active(snd(X)) -> snd(active(X))
38.34/11.02	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.34/11.02	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.34/11.02	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.34/11.02	active(head(X)) -> head(active(X))
38.34/11.02	active(tail(X)) -> tail(active(X))
38.34/11.02	active(sel(X1, X2)) -> sel(active(X1), X2)
38.34/11.02	active(sel(X1, X2)) -> sel(X1, active(X2))
38.34/11.02	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.34/11.02	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.34/11.02	active(take(X1, X2)) -> take(active(X1), X2)
38.34/11.02	active(take(X1, X2)) -> take(X1, active(X2))
38.34/11.02	proper(natsFrom(X)) -> natsFrom(proper(X))
38.34/11.02	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.34/11.02	proper(s(X)) -> s(proper(X))
38.34/11.02	proper(fst(X)) -> fst(proper(X))
38.34/11.02	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.34/11.02	proper(snd(X)) -> snd(proper(X))
38.34/11.02	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.34/11.02	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.34/11.02	proper(head(X)) -> head(proper(X))
38.34/11.02	proper(tail(X)) -> tail(proper(X))
38.34/11.02	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.34/11.02	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.34/11.02	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.34/11.02	
38.34/11.02	----------------------------------------
38.34/11.02	
38.34/11.02	(2)
38.34/11.02	Obligation:
38.34/11.02	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
38.34/11.04	
38.34/11.04	
38.34/11.04	The TRS R consists of the following rules:
38.34/11.04	
38.34/11.04	   natsFrom(mark(X)) -> mark(natsFrom(X))
38.34/11.04	   cons(mark(X1), X2) -> mark(cons(X1, X2))
38.34/11.04	   s(mark(X)) -> mark(s(X))
38.34/11.04	   fst(mark(X)) -> mark(fst(X))
38.34/11.04	   pair(mark(X1), X2) -> mark(pair(X1, X2))
38.34/11.04	   pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.34/11.04	   snd(mark(X)) -> mark(snd(X))
38.34/11.04	   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.34/11.04	   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.34/11.04	   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.34/11.04	   head(mark(X)) -> mark(head(X))
38.34/11.04	   tail(mark(X)) -> mark(tail(X))
38.34/11.04	   sel(mark(X1), X2) -> mark(sel(X1, X2))
38.34/11.04	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.34/11.04	   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.34/11.04	   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.34/11.04	   take(mark(X1), X2) -> mark(take(X1, X2))
38.34/11.04	   take(X1, mark(X2)) -> mark(take(X1, X2))
38.34/11.04	   proper(0) -> ok(0)
38.34/11.04	   proper(nil) -> ok(nil)
38.34/11.04	   natsFrom(ok(X)) -> ok(natsFrom(X))
38.34/11.04	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.34/11.04	   s(ok(X)) -> ok(s(X))
38.34/11.04	   fst(ok(X)) -> ok(fst(X))
38.34/11.04	   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.34/11.04	   snd(ok(X)) -> ok(snd(X))
38.34/11.04	   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.34/11.04	   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.34/11.04	   head(ok(X)) -> ok(head(X))
38.34/11.04	   tail(ok(X)) -> ok(tail(X))
38.34/11.04	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.34/11.04	   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.34/11.04	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.34/11.04	   top(mark(X)) -> top(proper(X))
38.34/11.04	   top(ok(X)) -> top(active(X))
38.34/11.04	
38.34/11.04	S is empty.
38.34/11.04	Rewrite Strategy: FULL
38.34/11.04	----------------------------------------
38.34/11.04	
38.34/11.04	(3) RelTrsToTrsProof (UPPER BOUND(ID))
38.34/11.04	transformed relative TRS to TRS
38.34/11.04	----------------------------------------
38.34/11.04	
38.34/11.04	(4)
38.34/11.04	Obligation:
38.34/11.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
38.34/11.04	
38.34/11.04	
38.34/11.04	The TRS R consists of the following rules:
38.34/11.04	
38.34/11.04	   natsFrom(mark(X)) -> mark(natsFrom(X))
38.34/11.04	   cons(mark(X1), X2) -> mark(cons(X1, X2))
38.34/11.04	   s(mark(X)) -> mark(s(X))
38.34/11.04	   fst(mark(X)) -> mark(fst(X))
38.34/11.04	   pair(mark(X1), X2) -> mark(pair(X1, X2))
38.34/11.04	   pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.34/11.04	   snd(mark(X)) -> mark(snd(X))
38.34/11.04	   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.34/11.04	   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.34/11.04	   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.34/11.04	   head(mark(X)) -> mark(head(X))
38.34/11.04	   tail(mark(X)) -> mark(tail(X))
38.34/11.04	   sel(mark(X1), X2) -> mark(sel(X1, X2))
38.34/11.04	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.34/11.04	   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.34/11.04	   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.34/11.04	   take(mark(X1), X2) -> mark(take(X1, X2))
38.34/11.04	   take(X1, mark(X2)) -> mark(take(X1, X2))
38.34/11.04	   proper(0) -> ok(0)
38.34/11.04	   proper(nil) -> ok(nil)
38.34/11.04	   natsFrom(ok(X)) -> ok(natsFrom(X))
38.34/11.04	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.34/11.04	   s(ok(X)) -> ok(s(X))
38.34/11.04	   fst(ok(X)) -> ok(fst(X))
38.34/11.04	   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.34/11.04	   snd(ok(X)) -> ok(snd(X))
38.34/11.04	   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.34/11.04	   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.34/11.04	   head(ok(X)) -> ok(head(X))
38.34/11.04	   tail(ok(X)) -> ok(tail(X))
38.34/11.04	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.34/11.04	   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.34/11.04	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.34/11.04	   top(mark(X)) -> top(proper(X))
38.34/11.04	   top(ok(X)) -> top(active(X))
38.34/11.04	
38.34/11.04	S is empty.
38.34/11.04	Rewrite Strategy: FULL
38.34/11.04	----------------------------------------
38.34/11.04	
38.34/11.04	(5) CpxTrsMatchBoundsTAProof (FINISHED)
38.34/11.04	A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 
38.34/11.04	
38.34/11.04	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
38.34/11.04	final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
38.34/11.04	transitions: 
38.34/11.04	mark0(0) -> 0
38.34/11.04	00() -> 0
38.34/11.04	ok0(0) -> 0
38.34/11.04	nil0() -> 0
38.34/11.04	active0(0) -> 0
38.34/11.04	natsFrom0(0) -> 1
38.34/11.04	cons0(0, 0) -> 2
38.34/11.04	s0(0) -> 3
38.34/11.04	fst0(0) -> 4
38.34/11.04	pair0(0, 0) -> 5
38.34/11.04	snd0(0) -> 6
38.34/11.04	splitAt0(0, 0) -> 7
38.34/11.04	u0(0, 0, 0, 0) -> 8
38.34/11.04	head0(0) -> 9
38.34/11.04	tail0(0) -> 10
38.34/11.04	sel0(0, 0) -> 11
38.34/11.04	afterNth0(0, 0) -> 12
38.34/11.04	take0(0, 0) -> 13
38.34/11.04	proper0(0) -> 14
38.34/11.04	top0(0) -> 15
38.34/11.04	natsFrom1(0) -> 16
38.34/11.04	mark1(16) -> 1
38.34/11.04	cons1(0, 0) -> 17
38.34/11.04	mark1(17) -> 2
38.34/11.04	s1(0) -> 18
38.34/11.04	mark1(18) -> 3
38.34/11.04	fst1(0) -> 19
38.34/11.04	mark1(19) -> 4
38.34/11.04	pair1(0, 0) -> 20
38.34/11.04	mark1(20) -> 5
38.34/11.04	snd1(0) -> 21
38.34/11.04	mark1(21) -> 6
38.34/11.04	splitAt1(0, 0) -> 22
38.34/11.04	mark1(22) -> 7
38.34/11.04	u1(0, 0, 0, 0) -> 23
38.34/11.04	mark1(23) -> 8
38.34/11.04	head1(0) -> 24
38.34/11.04	mark1(24) -> 9
38.34/11.04	tail1(0) -> 25
38.34/11.04	mark1(25) -> 10
38.34/11.04	sel1(0, 0) -> 26
38.34/11.04	mark1(26) -> 11
38.34/11.04	afterNth1(0, 0) -> 27
38.34/11.04	mark1(27) -> 12
38.34/11.04	take1(0, 0) -> 28
38.34/11.04	mark1(28) -> 13
38.34/11.04	01() -> 29
38.34/11.04	ok1(29) -> 14
38.34/11.04	nil1() -> 30
38.34/11.04	ok1(30) -> 14
38.34/11.04	natsFrom1(0) -> 31
38.34/11.04	ok1(31) -> 1
38.34/11.04	cons1(0, 0) -> 32
38.34/11.04	ok1(32) -> 2
38.34/11.04	s1(0) -> 33
38.34/11.04	ok1(33) -> 3
38.34/11.04	fst1(0) -> 34
38.34/11.04	ok1(34) -> 4
38.34/11.04	pair1(0, 0) -> 35
38.34/11.04	ok1(35) -> 5
38.34/11.04	snd1(0) -> 36
38.34/11.04	ok1(36) -> 6
38.34/11.04	splitAt1(0, 0) -> 37
38.34/11.04	ok1(37) -> 7
38.34/11.04	u1(0, 0, 0, 0) -> 38
38.34/11.04	ok1(38) -> 8
38.34/11.04	head1(0) -> 39
38.34/11.04	ok1(39) -> 9
38.34/11.04	tail1(0) -> 40
38.34/11.04	ok1(40) -> 10
38.34/11.04	sel1(0, 0) -> 41
38.34/11.04	ok1(41) -> 11
38.34/11.04	afterNth1(0, 0) -> 42
38.34/11.04	ok1(42) -> 12
38.34/11.04	take1(0, 0) -> 43
38.34/11.04	ok1(43) -> 13
38.34/11.04	proper1(0) -> 44
38.34/11.04	top1(44) -> 15
38.34/11.04	active1(0) -> 45
38.34/11.04	top1(45) -> 15
38.34/11.04	mark1(16) -> 16
38.34/11.04	mark1(16) -> 31
38.34/11.04	mark1(17) -> 17
38.34/11.04	mark1(17) -> 32
38.34/11.04	mark1(18) -> 18
38.34/11.04	mark1(18) -> 33
38.34/11.04	mark1(19) -> 19
38.34/11.04	mark1(19) -> 34
38.34/11.04	mark1(20) -> 20
38.34/11.04	mark1(20) -> 35
38.34/11.04	mark1(21) -> 21
38.34/11.04	mark1(21) -> 36
38.34/11.04	mark1(22) -> 22
38.34/11.04	mark1(22) -> 37
38.34/11.04	mark1(23) -> 23
38.34/11.04	mark1(23) -> 38
38.34/11.04	mark1(24) -> 24
38.34/11.04	mark1(24) -> 39
38.34/11.04	mark1(25) -> 25
38.34/11.04	mark1(25) -> 40
38.34/11.04	mark1(26) -> 26
38.34/11.04	mark1(26) -> 41
38.34/11.04	mark1(27) -> 27
38.34/11.04	mark1(27) -> 42
38.34/11.04	mark1(28) -> 28
38.34/11.04	mark1(28) -> 43
38.34/11.04	ok1(29) -> 44
38.34/11.04	ok1(30) -> 44
38.34/11.04	ok1(31) -> 16
38.34/11.04	ok1(31) -> 31
38.34/11.04	ok1(32) -> 17
38.34/11.04	ok1(32) -> 32
38.34/11.04	ok1(33) -> 18
38.34/11.04	ok1(33) -> 33
38.34/11.04	ok1(34) -> 19
38.34/11.04	ok1(34) -> 34
38.34/11.04	ok1(35) -> 20
38.34/11.04	ok1(35) -> 35
38.34/11.04	ok1(36) -> 21
38.34/11.04	ok1(36) -> 36
38.34/11.04	ok1(37) -> 22
38.34/11.04	ok1(37) -> 37
38.34/11.04	ok1(38) -> 23
38.34/11.04	ok1(38) -> 38
38.34/11.04	ok1(39) -> 24
38.34/11.04	ok1(39) -> 39
38.34/11.04	ok1(40) -> 25
38.34/11.04	ok1(40) -> 40
38.34/11.04	ok1(41) -> 26
38.34/11.04	ok1(41) -> 41
38.34/11.04	ok1(42) -> 27
38.34/11.04	ok1(42) -> 42
38.34/11.04	ok1(43) -> 28
38.34/11.04	ok1(43) -> 43
38.34/11.04	active2(29) -> 46
38.34/11.04	top2(46) -> 15
38.34/11.04	active2(30) -> 46
38.34/11.04	
38.34/11.04	----------------------------------------
38.34/11.04	
38.34/11.04	(6)
38.34/11.04	BOUNDS(1, n^1)
38.34/11.04	
38.34/11.04	----------------------------------------
38.34/11.04	
38.34/11.04	(7) RenamingProof (BOTH BOUNDS(ID, ID))
38.34/11.04	Renamed function symbols to avoid clashes with predefined symbol.
38.34/11.04	----------------------------------------
38.34/11.04	
38.34/11.04	(8)
38.34/11.04	Obligation:
38.34/11.04	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
38.34/11.04	
38.34/11.04	
38.34/11.04	The TRS R consists of the following rules:
38.34/11.04	
38.34/11.04	   active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.34/11.04	   active(fst(pair(XS, YS))) -> mark(XS)
38.34/11.04	   active(snd(pair(XS, YS))) -> mark(YS)
38.34/11.04	   active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.34/11.04	   active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.34/11.04	   active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.34/11.04	   active(head(cons(N, XS))) -> mark(N)
38.34/11.04	   active(tail(cons(N, XS))) -> mark(XS)
38.34/11.04	   active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.34/11.04	   active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.34/11.04	   active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.34/11.04	   active(natsFrom(X)) -> natsFrom(active(X))
38.34/11.04	   active(cons(X1, X2)) -> cons(active(X1), X2)
38.34/11.04	   active(s(X)) -> s(active(X))
38.34/11.04	   active(fst(X)) -> fst(active(X))
38.34/11.04	   active(pair(X1, X2)) -> pair(active(X1), X2)
38.34/11.04	   active(pair(X1, X2)) -> pair(X1, active(X2))
38.34/11.04	   active(snd(X)) -> snd(active(X))
38.34/11.04	   active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.34/11.04	   active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.34/11.04	   active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.34/11.04	   active(head(X)) -> head(active(X))
38.34/11.04	   active(tail(X)) -> tail(active(X))
38.34/11.04	   active(sel(X1, X2)) -> sel(active(X1), X2)
38.34/11.04	   active(sel(X1, X2)) -> sel(X1, active(X2))
38.34/11.04	   active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.34/11.04	   active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.34/11.04	   active(take(X1, X2)) -> take(active(X1), X2)
38.34/11.04	   active(take(X1, X2)) -> take(X1, active(X2))
38.34/11.04	   natsFrom(mark(X)) -> mark(natsFrom(X))
38.34/11.04	   cons(mark(X1), X2) -> mark(cons(X1, X2))
38.34/11.04	   s(mark(X)) -> mark(s(X))
38.34/11.04	   fst(mark(X)) -> mark(fst(X))
38.34/11.04	   pair(mark(X1), X2) -> mark(pair(X1, X2))
38.34/11.04	   pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.34/11.04	   snd(mark(X)) -> mark(snd(X))
38.34/11.04	   splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.34/11.04	   splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.34/11.04	   u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.34/11.04	   head(mark(X)) -> mark(head(X))
38.34/11.04	   tail(mark(X)) -> mark(tail(X))
38.34/11.04	   sel(mark(X1), X2) -> mark(sel(X1, X2))
38.34/11.04	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.34/11.04	   afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.34/11.04	   afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.34/11.04	   take(mark(X1), X2) -> mark(take(X1, X2))
38.34/11.04	   take(X1, mark(X2)) -> mark(take(X1, X2))
38.34/11.04	   proper(natsFrom(X)) -> natsFrom(proper(X))
38.34/11.04	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.34/11.04	   proper(s(X)) -> s(proper(X))
38.34/11.04	   proper(fst(X)) -> fst(proper(X))
38.34/11.04	   proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.34/11.04	   proper(snd(X)) -> snd(proper(X))
38.34/11.04	   proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.34/11.04	   proper(0') -> ok(0')
38.34/11.04	   proper(nil) -> ok(nil)
38.34/11.04	   proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.34/11.04	   proper(head(X)) -> head(proper(X))
38.34/11.04	   proper(tail(X)) -> tail(proper(X))
38.34/11.04	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.34/11.04	   proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.34/11.04	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.34/11.04	   natsFrom(ok(X)) -> ok(natsFrom(X))
38.34/11.04	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.34/11.04	   s(ok(X)) -> ok(s(X))
38.34/11.04	   fst(ok(X)) -> ok(fst(X))
38.34/11.04	   pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.34/11.04	   snd(ok(X)) -> ok(snd(X))
38.34/11.04	   splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.34/11.04	   u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.34/11.04	   head(ok(X)) -> ok(head(X))
38.34/11.04	   tail(ok(X)) -> ok(tail(X))
38.34/11.04	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.34/11.04	   afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.34/11.04	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.34/11.04	   top(mark(X)) -> top(proper(X))
38.34/11.04	   top(ok(X)) -> top(active(X))
38.34/11.04	
38.34/11.04	S is empty.
38.34/11.04	Rewrite Strategy: FULL
38.34/11.04	----------------------------------------
38.34/11.04	
38.34/11.04	(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
38.34/11.04	Infered types.
38.34/11.04	----------------------------------------
38.34/11.04	
38.34/11.04	(10)
38.34/11.04	Obligation:
38.34/11.04	TRS:
38.34/11.04	Rules:
38.34/11.04	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.34/11.04	active(fst(pair(XS, YS))) -> mark(XS)
38.34/11.04	active(snd(pair(XS, YS))) -> mark(YS)
38.34/11.04	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.34/11.04	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.34/11.04	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.34/11.04	active(head(cons(N, XS))) -> mark(N)
38.34/11.04	active(tail(cons(N, XS))) -> mark(XS)
38.34/11.04	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.34/11.04	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.34/11.04	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.34/11.04	active(natsFrom(X)) -> natsFrom(active(X))
38.34/11.04	active(cons(X1, X2)) -> cons(active(X1), X2)
38.34/11.04	active(s(X)) -> s(active(X))
38.34/11.04	active(fst(X)) -> fst(active(X))
38.34/11.04	active(pair(X1, X2)) -> pair(active(X1), X2)
38.34/11.04	active(pair(X1, X2)) -> pair(X1, active(X2))
38.34/11.04	active(snd(X)) -> snd(active(X))
38.34/11.04	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.34/11.04	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.34/11.04	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.34/11.04	active(head(X)) -> head(active(X))
38.34/11.04	active(tail(X)) -> tail(active(X))
38.34/11.04	active(sel(X1, X2)) -> sel(active(X1), X2)
38.34/11.04	active(sel(X1, X2)) -> sel(X1, active(X2))
38.34/11.04	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.34/11.04	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.34/11.04	active(take(X1, X2)) -> take(active(X1), X2)
38.34/11.04	active(take(X1, X2)) -> take(X1, active(X2))
38.34/11.04	natsFrom(mark(X)) -> mark(natsFrom(X))
38.34/11.04	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.34/11.04	s(mark(X)) -> mark(s(X))
38.34/11.04	fst(mark(X)) -> mark(fst(X))
38.34/11.04	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.34/11.04	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.34/11.04	snd(mark(X)) -> mark(snd(X))
38.34/11.04	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.34/11.04	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.34/11.04	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.34/11.04	head(mark(X)) -> mark(head(X))
38.34/11.04	tail(mark(X)) -> mark(tail(X))
38.34/11.04	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.34/11.04	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.34/11.04	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.34/11.04	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.34/11.04	take(mark(X1), X2) -> mark(take(X1, X2))
38.34/11.04	take(X1, mark(X2)) -> mark(take(X1, X2))
38.34/11.04	proper(natsFrom(X)) -> natsFrom(proper(X))
38.34/11.04	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.34/11.04	proper(s(X)) -> s(proper(X))
38.34/11.04	proper(fst(X)) -> fst(proper(X))
38.34/11.04	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.34/11.04	proper(snd(X)) -> snd(proper(X))
38.34/11.04	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.34/11.04	proper(0') -> ok(0')
38.34/11.04	proper(nil) -> ok(nil)
38.34/11.04	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.34/11.04	proper(head(X)) -> head(proper(X))
38.34/11.04	proper(tail(X)) -> tail(proper(X))
38.34/11.04	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.34/11.04	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.34/11.04	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.34/11.04	natsFrom(ok(X)) -> ok(natsFrom(X))
38.34/11.04	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.34/11.04	s(ok(X)) -> ok(s(X))
38.34/11.04	fst(ok(X)) -> ok(fst(X))
38.34/11.04	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.34/11.04	snd(ok(X)) -> ok(snd(X))
38.34/11.04	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.34/11.04	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.34/11.04	head(ok(X)) -> ok(head(X))
38.34/11.04	tail(ok(X)) -> ok(tail(X))
38.34/11.04	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.34/11.04	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.34/11.04	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.34/11.04	top(mark(X)) -> top(proper(X))
38.34/11.04	top(ok(X)) -> top(active(X))
38.34/11.04	
38.34/11.04	Types:
38.34/11.04	active :: mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	s :: mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	0' :: mark:0':nil:ok
38.34/11.04	nil :: mark:0':nil:ok
38.34/11.04	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	head :: mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.34/11.04	top :: mark:0':nil:ok -> top
38.34/11.04	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.34/11.04	hole_top2_0 :: top
38.34/11.04	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.34/11.04	
38.34/11.04	----------------------------------------
38.34/11.04	
38.34/11.04	(11) OrderProof (LOWER BOUND(ID))
38.34/11.04	Heuristically decided to analyse the following defined symbols:
38.34/11.04	active, cons, natsFrom, s, pair, u, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, top
38.34/11.04	
38.34/11.04	They will be analysed ascendingly in the following order:
38.34/11.04	cons < active
38.34/11.04	natsFrom < active
38.34/11.04	s < active
38.34/11.04	pair < active
38.34/11.04	u < active
38.34/11.04	splitAt < active
38.34/11.04	head < active
38.34/11.04	afterNth < active
38.34/11.04	fst < active
38.34/11.04	snd < active
38.34/11.04	tail < active
38.34/11.04	sel < active
38.34/11.04	take < active
38.34/11.04	active < top
38.34/11.04	cons < proper
38.34/11.04	natsFrom < proper
38.34/11.04	s < proper
38.34/11.04	pair < proper
38.34/11.04	u < proper
38.34/11.04	splitAt < proper
38.34/11.04	head < proper
38.34/11.04	afterNth < proper
38.34/11.04	fst < proper
38.34/11.04	snd < proper
38.34/11.04	tail < proper
38.34/11.04	sel < proper
38.34/11.04	take < proper
38.34/11.04	proper < top
38.34/11.04	
38.34/11.04	----------------------------------------
38.34/11.04	
38.34/11.04	(12)
38.34/11.04	Obligation:
38.34/11.04	TRS:
38.34/11.04	Rules:
38.34/11.04	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.34/11.04	active(fst(pair(XS, YS))) -> mark(XS)
38.34/11.04	active(snd(pair(XS, YS))) -> mark(YS)
38.34/11.04	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.34/11.04	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.34/11.04	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.34/11.04	active(head(cons(N, XS))) -> mark(N)
38.34/11.04	active(tail(cons(N, XS))) -> mark(XS)
38.34/11.04	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.34/11.04	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.34/11.04	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.34/11.04	active(natsFrom(X)) -> natsFrom(active(X))
38.34/11.04	active(cons(X1, X2)) -> cons(active(X1), X2)
38.34/11.04	active(s(X)) -> s(active(X))
38.34/11.04	active(fst(X)) -> fst(active(X))
38.34/11.04	active(pair(X1, X2)) -> pair(active(X1), X2)
38.34/11.04	active(pair(X1, X2)) -> pair(X1, active(X2))
38.34/11.04	active(snd(X)) -> snd(active(X))
38.34/11.04	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.34/11.04	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.34/11.04	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.34/11.04	active(head(X)) -> head(active(X))
38.34/11.04	active(tail(X)) -> tail(active(X))
38.34/11.04	active(sel(X1, X2)) -> sel(active(X1), X2)
38.34/11.04	active(sel(X1, X2)) -> sel(X1, active(X2))
38.34/11.04	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.34/11.04	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.34/11.04	active(take(X1, X2)) -> take(active(X1), X2)
38.34/11.04	active(take(X1, X2)) -> take(X1, active(X2))
38.34/11.04	natsFrom(mark(X)) -> mark(natsFrom(X))
38.34/11.04	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.34/11.04	s(mark(X)) -> mark(s(X))
38.34/11.04	fst(mark(X)) -> mark(fst(X))
38.34/11.04	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.04	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.04	snd(mark(X)) -> mark(snd(X))
38.45/11.04	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.04	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.04	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.04	head(mark(X)) -> mark(head(X))
38.45/11.04	tail(mark(X)) -> mark(tail(X))
38.45/11.04	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.04	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.04	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.04	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.04	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.04	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.04	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.04	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.04	proper(s(X)) -> s(proper(X))
38.45/11.04	proper(fst(X)) -> fst(proper(X))
38.45/11.04	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.04	proper(snd(X)) -> snd(proper(X))
38.45/11.04	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.04	proper(0') -> ok(0')
38.45/11.04	proper(nil) -> ok(nil)
38.45/11.04	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.04	proper(head(X)) -> head(proper(X))
38.45/11.04	proper(tail(X)) -> tail(proper(X))
38.45/11.04	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.04	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.04	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.04	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.04	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.04	s(ok(X)) -> ok(s(X))
38.45/11.04	fst(ok(X)) -> ok(fst(X))
38.45/11.04	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.04	snd(ok(X)) -> ok(snd(X))
38.45/11.04	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.04	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.04	head(ok(X)) -> ok(head(X))
38.45/11.04	tail(ok(X)) -> ok(tail(X))
38.45/11.04	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.04	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.04	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.04	top(mark(X)) -> top(proper(X))
38.45/11.04	top(ok(X)) -> top(active(X))
38.45/11.04	
38.45/11.04	Types:
38.45/11.04	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	0' :: mark:0':nil:ok
38.45/11.04	nil :: mark:0':nil:ok
38.45/11.04	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	top :: mark:0':nil:ok -> top
38.45/11.04	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.04	hole_top2_0 :: top
38.45/11.04	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.04	
38.45/11.04	
38.45/11.04	Generator Equations:
38.45/11.04	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.04	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.04	
38.45/11.04	
38.45/11.04	The following defined symbols remain to be analysed:
38.45/11.04	cons, active, natsFrom, s, pair, u, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, top
38.45/11.04	
38.45/11.04	They will be analysed ascendingly in the following order:
38.45/11.04	cons < active
38.45/11.04	natsFrom < active
38.45/11.04	s < active
38.45/11.04	pair < active
38.45/11.04	u < active
38.45/11.04	splitAt < active
38.45/11.04	head < active
38.45/11.04	afterNth < active
38.45/11.04	fst < active
38.45/11.04	snd < active
38.45/11.04	tail < active
38.45/11.04	sel < active
38.45/11.04	take < active
38.45/11.04	active < top
38.45/11.04	cons < proper
38.45/11.04	natsFrom < proper
38.45/11.04	s < proper
38.45/11.04	pair < proper
38.45/11.04	u < proper
38.45/11.04	splitAt < proper
38.45/11.04	head < proper
38.45/11.04	afterNth < proper
38.45/11.04	fst < proper
38.45/11.04	snd < proper
38.45/11.04	tail < proper
38.45/11.04	sel < proper
38.45/11.04	take < proper
38.45/11.04	proper < top
38.45/11.04	
38.45/11.04	----------------------------------------
38.45/11.04	
38.45/11.04	(13) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.04	Proved the following rewrite lemma:
38.45/11.04	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.04	
38.45/11.04	Induction Base:
38.45/11.04	cons(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))
38.45/11.04	
38.45/11.04	Induction Step:
38.45/11.04	cons(gen_mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1)
38.45/11.04	mark(cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b))) ->_IH
38.45/11.04	mark(*4_0)
38.45/11.04	
38.45/11.04	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.04	----------------------------------------
38.45/11.04	
38.45/11.04	(14)
38.45/11.04	Complex Obligation (BEST)
38.45/11.04	
38.45/11.04	----------------------------------------
38.45/11.04	
38.45/11.04	(15)
38.45/11.04	Obligation:
38.45/11.04	Proved the lower bound n^1 for the following obligation:
38.45/11.04	
38.45/11.04	TRS:
38.45/11.04	Rules:
38.45/11.04	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.04	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.04	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.04	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.04	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.04	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.04	active(head(cons(N, XS))) -> mark(N)
38.45/11.04	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.04	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.04	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.04	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.04	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.04	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.04	active(s(X)) -> s(active(X))
38.45/11.04	active(fst(X)) -> fst(active(X))
38.45/11.04	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.04	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.04	active(snd(X)) -> snd(active(X))
38.45/11.04	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.04	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.04	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.04	active(head(X)) -> head(active(X))
38.45/11.04	active(tail(X)) -> tail(active(X))
38.45/11.04	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.04	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.04	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.04	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.04	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.04	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.04	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.04	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.04	s(mark(X)) -> mark(s(X))
38.45/11.04	fst(mark(X)) -> mark(fst(X))
38.45/11.04	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.04	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.04	snd(mark(X)) -> mark(snd(X))
38.45/11.04	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.04	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.04	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.04	head(mark(X)) -> mark(head(X))
38.45/11.04	tail(mark(X)) -> mark(tail(X))
38.45/11.04	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.04	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.04	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.04	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.04	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.04	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.04	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.04	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.04	proper(s(X)) -> s(proper(X))
38.45/11.04	proper(fst(X)) -> fst(proper(X))
38.45/11.04	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.04	proper(snd(X)) -> snd(proper(X))
38.45/11.04	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.04	proper(0') -> ok(0')
38.45/11.04	proper(nil) -> ok(nil)
38.45/11.04	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.04	proper(head(X)) -> head(proper(X))
38.45/11.04	proper(tail(X)) -> tail(proper(X))
38.45/11.04	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.04	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.04	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.04	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.04	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.04	s(ok(X)) -> ok(s(X))
38.45/11.04	fst(ok(X)) -> ok(fst(X))
38.45/11.04	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.04	snd(ok(X)) -> ok(snd(X))
38.45/11.04	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.04	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.04	head(ok(X)) -> ok(head(X))
38.45/11.04	tail(ok(X)) -> ok(tail(X))
38.45/11.04	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.04	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.04	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.04	top(mark(X)) -> top(proper(X))
38.45/11.04	top(ok(X)) -> top(active(X))
38.45/11.04	
38.45/11.04	Types:
38.45/11.04	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	0' :: mark:0':nil:ok
38.45/11.04	nil :: mark:0':nil:ok
38.45/11.04	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	top :: mark:0':nil:ok -> top
38.45/11.04	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.04	hole_top2_0 :: top
38.45/11.04	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.04	
38.45/11.04	
38.45/11.04	Generator Equations:
38.45/11.04	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.04	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.04	
38.45/11.04	
38.45/11.04	The following defined symbols remain to be analysed:
38.45/11.04	cons, active, natsFrom, s, pair, u, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, top
38.45/11.04	
38.45/11.04	They will be analysed ascendingly in the following order:
38.45/11.04	cons < active
38.45/11.04	natsFrom < active
38.45/11.04	s < active
38.45/11.04	pair < active
38.45/11.04	u < active
38.45/11.04	splitAt < active
38.45/11.04	head < active
38.45/11.04	afterNth < active
38.45/11.04	fst < active
38.45/11.04	snd < active
38.45/11.04	tail < active
38.45/11.04	sel < active
38.45/11.04	take < active
38.45/11.04	active < top
38.45/11.04	cons < proper
38.45/11.04	natsFrom < proper
38.45/11.04	s < proper
38.45/11.04	pair < proper
38.45/11.04	u < proper
38.45/11.04	splitAt < proper
38.45/11.04	head < proper
38.45/11.04	afterNth < proper
38.45/11.04	fst < proper
38.45/11.04	snd < proper
38.45/11.04	tail < proper
38.45/11.04	sel < proper
38.45/11.04	take < proper
38.45/11.04	proper < top
38.45/11.04	
38.45/11.04	----------------------------------------
38.45/11.04	
38.45/11.04	(16) LowerBoundPropagationProof (FINISHED)
38.45/11.04	Propagated lower bound.
38.45/11.04	----------------------------------------
38.45/11.04	
38.45/11.04	(17)
38.45/11.04	BOUNDS(n^1, INF)
38.45/11.04	
38.45/11.04	----------------------------------------
38.45/11.04	
38.45/11.04	(18)
38.45/11.04	Obligation:
38.45/11.04	TRS:
38.45/11.04	Rules:
38.45/11.04	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.04	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.04	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.04	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.04	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.04	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.04	active(head(cons(N, XS))) -> mark(N)
38.45/11.04	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.04	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.04	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.04	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.04	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.04	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.04	active(s(X)) -> s(active(X))
38.45/11.04	active(fst(X)) -> fst(active(X))
38.45/11.04	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.04	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.04	active(snd(X)) -> snd(active(X))
38.45/11.04	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.04	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.04	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.04	active(head(X)) -> head(active(X))
38.45/11.04	active(tail(X)) -> tail(active(X))
38.45/11.04	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.04	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.04	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.04	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.04	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.04	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.04	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.04	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.04	s(mark(X)) -> mark(s(X))
38.45/11.04	fst(mark(X)) -> mark(fst(X))
38.45/11.04	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.04	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.04	snd(mark(X)) -> mark(snd(X))
38.45/11.04	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.04	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.04	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.04	head(mark(X)) -> mark(head(X))
38.45/11.04	tail(mark(X)) -> mark(tail(X))
38.45/11.04	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.04	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.04	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.04	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.04	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.04	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.04	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.04	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.04	proper(s(X)) -> s(proper(X))
38.45/11.04	proper(fst(X)) -> fst(proper(X))
38.45/11.04	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.04	proper(snd(X)) -> snd(proper(X))
38.45/11.04	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.04	proper(0') -> ok(0')
38.45/11.04	proper(nil) -> ok(nil)
38.45/11.04	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.04	proper(head(X)) -> head(proper(X))
38.45/11.04	proper(tail(X)) -> tail(proper(X))
38.45/11.04	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.04	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.04	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.04	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.04	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.04	s(ok(X)) -> ok(s(X))
38.45/11.04	fst(ok(X)) -> ok(fst(X))
38.45/11.04	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.04	snd(ok(X)) -> ok(snd(X))
38.45/11.04	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.04	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.04	head(ok(X)) -> ok(head(X))
38.45/11.04	tail(ok(X)) -> ok(tail(X))
38.45/11.04	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.04	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.04	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.04	top(mark(X)) -> top(proper(X))
38.45/11.04	top(ok(X)) -> top(active(X))
38.45/11.04	
38.45/11.04	Types:
38.45/11.04	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	0' :: mark:0':nil:ok
38.45/11.04	nil :: mark:0':nil:ok
38.45/11.04	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	top :: mark:0':nil:ok -> top
38.45/11.04	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.04	hole_top2_0 :: top
38.45/11.04	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.04	
38.45/11.04	
38.45/11.04	Lemmas:
38.45/11.04	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.04	
38.45/11.04	
38.45/11.04	Generator Equations:
38.45/11.04	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.04	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.04	
38.45/11.04	
38.45/11.04	The following defined symbols remain to be analysed:
38.45/11.04	natsFrom, active, s, pair, u, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, top
38.45/11.04	
38.45/11.04	They will be analysed ascendingly in the following order:
38.45/11.04	natsFrom < active
38.45/11.04	s < active
38.45/11.04	pair < active
38.45/11.04	u < active
38.45/11.04	splitAt < active
38.45/11.04	head < active
38.45/11.04	afterNth < active
38.45/11.04	fst < active
38.45/11.04	snd < active
38.45/11.04	tail < active
38.45/11.04	sel < active
38.45/11.04	take < active
38.45/11.04	active < top
38.45/11.04	natsFrom < proper
38.45/11.04	s < proper
38.45/11.04	pair < proper
38.45/11.04	u < proper
38.45/11.04	splitAt < proper
38.45/11.04	head < proper
38.45/11.04	afterNth < proper
38.45/11.04	fst < proper
38.45/11.04	snd < proper
38.45/11.04	tail < proper
38.45/11.04	sel < proper
38.45/11.04	take < proper
38.45/11.04	proper < top
38.45/11.04	
38.45/11.04	----------------------------------------
38.45/11.04	
38.45/11.04	(19) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.04	Proved the following rewrite lemma:
38.45/11.04	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.04	
38.45/11.04	Induction Base:
38.45/11.04	natsFrom(gen_mark:0':nil:ok3_0(+(1, 0)))
38.45/11.04	
38.45/11.04	Induction Step:
38.45/11.04	natsFrom(gen_mark:0':nil:ok3_0(+(1, +(n1518_0, 1)))) ->_R^Omega(1)
38.45/11.04	mark(natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0)))) ->_IH
38.45/11.04	mark(*4_0)
38.45/11.04	
38.45/11.04	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.04	----------------------------------------
38.45/11.04	
38.45/11.04	(20)
38.45/11.04	Obligation:
38.45/11.04	TRS:
38.45/11.04	Rules:
38.45/11.04	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.04	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.04	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.04	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.04	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.04	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.04	active(head(cons(N, XS))) -> mark(N)
38.45/11.04	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.04	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.04	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.04	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.04	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.04	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.04	active(s(X)) -> s(active(X))
38.45/11.04	active(fst(X)) -> fst(active(X))
38.45/11.04	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.04	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.04	active(snd(X)) -> snd(active(X))
38.45/11.04	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.04	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.04	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.04	active(head(X)) -> head(active(X))
38.45/11.04	active(tail(X)) -> tail(active(X))
38.45/11.04	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.04	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.04	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.04	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.04	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.04	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.04	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.04	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.04	s(mark(X)) -> mark(s(X))
38.45/11.04	fst(mark(X)) -> mark(fst(X))
38.45/11.04	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.04	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.04	snd(mark(X)) -> mark(snd(X))
38.45/11.04	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.04	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.04	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.04	head(mark(X)) -> mark(head(X))
38.45/11.04	tail(mark(X)) -> mark(tail(X))
38.45/11.04	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.04	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.04	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.04	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.04	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.04	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.04	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.04	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.04	proper(s(X)) -> s(proper(X))
38.45/11.04	proper(fst(X)) -> fst(proper(X))
38.45/11.04	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.04	proper(snd(X)) -> snd(proper(X))
38.45/11.04	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.04	proper(0') -> ok(0')
38.45/11.04	proper(nil) -> ok(nil)
38.45/11.04	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.04	proper(head(X)) -> head(proper(X))
38.45/11.04	proper(tail(X)) -> tail(proper(X))
38.45/11.04	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.04	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.04	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.04	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.04	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.04	s(ok(X)) -> ok(s(X))
38.45/11.04	fst(ok(X)) -> ok(fst(X))
38.45/11.04	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.04	snd(ok(X)) -> ok(snd(X))
38.45/11.04	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.04	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.04	head(ok(X)) -> ok(head(X))
38.45/11.04	tail(ok(X)) -> ok(tail(X))
38.45/11.04	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.04	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.04	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.04	top(mark(X)) -> top(proper(X))
38.45/11.04	top(ok(X)) -> top(active(X))
38.45/11.04	
38.45/11.04	Types:
38.45/11.04	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	0' :: mark:0':nil:ok
38.45/11.04	nil :: mark:0':nil:ok
38.45/11.04	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.04	top :: mark:0':nil:ok -> top
38.45/11.04	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.04	hole_top2_0 :: top
38.45/11.04	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.04	
38.45/11.04	
38.45/11.04	Lemmas:
38.45/11.04	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.04	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.04	
38.45/11.04	
38.45/11.04	Generator Equations:
38.45/11.04	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.04	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.04	
38.45/11.04	
38.45/11.04	The following defined symbols remain to be analysed:
38.45/11.04	s, active, pair, u, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, top
38.45/11.04	
38.45/11.04	They will be analysed ascendingly in the following order:
38.45/11.04	s < active
38.45/11.04	pair < active
38.45/11.04	u < active
38.45/11.04	splitAt < active
38.45/11.04	head < active
38.45/11.04	afterNth < active
38.45/11.04	fst < active
38.45/11.04	snd < active
38.45/11.04	tail < active
38.45/11.04	sel < active
38.45/11.04	take < active
38.45/11.04	active < top
38.45/11.04	s < proper
38.45/11.04	pair < proper
38.45/11.04	u < proper
38.45/11.04	splitAt < proper
38.45/11.04	head < proper
38.45/11.04	afterNth < proper
38.45/11.04	fst < proper
38.45/11.04	snd < proper
38.45/11.04	tail < proper
38.45/11.04	sel < proper
38.45/11.04	take < proper
38.45/11.04	proper < top
38.45/11.04	
38.45/11.04	----------------------------------------
38.45/11.04	
38.45/11.04	(21) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.04	Proved the following rewrite lemma:
38.45/11.04	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.04	
38.45/11.04	Induction Base:
38.45/11.04	s(gen_mark:0':nil:ok3_0(+(1, 0)))
38.45/11.04	
38.45/11.04	Induction Step:
38.45/11.04	s(gen_mark:0':nil:ok3_0(+(1, +(n2166_0, 1)))) ->_R^Omega(1)
38.45/11.04	mark(s(gen_mark:0':nil:ok3_0(+(1, n2166_0)))) ->_IH
38.45/11.04	mark(*4_0)
38.45/11.04	
38.45/11.04	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.04	----------------------------------------
38.45/11.04	
38.45/11.04	(22)
38.45/11.04	Obligation:
38.45/11.04	TRS:
38.45/11.04	Rules:
38.45/11.04	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.04	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.04	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.04	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.04	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.04	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.04	active(head(cons(N, XS))) -> mark(N)
38.45/11.04	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.04	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.04	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.04	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.04	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.04	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.04	active(s(X)) -> s(active(X))
38.45/11.04	active(fst(X)) -> fst(active(X))
38.45/11.04	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.05	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.05	active(snd(X)) -> snd(active(X))
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.05	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.05	active(head(X)) -> head(active(X))
38.45/11.05	active(tail(X)) -> tail(active(X))
38.45/11.05	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.05	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.05	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.05	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.05	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.05	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.05	s(mark(X)) -> mark(s(X))
38.45/11.05	fst(mark(X)) -> mark(fst(X))
38.45/11.05	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.05	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.05	snd(mark(X)) -> mark(snd(X))
38.45/11.05	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.05	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.05	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.05	head(mark(X)) -> mark(head(X))
38.45/11.05	tail(mark(X)) -> mark(tail(X))
38.45/11.05	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.05	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.05	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.05	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.05	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.05	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.05	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.05	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.05	proper(s(X)) -> s(proper(X))
38.45/11.05	proper(fst(X)) -> fst(proper(X))
38.45/11.05	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.05	proper(snd(X)) -> snd(proper(X))
38.45/11.05	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.05	proper(0') -> ok(0')
38.45/11.05	proper(nil) -> ok(nil)
38.45/11.05	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.05	proper(head(X)) -> head(proper(X))
38.45/11.05	proper(tail(X)) -> tail(proper(X))
38.45/11.05	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.05	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.05	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.05	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.05	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.05	s(ok(X)) -> ok(s(X))
38.45/11.05	fst(ok(X)) -> ok(fst(X))
38.45/11.05	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.05	snd(ok(X)) -> ok(snd(X))
38.45/11.05	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.05	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.05	head(ok(X)) -> ok(head(X))
38.45/11.05	tail(ok(X)) -> ok(tail(X))
38.45/11.05	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.05	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.05	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.05	top(mark(X)) -> top(proper(X))
38.45/11.05	top(ok(X)) -> top(active(X))
38.45/11.05	
38.45/11.05	Types:
38.45/11.05	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	0' :: mark:0':nil:ok
38.45/11.05	nil :: mark:0':nil:ok
38.45/11.05	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	top :: mark:0':nil:ok -> top
38.45/11.05	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.05	hole_top2_0 :: top
38.45/11.05	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.05	
38.45/11.05	
38.45/11.05	Lemmas:
38.45/11.05	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.05	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.05	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.05	
38.45/11.05	
38.45/11.05	Generator Equations:
38.45/11.05	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.05	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.05	
38.45/11.05	
38.45/11.05	The following defined symbols remain to be analysed:
38.45/11.05	pair, active, u, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, top
38.45/11.05	
38.45/11.05	They will be analysed ascendingly in the following order:
38.45/11.05	pair < active
38.45/11.05	u < active
38.45/11.05	splitAt < active
38.45/11.05	head < active
38.45/11.05	afterNth < active
38.45/11.05	fst < active
38.45/11.05	snd < active
38.45/11.05	tail < active
38.45/11.05	sel < active
38.45/11.05	take < active
38.45/11.05	active < top
38.45/11.05	pair < proper
38.45/11.05	u < proper
38.45/11.05	splitAt < proper
38.45/11.05	head < proper
38.45/11.05	afterNth < proper
38.45/11.05	fst < proper
38.45/11.05	snd < proper
38.45/11.05	tail < proper
38.45/11.05	sel < proper
38.45/11.05	take < proper
38.45/11.05	proper < top
38.45/11.05	
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(23) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.05	Proved the following rewrite lemma:
38.45/11.05	pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2915_0)
38.45/11.05	
38.45/11.05	Induction Base:
38.45/11.05	pair(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))
38.45/11.05	
38.45/11.05	Induction Step:
38.45/11.05	pair(gen_mark:0':nil:ok3_0(+(1, +(n2915_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1)
38.45/11.05	mark(pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b))) ->_IH
38.45/11.05	mark(*4_0)
38.45/11.05	
38.45/11.05	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(24)
38.45/11.05	Obligation:
38.45/11.05	TRS:
38.45/11.05	Rules:
38.45/11.05	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.05	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.05	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.05	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.05	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.05	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.05	active(head(cons(N, XS))) -> mark(N)
38.45/11.05	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.05	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.05	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.05	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.05	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.05	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.05	active(s(X)) -> s(active(X))
38.45/11.05	active(fst(X)) -> fst(active(X))
38.45/11.05	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.05	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.05	active(snd(X)) -> snd(active(X))
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.05	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.05	active(head(X)) -> head(active(X))
38.45/11.05	active(tail(X)) -> tail(active(X))
38.45/11.05	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.05	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.05	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.05	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.05	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.05	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.05	s(mark(X)) -> mark(s(X))
38.45/11.05	fst(mark(X)) -> mark(fst(X))
38.45/11.05	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.05	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.05	snd(mark(X)) -> mark(snd(X))
38.45/11.05	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.05	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.05	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.05	head(mark(X)) -> mark(head(X))
38.45/11.05	tail(mark(X)) -> mark(tail(X))
38.45/11.05	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.05	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.05	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.05	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.05	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.05	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.05	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.05	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.05	proper(s(X)) -> s(proper(X))
38.45/11.05	proper(fst(X)) -> fst(proper(X))
38.45/11.05	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.05	proper(snd(X)) -> snd(proper(X))
38.45/11.05	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.05	proper(0') -> ok(0')
38.45/11.05	proper(nil) -> ok(nil)
38.45/11.05	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.05	proper(head(X)) -> head(proper(X))
38.45/11.05	proper(tail(X)) -> tail(proper(X))
38.45/11.05	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.05	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.05	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.05	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.05	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.05	s(ok(X)) -> ok(s(X))
38.45/11.05	fst(ok(X)) -> ok(fst(X))
38.45/11.05	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.05	snd(ok(X)) -> ok(snd(X))
38.45/11.05	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.05	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.05	head(ok(X)) -> ok(head(X))
38.45/11.05	tail(ok(X)) -> ok(tail(X))
38.45/11.05	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.05	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.05	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.05	top(mark(X)) -> top(proper(X))
38.45/11.05	top(ok(X)) -> top(active(X))
38.45/11.05	
38.45/11.05	Types:
38.45/11.05	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	0' :: mark:0':nil:ok
38.45/11.05	nil :: mark:0':nil:ok
38.45/11.05	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	top :: mark:0':nil:ok -> top
38.45/11.05	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.05	hole_top2_0 :: top
38.45/11.05	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.05	
38.45/11.05	
38.45/11.05	Lemmas:
38.45/11.05	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.05	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.05	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.05	pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2915_0)
38.45/11.05	
38.45/11.05	
38.45/11.05	Generator Equations:
38.45/11.05	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.05	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.05	
38.45/11.05	
38.45/11.05	The following defined symbols remain to be analysed:
38.45/11.05	u, active, splitAt, head, afterNth, fst, snd, tail, sel, take, proper, top
38.45/11.05	
38.45/11.05	They will be analysed ascendingly in the following order:
38.45/11.05	u < active
38.45/11.05	splitAt < active
38.45/11.05	head < active
38.45/11.05	afterNth < active
38.45/11.05	fst < active
38.45/11.05	snd < active
38.45/11.05	tail < active
38.45/11.05	sel < active
38.45/11.05	take < active
38.45/11.05	active < top
38.45/11.05	u < proper
38.45/11.05	splitAt < proper
38.45/11.05	head < proper
38.45/11.05	afterNth < proper
38.45/11.05	fst < proper
38.45/11.05	snd < proper
38.45/11.05	tail < proper
38.45/11.05	sel < proper
38.45/11.05	take < proper
38.45/11.05	proper < top
38.45/11.05	
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(25) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.05	Proved the following rewrite lemma:
38.45/11.05	u(gen_mark:0':nil:ok3_0(+(1, n5349_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n5349_0)
38.45/11.05	
38.45/11.05	Induction Base:
38.45/11.05	u(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d))
38.45/11.05	
38.45/11.05	Induction Step:
38.45/11.05	u(gen_mark:0':nil:ok3_0(+(1, +(n5349_0, 1))), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) ->_R^Omega(1)
38.45/11.05	mark(u(gen_mark:0':nil:ok3_0(+(1, n5349_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d))) ->_IH
38.45/11.05	mark(*4_0)
38.45/11.05	
38.45/11.05	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(26)
38.45/11.05	Obligation:
38.45/11.05	TRS:
38.45/11.05	Rules:
38.45/11.05	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.05	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.05	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.05	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.05	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.05	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.05	active(head(cons(N, XS))) -> mark(N)
38.45/11.05	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.05	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.05	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.05	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.05	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.05	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.05	active(s(X)) -> s(active(X))
38.45/11.05	active(fst(X)) -> fst(active(X))
38.45/11.05	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.05	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.05	active(snd(X)) -> snd(active(X))
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.05	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.05	active(head(X)) -> head(active(X))
38.45/11.05	active(tail(X)) -> tail(active(X))
38.45/11.05	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.05	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.05	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.05	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.05	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.05	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.05	s(mark(X)) -> mark(s(X))
38.45/11.05	fst(mark(X)) -> mark(fst(X))
38.45/11.05	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.05	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.05	snd(mark(X)) -> mark(snd(X))
38.45/11.05	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.05	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.05	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.05	head(mark(X)) -> mark(head(X))
38.45/11.05	tail(mark(X)) -> mark(tail(X))
38.45/11.05	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.05	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.05	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.05	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.05	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.05	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.05	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.05	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.05	proper(s(X)) -> s(proper(X))
38.45/11.05	proper(fst(X)) -> fst(proper(X))
38.45/11.05	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.05	proper(snd(X)) -> snd(proper(X))
38.45/11.05	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.05	proper(0') -> ok(0')
38.45/11.05	proper(nil) -> ok(nil)
38.45/11.05	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.05	proper(head(X)) -> head(proper(X))
38.45/11.05	proper(tail(X)) -> tail(proper(X))
38.45/11.05	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.05	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.05	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.05	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.05	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.05	s(ok(X)) -> ok(s(X))
38.45/11.05	fst(ok(X)) -> ok(fst(X))
38.45/11.05	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.05	snd(ok(X)) -> ok(snd(X))
38.45/11.05	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.05	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.05	head(ok(X)) -> ok(head(X))
38.45/11.05	tail(ok(X)) -> ok(tail(X))
38.45/11.05	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.05	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.05	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.05	top(mark(X)) -> top(proper(X))
38.45/11.05	top(ok(X)) -> top(active(X))
38.45/11.05	
38.45/11.05	Types:
38.45/11.05	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	0' :: mark:0':nil:ok
38.45/11.05	nil :: mark:0':nil:ok
38.45/11.05	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	top :: mark:0':nil:ok -> top
38.45/11.05	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.05	hole_top2_0 :: top
38.45/11.05	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.05	
38.45/11.05	
38.45/11.05	Lemmas:
38.45/11.05	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.05	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.05	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.05	pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2915_0)
38.45/11.05	u(gen_mark:0':nil:ok3_0(+(1, n5349_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n5349_0)
38.45/11.05	
38.45/11.05	
38.45/11.05	Generator Equations:
38.45/11.05	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.05	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.05	
38.45/11.05	
38.45/11.05	The following defined symbols remain to be analysed:
38.45/11.05	splitAt, active, head, afterNth, fst, snd, tail, sel, take, proper, top
38.45/11.05	
38.45/11.05	They will be analysed ascendingly in the following order:
38.45/11.05	splitAt < active
38.45/11.05	head < active
38.45/11.05	afterNth < active
38.45/11.05	fst < active
38.45/11.05	snd < active
38.45/11.05	tail < active
38.45/11.05	sel < active
38.45/11.05	take < active
38.45/11.05	active < top
38.45/11.05	splitAt < proper
38.45/11.05	head < proper
38.45/11.05	afterNth < proper
38.45/11.05	fst < proper
38.45/11.05	snd < proper
38.45/11.05	tail < proper
38.45/11.05	sel < proper
38.45/11.05	take < proper
38.45/11.05	proper < top
38.45/11.05	
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(27) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.05	Proved the following rewrite lemma:
38.45/11.05	splitAt(gen_mark:0':nil:ok3_0(+(1, n10972_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n10972_0)
38.45/11.05	
38.45/11.05	Induction Base:
38.45/11.05	splitAt(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))
38.45/11.05	
38.45/11.05	Induction Step:
38.45/11.05	splitAt(gen_mark:0':nil:ok3_0(+(1, +(n10972_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1)
38.45/11.05	mark(splitAt(gen_mark:0':nil:ok3_0(+(1, n10972_0)), gen_mark:0':nil:ok3_0(b))) ->_IH
38.45/11.05	mark(*4_0)
38.45/11.05	
38.45/11.05	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(28)
38.45/11.05	Obligation:
38.45/11.05	TRS:
38.45/11.05	Rules:
38.45/11.05	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.05	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.05	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.05	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.05	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.05	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.05	active(head(cons(N, XS))) -> mark(N)
38.45/11.05	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.05	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.05	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.05	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.05	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.05	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.05	active(s(X)) -> s(active(X))
38.45/11.05	active(fst(X)) -> fst(active(X))
38.45/11.05	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.05	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.05	active(snd(X)) -> snd(active(X))
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.05	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.05	active(head(X)) -> head(active(X))
38.45/11.05	active(tail(X)) -> tail(active(X))
38.45/11.05	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.05	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.05	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.05	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.05	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.05	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.05	s(mark(X)) -> mark(s(X))
38.45/11.05	fst(mark(X)) -> mark(fst(X))
38.45/11.05	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.05	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.05	snd(mark(X)) -> mark(snd(X))
38.45/11.05	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.05	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.05	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.05	head(mark(X)) -> mark(head(X))
38.45/11.05	tail(mark(X)) -> mark(tail(X))
38.45/11.05	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.05	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.05	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.05	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.05	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.05	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.05	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.05	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.05	proper(s(X)) -> s(proper(X))
38.45/11.05	proper(fst(X)) -> fst(proper(X))
38.45/11.05	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.05	proper(snd(X)) -> snd(proper(X))
38.45/11.05	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.05	proper(0') -> ok(0')
38.45/11.05	proper(nil) -> ok(nil)
38.45/11.05	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.05	proper(head(X)) -> head(proper(X))
38.45/11.05	proper(tail(X)) -> tail(proper(X))
38.45/11.05	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.05	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.05	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.05	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.05	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.05	s(ok(X)) -> ok(s(X))
38.45/11.05	fst(ok(X)) -> ok(fst(X))
38.45/11.05	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.05	snd(ok(X)) -> ok(snd(X))
38.45/11.05	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.05	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.05	head(ok(X)) -> ok(head(X))
38.45/11.05	tail(ok(X)) -> ok(tail(X))
38.45/11.05	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.05	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.05	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.05	top(mark(X)) -> top(proper(X))
38.45/11.05	top(ok(X)) -> top(active(X))
38.45/11.05	
38.45/11.05	Types:
38.45/11.05	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	0' :: mark:0':nil:ok
38.45/11.05	nil :: mark:0':nil:ok
38.45/11.05	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	top :: mark:0':nil:ok -> top
38.45/11.05	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.05	hole_top2_0 :: top
38.45/11.05	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.05	
38.45/11.05	
38.45/11.05	Lemmas:
38.45/11.05	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.05	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.05	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.05	pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2915_0)
38.45/11.05	u(gen_mark:0':nil:ok3_0(+(1, n5349_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n5349_0)
38.45/11.05	splitAt(gen_mark:0':nil:ok3_0(+(1, n10972_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n10972_0)
38.45/11.05	
38.45/11.05	
38.45/11.05	Generator Equations:
38.45/11.05	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.05	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.05	
38.45/11.05	
38.45/11.05	The following defined symbols remain to be analysed:
38.45/11.05	head, active, afterNth, fst, snd, tail, sel, take, proper, top
38.45/11.05	
38.45/11.05	They will be analysed ascendingly in the following order:
38.45/11.05	head < active
38.45/11.05	afterNth < active
38.45/11.05	fst < active
38.45/11.05	snd < active
38.45/11.05	tail < active
38.45/11.05	sel < active
38.45/11.05	take < active
38.45/11.05	active < top
38.45/11.05	head < proper
38.45/11.05	afterNth < proper
38.45/11.05	fst < proper
38.45/11.05	snd < proper
38.45/11.05	tail < proper
38.45/11.05	sel < proper
38.45/11.05	take < proper
38.45/11.05	proper < top
38.45/11.05	
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(29) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.05	Proved the following rewrite lemma:
38.45/11.05	head(gen_mark:0':nil:ok3_0(+(1, n14214_0))) -> *4_0, rt in Omega(n14214_0)
38.45/11.05	
38.45/11.05	Induction Base:
38.45/11.05	head(gen_mark:0':nil:ok3_0(+(1, 0)))
38.45/11.05	
38.45/11.05	Induction Step:
38.45/11.05	head(gen_mark:0':nil:ok3_0(+(1, +(n14214_0, 1)))) ->_R^Omega(1)
38.45/11.05	mark(head(gen_mark:0':nil:ok3_0(+(1, n14214_0)))) ->_IH
38.45/11.05	mark(*4_0)
38.45/11.05	
38.45/11.05	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(30)
38.45/11.05	Obligation:
38.45/11.05	TRS:
38.45/11.05	Rules:
38.45/11.05	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.05	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.05	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.05	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.05	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.05	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.05	active(head(cons(N, XS))) -> mark(N)
38.45/11.05	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.05	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.05	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.05	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.05	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.05	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.05	active(s(X)) -> s(active(X))
38.45/11.05	active(fst(X)) -> fst(active(X))
38.45/11.05	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.05	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.05	active(snd(X)) -> snd(active(X))
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.05	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.05	active(head(X)) -> head(active(X))
38.45/11.05	active(tail(X)) -> tail(active(X))
38.45/11.05	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.05	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.05	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.05	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.05	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.05	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.05	s(mark(X)) -> mark(s(X))
38.45/11.05	fst(mark(X)) -> mark(fst(X))
38.45/11.05	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.05	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.05	snd(mark(X)) -> mark(snd(X))
38.45/11.05	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.05	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.05	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.05	head(mark(X)) -> mark(head(X))
38.45/11.05	tail(mark(X)) -> mark(tail(X))
38.45/11.05	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.05	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.05	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.05	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.05	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.05	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.05	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.05	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.05	proper(s(X)) -> s(proper(X))
38.45/11.05	proper(fst(X)) -> fst(proper(X))
38.45/11.05	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.05	proper(snd(X)) -> snd(proper(X))
38.45/11.05	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.05	proper(0') -> ok(0')
38.45/11.05	proper(nil) -> ok(nil)
38.45/11.05	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.05	proper(head(X)) -> head(proper(X))
38.45/11.05	proper(tail(X)) -> tail(proper(X))
38.45/11.05	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.05	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.05	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.05	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.05	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.05	s(ok(X)) -> ok(s(X))
38.45/11.05	fst(ok(X)) -> ok(fst(X))
38.45/11.05	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.05	snd(ok(X)) -> ok(snd(X))
38.45/11.05	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.05	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.05	head(ok(X)) -> ok(head(X))
38.45/11.05	tail(ok(X)) -> ok(tail(X))
38.45/11.05	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.05	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.05	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.05	top(mark(X)) -> top(proper(X))
38.45/11.05	top(ok(X)) -> top(active(X))
38.45/11.05	
38.45/11.05	Types:
38.45/11.05	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	0' :: mark:0':nil:ok
38.45/11.05	nil :: mark:0':nil:ok
38.45/11.05	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	top :: mark:0':nil:ok -> top
38.45/11.05	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.05	hole_top2_0 :: top
38.45/11.05	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.05	
38.45/11.05	
38.45/11.05	Lemmas:
38.45/11.05	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.05	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.05	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.05	pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2915_0)
38.45/11.05	u(gen_mark:0':nil:ok3_0(+(1, n5349_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n5349_0)
38.45/11.05	splitAt(gen_mark:0':nil:ok3_0(+(1, n10972_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n10972_0)
38.45/11.05	head(gen_mark:0':nil:ok3_0(+(1, n14214_0))) -> *4_0, rt in Omega(n14214_0)
38.45/11.05	
38.45/11.05	
38.45/11.05	Generator Equations:
38.45/11.05	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.05	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.05	
38.45/11.05	
38.45/11.05	The following defined symbols remain to be analysed:
38.45/11.05	afterNth, active, fst, snd, tail, sel, take, proper, top
38.45/11.05	
38.45/11.05	They will be analysed ascendingly in the following order:
38.45/11.05	afterNth < active
38.45/11.05	fst < active
38.45/11.05	snd < active
38.45/11.05	tail < active
38.45/11.05	sel < active
38.45/11.05	take < active
38.45/11.05	active < top
38.45/11.05	afterNth < proper
38.45/11.05	fst < proper
38.45/11.05	snd < proper
38.45/11.05	tail < proper
38.45/11.05	sel < proper
38.45/11.05	take < proper
38.45/11.05	proper < top
38.45/11.05	
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(31) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.05	Proved the following rewrite lemma:
38.45/11.05	afterNth(gen_mark:0':nil:ok3_0(+(1, n15612_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n15612_0)
38.45/11.05	
38.45/11.05	Induction Base:
38.45/11.05	afterNth(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))
38.45/11.05	
38.45/11.05	Induction Step:
38.45/11.05	afterNth(gen_mark:0':nil:ok3_0(+(1, +(n15612_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1)
38.45/11.05	mark(afterNth(gen_mark:0':nil:ok3_0(+(1, n15612_0)), gen_mark:0':nil:ok3_0(b))) ->_IH
38.45/11.05	mark(*4_0)
38.45/11.05	
38.45/11.05	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(32)
38.45/11.05	Obligation:
38.45/11.05	TRS:
38.45/11.05	Rules:
38.45/11.05	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.05	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.05	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.05	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.05	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.05	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.05	active(head(cons(N, XS))) -> mark(N)
38.45/11.05	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.05	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.05	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.05	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.05	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.05	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.05	active(s(X)) -> s(active(X))
38.45/11.05	active(fst(X)) -> fst(active(X))
38.45/11.05	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.05	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.05	active(snd(X)) -> snd(active(X))
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.05	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.05	active(head(X)) -> head(active(X))
38.45/11.05	active(tail(X)) -> tail(active(X))
38.45/11.05	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.05	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.05	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.05	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.05	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.05	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.05	s(mark(X)) -> mark(s(X))
38.45/11.05	fst(mark(X)) -> mark(fst(X))
38.45/11.05	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.05	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.05	snd(mark(X)) -> mark(snd(X))
38.45/11.05	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.05	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.05	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.05	head(mark(X)) -> mark(head(X))
38.45/11.05	tail(mark(X)) -> mark(tail(X))
38.45/11.05	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.05	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.05	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.05	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.05	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.05	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.05	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.05	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.05	proper(s(X)) -> s(proper(X))
38.45/11.05	proper(fst(X)) -> fst(proper(X))
38.45/11.05	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.05	proper(snd(X)) -> snd(proper(X))
38.45/11.05	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.05	proper(0') -> ok(0')
38.45/11.05	proper(nil) -> ok(nil)
38.45/11.05	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.05	proper(head(X)) -> head(proper(X))
38.45/11.05	proper(tail(X)) -> tail(proper(X))
38.45/11.05	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.05	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.05	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.05	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.05	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.05	s(ok(X)) -> ok(s(X))
38.45/11.05	fst(ok(X)) -> ok(fst(X))
38.45/11.05	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.05	snd(ok(X)) -> ok(snd(X))
38.45/11.05	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.05	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.05	head(ok(X)) -> ok(head(X))
38.45/11.05	tail(ok(X)) -> ok(tail(X))
38.45/11.05	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.05	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.05	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.05	top(mark(X)) -> top(proper(X))
38.45/11.05	top(ok(X)) -> top(active(X))
38.45/11.05	
38.45/11.05	Types:
38.45/11.05	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	0' :: mark:0':nil:ok
38.45/11.05	nil :: mark:0':nil:ok
38.45/11.05	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	top :: mark:0':nil:ok -> top
38.45/11.05	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.05	hole_top2_0 :: top
38.45/11.05	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.05	
38.45/11.05	
38.45/11.05	Lemmas:
38.45/11.05	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.05	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.05	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.05	pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2915_0)
38.45/11.05	u(gen_mark:0':nil:ok3_0(+(1, n5349_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n5349_0)
38.45/11.05	splitAt(gen_mark:0':nil:ok3_0(+(1, n10972_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n10972_0)
38.45/11.05	head(gen_mark:0':nil:ok3_0(+(1, n14214_0))) -> *4_0, rt in Omega(n14214_0)
38.45/11.05	afterNth(gen_mark:0':nil:ok3_0(+(1, n15612_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n15612_0)
38.45/11.05	
38.45/11.05	
38.45/11.05	Generator Equations:
38.45/11.05	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.05	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.05	
38.45/11.05	
38.45/11.05	The following defined symbols remain to be analysed:
38.45/11.05	fst, active, snd, tail, sel, take, proper, top
38.45/11.05	
38.45/11.05	They will be analysed ascendingly in the following order:
38.45/11.05	fst < active
38.45/11.05	snd < active
38.45/11.05	tail < active
38.45/11.05	sel < active
38.45/11.05	take < active
38.45/11.05	active < top
38.45/11.05	fst < proper
38.45/11.05	snd < proper
38.45/11.05	tail < proper
38.45/11.05	sel < proper
38.45/11.05	take < proper
38.45/11.05	proper < top
38.45/11.05	
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(33) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.05	Proved the following rewrite lemma:
38.45/11.05	fst(gen_mark:0':nil:ok3_0(+(1, n19368_0))) -> *4_0, rt in Omega(n19368_0)
38.45/11.05	
38.45/11.05	Induction Base:
38.45/11.05	fst(gen_mark:0':nil:ok3_0(+(1, 0)))
38.45/11.05	
38.45/11.05	Induction Step:
38.45/11.05	fst(gen_mark:0':nil:ok3_0(+(1, +(n19368_0, 1)))) ->_R^Omega(1)
38.45/11.05	mark(fst(gen_mark:0':nil:ok3_0(+(1, n19368_0)))) ->_IH
38.45/11.05	mark(*4_0)
38.45/11.05	
38.45/11.05	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(34)
38.45/11.05	Obligation:
38.45/11.05	TRS:
38.45/11.05	Rules:
38.45/11.05	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.05	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.05	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.05	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.05	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.05	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.05	active(head(cons(N, XS))) -> mark(N)
38.45/11.05	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.05	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.05	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.05	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.05	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.05	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.05	active(s(X)) -> s(active(X))
38.45/11.05	active(fst(X)) -> fst(active(X))
38.45/11.05	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.05	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.05	active(snd(X)) -> snd(active(X))
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.05	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.05	active(head(X)) -> head(active(X))
38.45/11.05	active(tail(X)) -> tail(active(X))
38.45/11.05	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.05	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.05	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.05	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.05	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.05	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.05	s(mark(X)) -> mark(s(X))
38.45/11.05	fst(mark(X)) -> mark(fst(X))
38.45/11.05	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.05	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.05	snd(mark(X)) -> mark(snd(X))
38.45/11.05	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.05	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.05	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.05	head(mark(X)) -> mark(head(X))
38.45/11.05	tail(mark(X)) -> mark(tail(X))
38.45/11.05	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.05	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.05	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.05	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.05	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.05	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.05	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.05	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.05	proper(s(X)) -> s(proper(X))
38.45/11.05	proper(fst(X)) -> fst(proper(X))
38.45/11.05	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.05	proper(snd(X)) -> snd(proper(X))
38.45/11.05	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.05	proper(0') -> ok(0')
38.45/11.05	proper(nil) -> ok(nil)
38.45/11.05	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.05	proper(head(X)) -> head(proper(X))
38.45/11.05	proper(tail(X)) -> tail(proper(X))
38.45/11.05	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.05	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.05	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.05	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.05	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.05	s(ok(X)) -> ok(s(X))
38.45/11.05	fst(ok(X)) -> ok(fst(X))
38.45/11.05	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.05	snd(ok(X)) -> ok(snd(X))
38.45/11.05	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.05	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.05	head(ok(X)) -> ok(head(X))
38.45/11.05	tail(ok(X)) -> ok(tail(X))
38.45/11.05	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.05	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.05	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.05	top(mark(X)) -> top(proper(X))
38.45/11.05	top(ok(X)) -> top(active(X))
38.45/11.05	
38.45/11.05	Types:
38.45/11.05	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	0' :: mark:0':nil:ok
38.45/11.05	nil :: mark:0':nil:ok
38.45/11.05	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	top :: mark:0':nil:ok -> top
38.45/11.05	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.05	hole_top2_0 :: top
38.45/11.05	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.05	
38.45/11.05	
38.45/11.05	Lemmas:
38.45/11.05	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.05	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.05	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.05	pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2915_0)
38.45/11.05	u(gen_mark:0':nil:ok3_0(+(1, n5349_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n5349_0)
38.45/11.05	splitAt(gen_mark:0':nil:ok3_0(+(1, n10972_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n10972_0)
38.45/11.05	head(gen_mark:0':nil:ok3_0(+(1, n14214_0))) -> *4_0, rt in Omega(n14214_0)
38.45/11.05	afterNth(gen_mark:0':nil:ok3_0(+(1, n15612_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n15612_0)
38.45/11.05	fst(gen_mark:0':nil:ok3_0(+(1, n19368_0))) -> *4_0, rt in Omega(n19368_0)
38.45/11.05	
38.45/11.05	
38.45/11.05	Generator Equations:
38.45/11.05	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.05	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.05	
38.45/11.05	
38.45/11.05	The following defined symbols remain to be analysed:
38.45/11.05	snd, active, tail, sel, take, proper, top
38.45/11.05	
38.45/11.05	They will be analysed ascendingly in the following order:
38.45/11.05	snd < active
38.45/11.05	tail < active
38.45/11.05	sel < active
38.45/11.05	take < active
38.45/11.05	active < top
38.45/11.05	snd < proper
38.45/11.05	tail < proper
38.45/11.05	sel < proper
38.45/11.05	take < proper
38.45/11.05	proper < top
38.45/11.05	
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(35) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.05	Proved the following rewrite lemma:
38.45/11.05	snd(gen_mark:0':nil:ok3_0(+(1, n21017_0))) -> *4_0, rt in Omega(n21017_0)
38.45/11.05	
38.45/11.05	Induction Base:
38.45/11.05	snd(gen_mark:0':nil:ok3_0(+(1, 0)))
38.45/11.05	
38.45/11.05	Induction Step:
38.45/11.05	snd(gen_mark:0':nil:ok3_0(+(1, +(n21017_0, 1)))) ->_R^Omega(1)
38.45/11.05	mark(snd(gen_mark:0':nil:ok3_0(+(1, n21017_0)))) ->_IH
38.45/11.05	mark(*4_0)
38.45/11.05	
38.45/11.05	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(36)
38.45/11.05	Obligation:
38.45/11.05	TRS:
38.45/11.05	Rules:
38.45/11.05	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.05	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.05	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.05	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.05	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.05	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.05	active(head(cons(N, XS))) -> mark(N)
38.45/11.05	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.05	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.05	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.05	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.05	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.05	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.05	active(s(X)) -> s(active(X))
38.45/11.05	active(fst(X)) -> fst(active(X))
38.45/11.05	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.05	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.05	active(snd(X)) -> snd(active(X))
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.05	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.05	active(head(X)) -> head(active(X))
38.45/11.05	active(tail(X)) -> tail(active(X))
38.45/11.05	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.05	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.05	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.05	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.05	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.05	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.05	s(mark(X)) -> mark(s(X))
38.45/11.05	fst(mark(X)) -> mark(fst(X))
38.45/11.05	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.05	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.05	snd(mark(X)) -> mark(snd(X))
38.45/11.05	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.05	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.05	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.05	head(mark(X)) -> mark(head(X))
38.45/11.05	tail(mark(X)) -> mark(tail(X))
38.45/11.05	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.05	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.05	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.05	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.05	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.05	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.05	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.05	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.05	proper(s(X)) -> s(proper(X))
38.45/11.05	proper(fst(X)) -> fst(proper(X))
38.45/11.05	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.05	proper(snd(X)) -> snd(proper(X))
38.45/11.05	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.05	proper(0') -> ok(0')
38.45/11.05	proper(nil) -> ok(nil)
38.45/11.05	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.05	proper(head(X)) -> head(proper(X))
38.45/11.05	proper(tail(X)) -> tail(proper(X))
38.45/11.05	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.05	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.05	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.05	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.05	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.05	s(ok(X)) -> ok(s(X))
38.45/11.05	fst(ok(X)) -> ok(fst(X))
38.45/11.05	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.05	snd(ok(X)) -> ok(snd(X))
38.45/11.05	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.05	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.05	head(ok(X)) -> ok(head(X))
38.45/11.05	tail(ok(X)) -> ok(tail(X))
38.45/11.05	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.05	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.05	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.05	top(mark(X)) -> top(proper(X))
38.45/11.05	top(ok(X)) -> top(active(X))
38.45/11.05	
38.45/11.05	Types:
38.45/11.05	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	0' :: mark:0':nil:ok
38.45/11.05	nil :: mark:0':nil:ok
38.45/11.05	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	top :: mark:0':nil:ok -> top
38.45/11.05	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.05	hole_top2_0 :: top
38.45/11.05	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.05	
38.45/11.05	
38.45/11.05	Lemmas:
38.45/11.05	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.05	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.05	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.05	pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2915_0)
38.45/11.05	u(gen_mark:0':nil:ok3_0(+(1, n5349_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n5349_0)
38.45/11.05	splitAt(gen_mark:0':nil:ok3_0(+(1, n10972_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n10972_0)
38.45/11.05	head(gen_mark:0':nil:ok3_0(+(1, n14214_0))) -> *4_0, rt in Omega(n14214_0)
38.45/11.05	afterNth(gen_mark:0':nil:ok3_0(+(1, n15612_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n15612_0)
38.45/11.05	fst(gen_mark:0':nil:ok3_0(+(1, n19368_0))) -> *4_0, rt in Omega(n19368_0)
38.45/11.05	snd(gen_mark:0':nil:ok3_0(+(1, n21017_0))) -> *4_0, rt in Omega(n21017_0)
38.45/11.05	
38.45/11.05	
38.45/11.05	Generator Equations:
38.45/11.05	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.05	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.05	
38.45/11.05	
38.45/11.05	The following defined symbols remain to be analysed:
38.45/11.05	tail, active, sel, take, proper, top
38.45/11.05	
38.45/11.05	They will be analysed ascendingly in the following order:
38.45/11.05	tail < active
38.45/11.05	sel < active
38.45/11.05	take < active
38.45/11.05	active < top
38.45/11.05	tail < proper
38.45/11.05	sel < proper
38.45/11.05	take < proper
38.45/11.05	proper < top
38.45/11.05	
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(37) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.05	Proved the following rewrite lemma:
38.45/11.05	tail(gen_mark:0':nil:ok3_0(+(1, n22767_0))) -> *4_0, rt in Omega(n22767_0)
38.45/11.05	
38.45/11.05	Induction Base:
38.45/11.05	tail(gen_mark:0':nil:ok3_0(+(1, 0)))
38.45/11.05	
38.45/11.05	Induction Step:
38.45/11.05	tail(gen_mark:0':nil:ok3_0(+(1, +(n22767_0, 1)))) ->_R^Omega(1)
38.45/11.05	mark(tail(gen_mark:0':nil:ok3_0(+(1, n22767_0)))) ->_IH
38.45/11.05	mark(*4_0)
38.45/11.05	
38.45/11.05	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(38)
38.45/11.05	Obligation:
38.45/11.05	TRS:
38.45/11.05	Rules:
38.45/11.05	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.05	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.05	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.05	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.05	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.05	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.05	active(head(cons(N, XS))) -> mark(N)
38.45/11.05	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.05	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.05	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.05	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.05	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.05	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.05	active(s(X)) -> s(active(X))
38.45/11.05	active(fst(X)) -> fst(active(X))
38.45/11.05	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.05	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.05	active(snd(X)) -> snd(active(X))
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.05	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.05	active(head(X)) -> head(active(X))
38.45/11.05	active(tail(X)) -> tail(active(X))
38.45/11.05	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.05	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.05	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.05	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.05	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.05	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.05	s(mark(X)) -> mark(s(X))
38.45/11.05	fst(mark(X)) -> mark(fst(X))
38.45/11.05	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.05	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.05	snd(mark(X)) -> mark(snd(X))
38.45/11.05	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.05	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.05	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.05	head(mark(X)) -> mark(head(X))
38.45/11.05	tail(mark(X)) -> mark(tail(X))
38.45/11.05	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.05	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.05	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.05	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.05	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.05	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.05	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.05	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.05	proper(s(X)) -> s(proper(X))
38.45/11.05	proper(fst(X)) -> fst(proper(X))
38.45/11.05	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.05	proper(snd(X)) -> snd(proper(X))
38.45/11.05	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.05	proper(0') -> ok(0')
38.45/11.05	proper(nil) -> ok(nil)
38.45/11.05	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.05	proper(head(X)) -> head(proper(X))
38.45/11.05	proper(tail(X)) -> tail(proper(X))
38.45/11.05	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.05	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.05	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.05	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.05	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.05	s(ok(X)) -> ok(s(X))
38.45/11.05	fst(ok(X)) -> ok(fst(X))
38.45/11.05	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.05	snd(ok(X)) -> ok(snd(X))
38.45/11.05	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.05	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.05	head(ok(X)) -> ok(head(X))
38.45/11.05	tail(ok(X)) -> ok(tail(X))
38.45/11.05	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.05	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.05	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.05	top(mark(X)) -> top(proper(X))
38.45/11.05	top(ok(X)) -> top(active(X))
38.45/11.05	
38.45/11.05	Types:
38.45/11.05	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	0' :: mark:0':nil:ok
38.45/11.05	nil :: mark:0':nil:ok
38.45/11.05	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	top :: mark:0':nil:ok -> top
38.45/11.05	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.05	hole_top2_0 :: top
38.45/11.05	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.05	
38.45/11.05	
38.45/11.05	Lemmas:
38.45/11.05	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.05	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.05	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.05	pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2915_0)
38.45/11.05	u(gen_mark:0':nil:ok3_0(+(1, n5349_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n5349_0)
38.45/11.05	splitAt(gen_mark:0':nil:ok3_0(+(1, n10972_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n10972_0)
38.45/11.05	head(gen_mark:0':nil:ok3_0(+(1, n14214_0))) -> *4_0, rt in Omega(n14214_0)
38.45/11.05	afterNth(gen_mark:0':nil:ok3_0(+(1, n15612_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n15612_0)
38.45/11.05	fst(gen_mark:0':nil:ok3_0(+(1, n19368_0))) -> *4_0, rt in Omega(n19368_0)
38.45/11.05	snd(gen_mark:0':nil:ok3_0(+(1, n21017_0))) -> *4_0, rt in Omega(n21017_0)
38.45/11.05	tail(gen_mark:0':nil:ok3_0(+(1, n22767_0))) -> *4_0, rt in Omega(n22767_0)
38.45/11.05	
38.45/11.05	
38.45/11.05	Generator Equations:
38.45/11.05	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.05	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.05	
38.45/11.05	
38.45/11.05	The following defined symbols remain to be analysed:
38.45/11.05	sel, active, take, proper, top
38.45/11.05	
38.45/11.05	They will be analysed ascendingly in the following order:
38.45/11.05	sel < active
38.45/11.05	take < active
38.45/11.05	active < top
38.45/11.05	sel < proper
38.45/11.05	take < proper
38.45/11.05	proper < top
38.45/11.05	
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(39) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.05	Proved the following rewrite lemma:
38.45/11.05	sel(gen_mark:0':nil:ok3_0(+(1, n24618_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n24618_0)
38.45/11.05	
38.45/11.05	Induction Base:
38.45/11.05	sel(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))
38.45/11.05	
38.45/11.05	Induction Step:
38.45/11.05	sel(gen_mark:0':nil:ok3_0(+(1, +(n24618_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1)
38.45/11.05	mark(sel(gen_mark:0':nil:ok3_0(+(1, n24618_0)), gen_mark:0':nil:ok3_0(b))) ->_IH
38.45/11.05	mark(*4_0)
38.45/11.05	
38.45/11.05	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.05	----------------------------------------
38.45/11.05	
38.45/11.05	(40)
38.45/11.05	Obligation:
38.45/11.05	TRS:
38.45/11.05	Rules:
38.45/11.05	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.05	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.05	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.05	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.05	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.05	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.05	active(head(cons(N, XS))) -> mark(N)
38.45/11.05	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.05	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.05	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.05	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.05	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.05	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.05	active(s(X)) -> s(active(X))
38.45/11.05	active(fst(X)) -> fst(active(X))
38.45/11.05	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.05	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.05	active(snd(X)) -> snd(active(X))
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.05	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.05	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.05	active(head(X)) -> head(active(X))
38.45/11.05	active(tail(X)) -> tail(active(X))
38.45/11.05	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.05	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.05	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.05	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.05	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.05	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.05	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.05	s(mark(X)) -> mark(s(X))
38.45/11.05	fst(mark(X)) -> mark(fst(X))
38.45/11.05	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.05	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.05	snd(mark(X)) -> mark(snd(X))
38.45/11.05	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.05	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.05	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.05	head(mark(X)) -> mark(head(X))
38.45/11.05	tail(mark(X)) -> mark(tail(X))
38.45/11.05	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.05	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.05	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.05	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.05	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.05	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.05	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.05	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.05	proper(s(X)) -> s(proper(X))
38.45/11.05	proper(fst(X)) -> fst(proper(X))
38.45/11.05	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.05	proper(snd(X)) -> snd(proper(X))
38.45/11.05	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.05	proper(0') -> ok(0')
38.45/11.05	proper(nil) -> ok(nil)
38.45/11.05	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.05	proper(head(X)) -> head(proper(X))
38.45/11.05	proper(tail(X)) -> tail(proper(X))
38.45/11.05	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.05	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.05	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.05	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.05	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.05	s(ok(X)) -> ok(s(X))
38.45/11.05	fst(ok(X)) -> ok(fst(X))
38.45/11.05	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.05	snd(ok(X)) -> ok(snd(X))
38.45/11.05	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.05	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.05	head(ok(X)) -> ok(head(X))
38.45/11.05	tail(ok(X)) -> ok(tail(X))
38.45/11.05	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.05	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.05	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.05	top(mark(X)) -> top(proper(X))
38.45/11.05	top(ok(X)) -> top(active(X))
38.45/11.05	
38.45/11.05	Types:
38.45/11.05	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.05	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	0' :: mark:0':nil:ok
38.45/11.06	nil :: mark:0':nil:ok
38.45/11.06	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	top :: mark:0':nil:ok -> top
38.45/11.06	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.06	hole_top2_0 :: top
38.45/11.06	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.06	
38.45/11.06	
38.45/11.06	Lemmas:
38.45/11.06	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.06	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.06	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.06	pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2915_0)
38.45/11.06	u(gen_mark:0':nil:ok3_0(+(1, n5349_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n5349_0)
38.45/11.06	splitAt(gen_mark:0':nil:ok3_0(+(1, n10972_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n10972_0)
38.45/11.06	head(gen_mark:0':nil:ok3_0(+(1, n14214_0))) -> *4_0, rt in Omega(n14214_0)
38.45/11.06	afterNth(gen_mark:0':nil:ok3_0(+(1, n15612_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n15612_0)
38.45/11.06	fst(gen_mark:0':nil:ok3_0(+(1, n19368_0))) -> *4_0, rt in Omega(n19368_0)
38.45/11.06	snd(gen_mark:0':nil:ok3_0(+(1, n21017_0))) -> *4_0, rt in Omega(n21017_0)
38.45/11.06	tail(gen_mark:0':nil:ok3_0(+(1, n22767_0))) -> *4_0, rt in Omega(n22767_0)
38.45/11.06	sel(gen_mark:0':nil:ok3_0(+(1, n24618_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n24618_0)
38.45/11.06	
38.45/11.06	
38.45/11.06	Generator Equations:
38.45/11.06	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.06	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.06	
38.45/11.06	
38.45/11.06	The following defined symbols remain to be analysed:
38.45/11.06	take, active, proper, top
38.45/11.06	
38.45/11.06	They will be analysed ascendingly in the following order:
38.45/11.06	take < active
38.45/11.06	active < top
38.45/11.06	take < proper
38.45/11.06	proper < top
38.45/11.06	
38.45/11.06	----------------------------------------
38.45/11.06	
38.45/11.06	(41) RewriteLemmaProof (LOWER BOUND(ID))
38.45/11.06	Proved the following rewrite lemma:
38.45/11.06	take(gen_mark:0':nil:ok3_0(+(1, n29304_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n29304_0)
38.45/11.06	
38.45/11.06	Induction Base:
38.45/11.06	take(gen_mark:0':nil:ok3_0(+(1, 0)), gen_mark:0':nil:ok3_0(b))
38.45/11.06	
38.45/11.06	Induction Step:
38.45/11.06	take(gen_mark:0':nil:ok3_0(+(1, +(n29304_0, 1))), gen_mark:0':nil:ok3_0(b)) ->_R^Omega(1)
38.45/11.06	mark(take(gen_mark:0':nil:ok3_0(+(1, n29304_0)), gen_mark:0':nil:ok3_0(b))) ->_IH
38.45/11.06	mark(*4_0)
38.45/11.06	
38.45/11.06	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
38.45/11.06	----------------------------------------
38.45/11.06	
38.45/11.06	(42)
38.45/11.06	Obligation:
38.45/11.06	TRS:
38.45/11.06	Rules:
38.45/11.06	active(natsFrom(N)) -> mark(cons(N, natsFrom(s(N))))
38.45/11.06	active(fst(pair(XS, YS))) -> mark(XS)
38.45/11.06	active(snd(pair(XS, YS))) -> mark(YS)
38.45/11.06	active(splitAt(0', XS)) -> mark(pair(nil, XS))
38.45/11.06	active(splitAt(s(N), cons(X, XS))) -> mark(u(splitAt(N, XS), N, X, XS))
38.45/11.06	active(u(pair(YS, ZS), N, X, XS)) -> mark(pair(cons(X, YS), ZS))
38.45/11.06	active(head(cons(N, XS))) -> mark(N)
38.45/11.06	active(tail(cons(N, XS))) -> mark(XS)
38.45/11.06	active(sel(N, XS)) -> mark(head(afterNth(N, XS)))
38.45/11.06	active(take(N, XS)) -> mark(fst(splitAt(N, XS)))
38.45/11.06	active(afterNth(N, XS)) -> mark(snd(splitAt(N, XS)))
38.45/11.06	active(natsFrom(X)) -> natsFrom(active(X))
38.45/11.06	active(cons(X1, X2)) -> cons(active(X1), X2)
38.45/11.06	active(s(X)) -> s(active(X))
38.45/11.06	active(fst(X)) -> fst(active(X))
38.45/11.06	active(pair(X1, X2)) -> pair(active(X1), X2)
38.45/11.06	active(pair(X1, X2)) -> pair(X1, active(X2))
38.45/11.06	active(snd(X)) -> snd(active(X))
38.45/11.06	active(splitAt(X1, X2)) -> splitAt(active(X1), X2)
38.45/11.06	active(splitAt(X1, X2)) -> splitAt(X1, active(X2))
38.45/11.06	active(u(X1, X2, X3, X4)) -> u(active(X1), X2, X3, X4)
38.45/11.06	active(head(X)) -> head(active(X))
38.45/11.06	active(tail(X)) -> tail(active(X))
38.45/11.06	active(sel(X1, X2)) -> sel(active(X1), X2)
38.45/11.06	active(sel(X1, X2)) -> sel(X1, active(X2))
38.45/11.06	active(afterNth(X1, X2)) -> afterNth(active(X1), X2)
38.45/11.06	active(afterNth(X1, X2)) -> afterNth(X1, active(X2))
38.45/11.06	active(take(X1, X2)) -> take(active(X1), X2)
38.45/11.06	active(take(X1, X2)) -> take(X1, active(X2))
38.45/11.06	natsFrom(mark(X)) -> mark(natsFrom(X))
38.45/11.06	cons(mark(X1), X2) -> mark(cons(X1, X2))
38.45/11.06	s(mark(X)) -> mark(s(X))
38.45/11.06	fst(mark(X)) -> mark(fst(X))
38.45/11.06	pair(mark(X1), X2) -> mark(pair(X1, X2))
38.45/11.06	pair(X1, mark(X2)) -> mark(pair(X1, X2))
38.45/11.06	snd(mark(X)) -> mark(snd(X))
38.45/11.06	splitAt(mark(X1), X2) -> mark(splitAt(X1, X2))
38.45/11.06	splitAt(X1, mark(X2)) -> mark(splitAt(X1, X2))
38.45/11.06	u(mark(X1), X2, X3, X4) -> mark(u(X1, X2, X3, X4))
38.45/11.06	head(mark(X)) -> mark(head(X))
38.45/11.06	tail(mark(X)) -> mark(tail(X))
38.45/11.06	sel(mark(X1), X2) -> mark(sel(X1, X2))
38.45/11.06	sel(X1, mark(X2)) -> mark(sel(X1, X2))
38.45/11.06	afterNth(mark(X1), X2) -> mark(afterNth(X1, X2))
38.45/11.06	afterNth(X1, mark(X2)) -> mark(afterNth(X1, X2))
38.45/11.06	take(mark(X1), X2) -> mark(take(X1, X2))
38.45/11.06	take(X1, mark(X2)) -> mark(take(X1, X2))
38.45/11.06	proper(natsFrom(X)) -> natsFrom(proper(X))
38.45/11.06	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
38.45/11.06	proper(s(X)) -> s(proper(X))
38.45/11.06	proper(fst(X)) -> fst(proper(X))
38.45/11.06	proper(pair(X1, X2)) -> pair(proper(X1), proper(X2))
38.45/11.06	proper(snd(X)) -> snd(proper(X))
38.45/11.06	proper(splitAt(X1, X2)) -> splitAt(proper(X1), proper(X2))
38.45/11.06	proper(0') -> ok(0')
38.45/11.06	proper(nil) -> ok(nil)
38.45/11.06	proper(u(X1, X2, X3, X4)) -> u(proper(X1), proper(X2), proper(X3), proper(X4))
38.45/11.06	proper(head(X)) -> head(proper(X))
38.45/11.06	proper(tail(X)) -> tail(proper(X))
38.45/11.06	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
38.45/11.06	proper(afterNth(X1, X2)) -> afterNth(proper(X1), proper(X2))
38.45/11.06	proper(take(X1, X2)) -> take(proper(X1), proper(X2))
38.45/11.06	natsFrom(ok(X)) -> ok(natsFrom(X))
38.45/11.06	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
38.45/11.06	s(ok(X)) -> ok(s(X))
38.45/11.06	fst(ok(X)) -> ok(fst(X))
38.45/11.06	pair(ok(X1), ok(X2)) -> ok(pair(X1, X2))
38.45/11.06	snd(ok(X)) -> ok(snd(X))
38.45/11.06	splitAt(ok(X1), ok(X2)) -> ok(splitAt(X1, X2))
38.45/11.06	u(ok(X1), ok(X2), ok(X3), ok(X4)) -> ok(u(X1, X2, X3, X4))
38.45/11.06	head(ok(X)) -> ok(head(X))
38.45/11.06	tail(ok(X)) -> ok(tail(X))
38.45/11.06	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
38.45/11.06	afterNth(ok(X1), ok(X2)) -> ok(afterNth(X1, X2))
38.45/11.06	take(ok(X1), ok(X2)) -> ok(take(X1, X2))
38.45/11.06	top(mark(X)) -> top(proper(X))
38.45/11.06	top(ok(X)) -> top(active(X))
38.45/11.06	
38.45/11.06	Types:
38.45/11.06	active :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	natsFrom :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	mark :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	cons :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	s :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	fst :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	pair :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	snd :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	splitAt :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	0' :: mark:0':nil:ok
38.45/11.06	nil :: mark:0':nil:ok
38.45/11.06	u :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	head :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	tail :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	sel :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	afterNth :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	take :: mark:0':nil:ok -> mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	proper :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	ok :: mark:0':nil:ok -> mark:0':nil:ok
38.45/11.06	top :: mark:0':nil:ok -> top
38.45/11.06	hole_mark:0':nil:ok1_0 :: mark:0':nil:ok
38.45/11.06	hole_top2_0 :: top
38.45/11.06	gen_mark:0':nil:ok3_0 :: Nat -> mark:0':nil:ok
38.45/11.06	
38.45/11.06	
38.45/11.06	Lemmas:
38.45/11.06	cons(gen_mark:0':nil:ok3_0(+(1, n5_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
38.45/11.06	natsFrom(gen_mark:0':nil:ok3_0(+(1, n1518_0))) -> *4_0, rt in Omega(n1518_0)
38.45/11.06	s(gen_mark:0':nil:ok3_0(+(1, n2166_0))) -> *4_0, rt in Omega(n2166_0)
38.45/11.06	pair(gen_mark:0':nil:ok3_0(+(1, n2915_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n2915_0)
38.45/11.06	u(gen_mark:0':nil:ok3_0(+(1, n5349_0)), gen_mark:0':nil:ok3_0(b), gen_mark:0':nil:ok3_0(c), gen_mark:0':nil:ok3_0(d)) -> *4_0, rt in Omega(n5349_0)
38.45/11.06	splitAt(gen_mark:0':nil:ok3_0(+(1, n10972_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n10972_0)
38.45/11.06	head(gen_mark:0':nil:ok3_0(+(1, n14214_0))) -> *4_0, rt in Omega(n14214_0)
38.45/11.06	afterNth(gen_mark:0':nil:ok3_0(+(1, n15612_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n15612_0)
38.45/11.06	fst(gen_mark:0':nil:ok3_0(+(1, n19368_0))) -> *4_0, rt in Omega(n19368_0)
38.45/11.06	snd(gen_mark:0':nil:ok3_0(+(1, n21017_0))) -> *4_0, rt in Omega(n21017_0)
38.45/11.06	tail(gen_mark:0':nil:ok3_0(+(1, n22767_0))) -> *4_0, rt in Omega(n22767_0)
38.45/11.06	sel(gen_mark:0':nil:ok3_0(+(1, n24618_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n24618_0)
38.45/11.06	take(gen_mark:0':nil:ok3_0(+(1, n29304_0)), gen_mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n29304_0)
38.45/11.06	
38.45/11.06	
38.45/11.06	Generator Equations:
38.45/11.06	gen_mark:0':nil:ok3_0(0) <=> 0'
38.45/11.06	gen_mark:0':nil:ok3_0(+(x, 1)) <=> mark(gen_mark:0':nil:ok3_0(x))
38.45/11.06	
38.45/11.06	
38.45/11.06	The following defined symbols remain to be analysed:
38.45/11.06	active, proper, top
38.45/11.06	
38.45/11.06	They will be analysed ascendingly in the following order:
38.45/11.06	active < top
38.45/11.06	proper < top
38.45/11.09	EOF