25.81/9.51	WORST_CASE(Omega(n^1), O(n^1))
25.81/9.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
25.81/9.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
25.81/9.52	
25.81/9.52	
25.81/9.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
25.81/9.52	
25.81/9.52	(0) CpxTRS
25.81/9.52	(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
25.81/9.52	(2) CpxTRS
25.81/9.52	    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
25.81/9.52	    (4) CpxTRS
25.81/9.52	    (5) CpxTrsMatchBoundsTAProof [FINISHED, 71 ms]
25.81/9.52	    (6) BOUNDS(1, n^1)
25.81/9.52	(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
25.81/9.52	(8) CpxTRS
25.81/9.52	    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
25.81/9.52	    (10) typed CpxTrs
25.81/9.52	    (11) OrderProof [LOWER BOUND(ID), 0 ms]
25.81/9.52	    (12) typed CpxTrs
25.81/9.52	    (13) RewriteLemmaProof [LOWER BOUND(ID), 453 ms]
25.81/9.52	    (14) BEST
25.81/9.52	        (15) proven lower bound
25.81/9.52	            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
25.81/9.52	            (17) BOUNDS(n^1, INF)
25.81/9.52	        (18) typed CpxTrs
25.81/9.52	            (19) RewriteLemmaProof [LOWER BOUND(ID), 97 ms]
25.81/9.52	            (20) typed CpxTrs
25.81/9.52	            (21) RewriteLemmaProof [LOWER BOUND(ID), 61 ms]
25.81/9.52	            (22) typed CpxTrs
25.81/9.52	            (23) RewriteLemmaProof [LOWER BOUND(ID), 97 ms]
25.81/9.52	            (24) typed CpxTrs
25.81/9.52	            (25) RewriteLemmaProof [LOWER BOUND(ID), 41 ms]
25.81/9.52	            (26) typed CpxTrs
25.81/9.52	
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(0)
25.81/9.52	Obligation:
25.81/9.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).
25.81/9.52	
25.81/9.52	
25.81/9.52	The TRS R consists of the following rules:
25.81/9.52	
25.81/9.52	   active(f(X)) -> mark(cons(X, f(g(X))))
25.81/9.52	   active(g(0)) -> mark(s(0))
25.81/9.52	   active(g(s(X))) -> mark(s(s(g(X))))
25.81/9.52	   active(sel(0, cons(X, Y))) -> mark(X)
25.81/9.52	   active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
25.81/9.52	   active(f(X)) -> f(active(X))
25.81/9.52	   active(cons(X1, X2)) -> cons(active(X1), X2)
25.81/9.52	   active(g(X)) -> g(active(X))
25.81/9.52	   active(s(X)) -> s(active(X))
25.81/9.52	   active(sel(X1, X2)) -> sel(active(X1), X2)
25.81/9.52	   active(sel(X1, X2)) -> sel(X1, active(X2))
25.81/9.52	   f(mark(X)) -> mark(f(X))
25.81/9.52	   cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	   g(mark(X)) -> mark(g(X))
25.81/9.52	   s(mark(X)) -> mark(s(X))
25.81/9.52	   sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	   proper(f(X)) -> f(proper(X))
25.81/9.52	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
25.81/9.52	   proper(g(X)) -> g(proper(X))
25.81/9.52	   proper(0) -> ok(0)
25.81/9.52	   proper(s(X)) -> s(proper(X))
25.81/9.52	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
25.81/9.52	   f(ok(X)) -> ok(f(X))
25.81/9.52	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	   g(ok(X)) -> ok(g(X))
25.81/9.52	   s(ok(X)) -> ok(s(X))
25.81/9.52	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	   top(mark(X)) -> top(proper(X))
25.81/9.52	   top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	S is empty.
25.81/9.52	Rewrite Strategy: FULL
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
25.81/9.52	The following defined symbols can occur below the 0th argument of top: proper, active
25.81/9.52	The following defined symbols can occur below the 0th argument of proper: proper, active
25.81/9.52	The following defined symbols can occur below the 0th argument of active: proper, active
25.81/9.52	
25.81/9.52	Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
25.81/9.52	active(f(X)) -> mark(cons(X, f(g(X))))
25.81/9.52	active(g(0)) -> mark(s(0))
25.81/9.52	active(g(s(X))) -> mark(s(s(g(X))))
25.81/9.52	active(sel(0, cons(X, Y))) -> mark(X)
25.81/9.52	active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
25.81/9.52	active(f(X)) -> f(active(X))
25.81/9.52	active(cons(X1, X2)) -> cons(active(X1), X2)
25.81/9.52	active(g(X)) -> g(active(X))
25.81/9.52	active(s(X)) -> s(active(X))
25.81/9.52	active(sel(X1, X2)) -> sel(active(X1), X2)
25.81/9.52	active(sel(X1, X2)) -> sel(X1, active(X2))
25.81/9.52	proper(f(X)) -> f(proper(X))
25.81/9.52	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
25.81/9.52	proper(g(X)) -> g(proper(X))
25.81/9.52	proper(s(X)) -> s(proper(X))
25.81/9.52	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(2)
25.81/9.52	Obligation:
25.81/9.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
25.81/9.52	
25.81/9.52	
25.81/9.52	The TRS R consists of the following rules:
25.81/9.52	
25.81/9.52	   f(mark(X)) -> mark(f(X))
25.81/9.52	   cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	   g(mark(X)) -> mark(g(X))
25.81/9.52	   s(mark(X)) -> mark(s(X))
25.81/9.52	   sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	   proper(0) -> ok(0)
25.81/9.52	   f(ok(X)) -> ok(f(X))
25.81/9.52	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	   g(ok(X)) -> ok(g(X))
25.81/9.52	   s(ok(X)) -> ok(s(X))
25.81/9.52	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	   top(mark(X)) -> top(proper(X))
25.81/9.52	   top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	S is empty.
25.81/9.52	Rewrite Strategy: FULL
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(3) RelTrsToTrsProof (UPPER BOUND(ID))
25.81/9.52	transformed relative TRS to TRS
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(4)
25.81/9.52	Obligation:
25.81/9.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).
25.81/9.52	
25.81/9.52	
25.81/9.52	The TRS R consists of the following rules:
25.81/9.52	
25.81/9.52	   f(mark(X)) -> mark(f(X))
25.81/9.52	   cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	   g(mark(X)) -> mark(g(X))
25.81/9.52	   s(mark(X)) -> mark(s(X))
25.81/9.52	   sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	   proper(0) -> ok(0)
25.81/9.52	   f(ok(X)) -> ok(f(X))
25.81/9.52	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	   g(ok(X)) -> ok(g(X))
25.81/9.52	   s(ok(X)) -> ok(s(X))
25.81/9.52	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	   top(mark(X)) -> top(proper(X))
25.81/9.52	   top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	S is empty.
25.81/9.52	Rewrite Strategy: FULL
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(5) CpxTrsMatchBoundsTAProof (FINISHED)
25.81/9.52	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. 
25.81/9.52	
25.81/9.52	The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
25.81/9.52	final states : [1, 2, 3, 4, 5, 6, 7]
25.81/9.52	transitions: 
25.81/9.52	mark0(0) -> 0
25.81/9.52	00() -> 0
25.81/9.52	ok0(0) -> 0
25.81/9.52	active0(0) -> 0
25.81/9.52	f0(0) -> 1
25.81/9.52	cons0(0, 0) -> 2
25.81/9.52	g0(0) -> 3
25.81/9.52	s0(0) -> 4
25.81/9.52	sel0(0, 0) -> 5
25.81/9.52	proper0(0) -> 6
25.81/9.52	top0(0) -> 7
25.81/9.52	f1(0) -> 8
25.81/9.52	mark1(8) -> 1
25.81/9.52	cons1(0, 0) -> 9
25.81/9.52	mark1(9) -> 2
25.81/9.52	g1(0) -> 10
25.81/9.52	mark1(10) -> 3
25.81/9.52	s1(0) -> 11
25.81/9.52	mark1(11) -> 4
25.81/9.52	sel1(0, 0) -> 12
25.81/9.52	mark1(12) -> 5
25.81/9.52	01() -> 13
25.81/9.52	ok1(13) -> 6
25.81/9.52	f1(0) -> 14
25.81/9.52	ok1(14) -> 1
25.81/9.52	cons1(0, 0) -> 15
25.81/9.52	ok1(15) -> 2
25.81/9.52	g1(0) -> 16
25.81/9.52	ok1(16) -> 3
25.81/9.52	s1(0) -> 17
25.81/9.52	ok1(17) -> 4
25.81/9.52	sel1(0, 0) -> 18
25.81/9.52	ok1(18) -> 5
25.81/9.52	proper1(0) -> 19
25.81/9.52	top1(19) -> 7
25.81/9.52	active1(0) -> 20
25.81/9.52	top1(20) -> 7
25.81/9.52	mark1(8) -> 8
25.81/9.52	mark1(8) -> 14
25.81/9.52	mark1(9) -> 9
25.81/9.52	mark1(9) -> 15
25.81/9.52	mark1(10) -> 10
25.81/9.52	mark1(10) -> 16
25.81/9.52	mark1(11) -> 11
25.81/9.52	mark1(11) -> 17
25.81/9.52	mark1(12) -> 12
25.81/9.52	mark1(12) -> 18
25.81/9.52	ok1(13) -> 19
25.81/9.52	ok1(14) -> 8
25.81/9.52	ok1(14) -> 14
25.81/9.52	ok1(15) -> 9
25.81/9.52	ok1(15) -> 15
25.81/9.52	ok1(16) -> 10
25.81/9.52	ok1(16) -> 16
25.81/9.52	ok1(17) -> 11
25.81/9.52	ok1(17) -> 17
25.81/9.52	ok1(18) -> 12
25.81/9.52	ok1(18) -> 18
25.81/9.52	active2(13) -> 21
25.81/9.52	top2(21) -> 7
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(6)
25.81/9.52	BOUNDS(1, n^1)
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(7) RenamingProof (BOTH BOUNDS(ID, ID))
25.81/9.52	Renamed function symbols to avoid clashes with predefined symbol.
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(8)
25.81/9.52	Obligation:
25.81/9.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
25.81/9.52	
25.81/9.52	
25.81/9.52	The TRS R consists of the following rules:
25.81/9.52	
25.81/9.52	   active(f(X)) -> mark(cons(X, f(g(X))))
25.81/9.52	   active(g(0')) -> mark(s(0'))
25.81/9.52	   active(g(s(X))) -> mark(s(s(g(X))))
25.81/9.52	   active(sel(0', cons(X, Y))) -> mark(X)
25.81/9.52	   active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
25.81/9.52	   active(f(X)) -> f(active(X))
25.81/9.52	   active(cons(X1, X2)) -> cons(active(X1), X2)
25.81/9.52	   active(g(X)) -> g(active(X))
25.81/9.52	   active(s(X)) -> s(active(X))
25.81/9.52	   active(sel(X1, X2)) -> sel(active(X1), X2)
25.81/9.52	   active(sel(X1, X2)) -> sel(X1, active(X2))
25.81/9.52	   f(mark(X)) -> mark(f(X))
25.81/9.52	   cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	   g(mark(X)) -> mark(g(X))
25.81/9.52	   s(mark(X)) -> mark(s(X))
25.81/9.52	   sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	   sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	   proper(f(X)) -> f(proper(X))
25.81/9.52	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
25.81/9.52	   proper(g(X)) -> g(proper(X))
25.81/9.52	   proper(0') -> ok(0')
25.81/9.52	   proper(s(X)) -> s(proper(X))
25.81/9.52	   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
25.81/9.52	   f(ok(X)) -> ok(f(X))
25.81/9.52	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	   g(ok(X)) -> ok(g(X))
25.81/9.52	   s(ok(X)) -> ok(s(X))
25.81/9.52	   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	   top(mark(X)) -> top(proper(X))
25.81/9.52	   top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	S is empty.
25.81/9.52	Rewrite Strategy: FULL
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
25.81/9.52	Infered types.
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(10)
25.81/9.52	Obligation:
25.81/9.52	TRS:
25.81/9.52	Rules:
25.81/9.52	active(f(X)) -> mark(cons(X, f(g(X))))
25.81/9.52	active(g(0')) -> mark(s(0'))
25.81/9.52	active(g(s(X))) -> mark(s(s(g(X))))
25.81/9.52	active(sel(0', cons(X, Y))) -> mark(X)
25.81/9.52	active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
25.81/9.52	active(f(X)) -> f(active(X))
25.81/9.52	active(cons(X1, X2)) -> cons(active(X1), X2)
25.81/9.52	active(g(X)) -> g(active(X))
25.81/9.52	active(s(X)) -> s(active(X))
25.81/9.52	active(sel(X1, X2)) -> sel(active(X1), X2)
25.81/9.52	active(sel(X1, X2)) -> sel(X1, active(X2))
25.81/9.52	f(mark(X)) -> mark(f(X))
25.81/9.52	cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	g(mark(X)) -> mark(g(X))
25.81/9.52	s(mark(X)) -> mark(s(X))
25.81/9.52	sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	proper(f(X)) -> f(proper(X))
25.81/9.52	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
25.81/9.52	proper(g(X)) -> g(proper(X))
25.81/9.52	proper(0') -> ok(0')
25.81/9.52	proper(s(X)) -> s(proper(X))
25.81/9.52	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
25.81/9.52	f(ok(X)) -> ok(f(X))
25.81/9.52	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	g(ok(X)) -> ok(g(X))
25.81/9.52	s(ok(X)) -> ok(s(X))
25.81/9.52	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	top(mark(X)) -> top(proper(X))
25.81/9.52	top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	Types:
25.81/9.52	active :: mark:0':ok -> mark:0':ok
25.81/9.52	f :: mark:0':ok -> mark:0':ok
25.81/9.52	mark :: mark:0':ok -> mark:0':ok
25.81/9.52	cons :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	g :: mark:0':ok -> mark:0':ok
25.81/9.52	0' :: mark:0':ok
25.81/9.52	s :: mark:0':ok -> mark:0':ok
25.81/9.52	sel :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	proper :: mark:0':ok -> mark:0':ok
25.81/9.52	ok :: mark:0':ok -> mark:0':ok
25.81/9.52	top :: mark:0':ok -> top
25.81/9.52	hole_mark:0':ok1_0 :: mark:0':ok
25.81/9.52	hole_top2_0 :: top
25.81/9.52	gen_mark:0':ok3_0 :: Nat -> mark:0':ok
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(11) OrderProof (LOWER BOUND(ID))
25.81/9.52	Heuristically decided to analyse the following defined symbols:
25.81/9.52	active, cons, f, g, s, sel, proper, top
25.81/9.52	
25.81/9.52	They will be analysed ascendingly in the following order:
25.81/9.52	cons < active
25.81/9.52	f < active
25.81/9.52	g < active
25.81/9.52	s < active
25.81/9.52	sel < active
25.81/9.52	active < top
25.81/9.52	cons < proper
25.81/9.52	f < proper
25.81/9.52	g < proper
25.81/9.52	s < proper
25.81/9.52	sel < proper
25.81/9.52	proper < top
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(12)
25.81/9.52	Obligation:
25.81/9.52	TRS:
25.81/9.52	Rules:
25.81/9.52	active(f(X)) -> mark(cons(X, f(g(X))))
25.81/9.52	active(g(0')) -> mark(s(0'))
25.81/9.52	active(g(s(X))) -> mark(s(s(g(X))))
25.81/9.52	active(sel(0', cons(X, Y))) -> mark(X)
25.81/9.52	active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
25.81/9.52	active(f(X)) -> f(active(X))
25.81/9.52	active(cons(X1, X2)) -> cons(active(X1), X2)
25.81/9.52	active(g(X)) -> g(active(X))
25.81/9.52	active(s(X)) -> s(active(X))
25.81/9.52	active(sel(X1, X2)) -> sel(active(X1), X2)
25.81/9.52	active(sel(X1, X2)) -> sel(X1, active(X2))
25.81/9.52	f(mark(X)) -> mark(f(X))
25.81/9.52	cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	g(mark(X)) -> mark(g(X))
25.81/9.52	s(mark(X)) -> mark(s(X))
25.81/9.52	sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	proper(f(X)) -> f(proper(X))
25.81/9.52	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
25.81/9.52	proper(g(X)) -> g(proper(X))
25.81/9.52	proper(0') -> ok(0')
25.81/9.52	proper(s(X)) -> s(proper(X))
25.81/9.52	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
25.81/9.52	f(ok(X)) -> ok(f(X))
25.81/9.52	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	g(ok(X)) -> ok(g(X))
25.81/9.52	s(ok(X)) -> ok(s(X))
25.81/9.52	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	top(mark(X)) -> top(proper(X))
25.81/9.52	top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	Types:
25.81/9.52	active :: mark:0':ok -> mark:0':ok
25.81/9.52	f :: mark:0':ok -> mark:0':ok
25.81/9.52	mark :: mark:0':ok -> mark:0':ok
25.81/9.52	cons :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	g :: mark:0':ok -> mark:0':ok
25.81/9.52	0' :: mark:0':ok
25.81/9.52	s :: mark:0':ok -> mark:0':ok
25.81/9.52	sel :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	proper :: mark:0':ok -> mark:0':ok
25.81/9.52	ok :: mark:0':ok -> mark:0':ok
25.81/9.52	top :: mark:0':ok -> top
25.81/9.52	hole_mark:0':ok1_0 :: mark:0':ok
25.81/9.52	hole_top2_0 :: top
25.81/9.52	gen_mark:0':ok3_0 :: Nat -> mark:0':ok
25.81/9.52	
25.81/9.52	
25.81/9.52	Generator Equations:
25.81/9.52	gen_mark:0':ok3_0(0) <=> 0'
25.81/9.52	gen_mark:0':ok3_0(+(x, 1)) <=> mark(gen_mark:0':ok3_0(x))
25.81/9.52	
25.81/9.52	
25.81/9.52	The following defined symbols remain to be analysed:
25.81/9.52	cons, active, f, g, s, sel, proper, top
25.81/9.52	
25.81/9.52	They will be analysed ascendingly in the following order:
25.81/9.52	cons < active
25.81/9.52	f < active
25.81/9.52	g < active
25.81/9.52	s < active
25.81/9.52	sel < active
25.81/9.52	active < top
25.81/9.52	cons < proper
25.81/9.52	f < proper
25.81/9.52	g < proper
25.81/9.52	s < proper
25.81/9.52	sel < proper
25.81/9.52	proper < top
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(13) RewriteLemmaProof (LOWER BOUND(ID))
25.81/9.52	Proved the following rewrite lemma:
25.81/9.52	cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
25.81/9.52	
25.81/9.52	Induction Base:
25.81/9.52	cons(gen_mark:0':ok3_0(+(1, 0)), gen_mark:0':ok3_0(b))
25.81/9.52	
25.81/9.52	Induction Step:
25.81/9.52	cons(gen_mark:0':ok3_0(+(1, +(n5_0, 1))), gen_mark:0':ok3_0(b)) ->_R^Omega(1)
25.81/9.52	mark(cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b))) ->_IH
25.81/9.52	mark(*4_0)
25.81/9.52	
25.81/9.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(14)
25.81/9.52	Complex Obligation (BEST)
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(15)
25.81/9.52	Obligation:
25.81/9.52	Proved the lower bound n^1 for the following obligation:
25.81/9.52	
25.81/9.52	TRS:
25.81/9.52	Rules:
25.81/9.52	active(f(X)) -> mark(cons(X, f(g(X))))
25.81/9.52	active(g(0')) -> mark(s(0'))
25.81/9.52	active(g(s(X))) -> mark(s(s(g(X))))
25.81/9.52	active(sel(0', cons(X, Y))) -> mark(X)
25.81/9.52	active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
25.81/9.52	active(f(X)) -> f(active(X))
25.81/9.52	active(cons(X1, X2)) -> cons(active(X1), X2)
25.81/9.52	active(g(X)) -> g(active(X))
25.81/9.52	active(s(X)) -> s(active(X))
25.81/9.52	active(sel(X1, X2)) -> sel(active(X1), X2)
25.81/9.52	active(sel(X1, X2)) -> sel(X1, active(X2))
25.81/9.52	f(mark(X)) -> mark(f(X))
25.81/9.52	cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	g(mark(X)) -> mark(g(X))
25.81/9.52	s(mark(X)) -> mark(s(X))
25.81/9.52	sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	proper(f(X)) -> f(proper(X))
25.81/9.52	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
25.81/9.52	proper(g(X)) -> g(proper(X))
25.81/9.52	proper(0') -> ok(0')
25.81/9.52	proper(s(X)) -> s(proper(X))
25.81/9.52	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
25.81/9.52	f(ok(X)) -> ok(f(X))
25.81/9.52	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	g(ok(X)) -> ok(g(X))
25.81/9.52	s(ok(X)) -> ok(s(X))
25.81/9.52	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	top(mark(X)) -> top(proper(X))
25.81/9.52	top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	Types:
25.81/9.52	active :: mark:0':ok -> mark:0':ok
25.81/9.52	f :: mark:0':ok -> mark:0':ok
25.81/9.52	mark :: mark:0':ok -> mark:0':ok
25.81/9.52	cons :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	g :: mark:0':ok -> mark:0':ok
25.81/9.52	0' :: mark:0':ok
25.81/9.52	s :: mark:0':ok -> mark:0':ok
25.81/9.52	sel :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	proper :: mark:0':ok -> mark:0':ok
25.81/9.52	ok :: mark:0':ok -> mark:0':ok
25.81/9.52	top :: mark:0':ok -> top
25.81/9.52	hole_mark:0':ok1_0 :: mark:0':ok
25.81/9.52	hole_top2_0 :: top
25.81/9.52	gen_mark:0':ok3_0 :: Nat -> mark:0':ok
25.81/9.52	
25.81/9.52	
25.81/9.52	Generator Equations:
25.81/9.52	gen_mark:0':ok3_0(0) <=> 0'
25.81/9.52	gen_mark:0':ok3_0(+(x, 1)) <=> mark(gen_mark:0':ok3_0(x))
25.81/9.52	
25.81/9.52	
25.81/9.52	The following defined symbols remain to be analysed:
25.81/9.52	cons, active, f, g, s, sel, proper, top
25.81/9.52	
25.81/9.52	They will be analysed ascendingly in the following order:
25.81/9.52	cons < active
25.81/9.52	f < active
25.81/9.52	g < active
25.81/9.52	s < active
25.81/9.52	sel < active
25.81/9.52	active < top
25.81/9.52	cons < proper
25.81/9.52	f < proper
25.81/9.52	g < proper
25.81/9.52	s < proper
25.81/9.52	sel < proper
25.81/9.52	proper < top
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(16) LowerBoundPropagationProof (FINISHED)
25.81/9.52	Propagated lower bound.
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(17)
25.81/9.52	BOUNDS(n^1, INF)
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(18)
25.81/9.52	Obligation:
25.81/9.52	TRS:
25.81/9.52	Rules:
25.81/9.52	active(f(X)) -> mark(cons(X, f(g(X))))
25.81/9.52	active(g(0')) -> mark(s(0'))
25.81/9.52	active(g(s(X))) -> mark(s(s(g(X))))
25.81/9.52	active(sel(0', cons(X, Y))) -> mark(X)
25.81/9.52	active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
25.81/9.52	active(f(X)) -> f(active(X))
25.81/9.52	active(cons(X1, X2)) -> cons(active(X1), X2)
25.81/9.52	active(g(X)) -> g(active(X))
25.81/9.52	active(s(X)) -> s(active(X))
25.81/9.52	active(sel(X1, X2)) -> sel(active(X1), X2)
25.81/9.52	active(sel(X1, X2)) -> sel(X1, active(X2))
25.81/9.52	f(mark(X)) -> mark(f(X))
25.81/9.52	cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	g(mark(X)) -> mark(g(X))
25.81/9.52	s(mark(X)) -> mark(s(X))
25.81/9.52	sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	proper(f(X)) -> f(proper(X))
25.81/9.52	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
25.81/9.52	proper(g(X)) -> g(proper(X))
25.81/9.52	proper(0') -> ok(0')
25.81/9.52	proper(s(X)) -> s(proper(X))
25.81/9.52	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
25.81/9.52	f(ok(X)) -> ok(f(X))
25.81/9.52	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	g(ok(X)) -> ok(g(X))
25.81/9.52	s(ok(X)) -> ok(s(X))
25.81/9.52	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	top(mark(X)) -> top(proper(X))
25.81/9.52	top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	Types:
25.81/9.52	active :: mark:0':ok -> mark:0':ok
25.81/9.52	f :: mark:0':ok -> mark:0':ok
25.81/9.52	mark :: mark:0':ok -> mark:0':ok
25.81/9.52	cons :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	g :: mark:0':ok -> mark:0':ok
25.81/9.52	0' :: mark:0':ok
25.81/9.52	s :: mark:0':ok -> mark:0':ok
25.81/9.52	sel :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	proper :: mark:0':ok -> mark:0':ok
25.81/9.52	ok :: mark:0':ok -> mark:0':ok
25.81/9.52	top :: mark:0':ok -> top
25.81/9.52	hole_mark:0':ok1_0 :: mark:0':ok
25.81/9.52	hole_top2_0 :: top
25.81/9.52	gen_mark:0':ok3_0 :: Nat -> mark:0':ok
25.81/9.52	
25.81/9.52	
25.81/9.52	Lemmas:
25.81/9.52	cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
25.81/9.52	
25.81/9.52	
25.81/9.52	Generator Equations:
25.81/9.52	gen_mark:0':ok3_0(0) <=> 0'
25.81/9.52	gen_mark:0':ok3_0(+(x, 1)) <=> mark(gen_mark:0':ok3_0(x))
25.81/9.52	
25.81/9.52	
25.81/9.52	The following defined symbols remain to be analysed:
25.81/9.52	f, active, g, s, sel, proper, top
25.81/9.52	
25.81/9.52	They will be analysed ascendingly in the following order:
25.81/9.52	f < active
25.81/9.52	g < active
25.81/9.52	s < active
25.81/9.52	sel < active
25.81/9.52	active < top
25.81/9.52	f < proper
25.81/9.52	g < proper
25.81/9.52	s < proper
25.81/9.52	sel < proper
25.81/9.52	proper < top
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(19) RewriteLemmaProof (LOWER BOUND(ID))
25.81/9.52	Proved the following rewrite lemma:
25.81/9.52	f(gen_mark:0':ok3_0(+(1, n860_0))) -> *4_0, rt in Omega(n860_0)
25.81/9.52	
25.81/9.52	Induction Base:
25.81/9.52	f(gen_mark:0':ok3_0(+(1, 0)))
25.81/9.52	
25.81/9.52	Induction Step:
25.81/9.52	f(gen_mark:0':ok3_0(+(1, +(n860_0, 1)))) ->_R^Omega(1)
25.81/9.52	mark(f(gen_mark:0':ok3_0(+(1, n860_0)))) ->_IH
25.81/9.52	mark(*4_0)
25.81/9.52	
25.81/9.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(20)
25.81/9.52	Obligation:
25.81/9.52	TRS:
25.81/9.52	Rules:
25.81/9.52	active(f(X)) -> mark(cons(X, f(g(X))))
25.81/9.52	active(g(0')) -> mark(s(0'))
25.81/9.52	active(g(s(X))) -> mark(s(s(g(X))))
25.81/9.52	active(sel(0', cons(X, Y))) -> mark(X)
25.81/9.52	active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
25.81/9.52	active(f(X)) -> f(active(X))
25.81/9.52	active(cons(X1, X2)) -> cons(active(X1), X2)
25.81/9.52	active(g(X)) -> g(active(X))
25.81/9.52	active(s(X)) -> s(active(X))
25.81/9.52	active(sel(X1, X2)) -> sel(active(X1), X2)
25.81/9.52	active(sel(X1, X2)) -> sel(X1, active(X2))
25.81/9.52	f(mark(X)) -> mark(f(X))
25.81/9.52	cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	g(mark(X)) -> mark(g(X))
25.81/9.52	s(mark(X)) -> mark(s(X))
25.81/9.52	sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	proper(f(X)) -> f(proper(X))
25.81/9.52	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
25.81/9.52	proper(g(X)) -> g(proper(X))
25.81/9.52	proper(0') -> ok(0')
25.81/9.52	proper(s(X)) -> s(proper(X))
25.81/9.52	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
25.81/9.52	f(ok(X)) -> ok(f(X))
25.81/9.52	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	g(ok(X)) -> ok(g(X))
25.81/9.52	s(ok(X)) -> ok(s(X))
25.81/9.52	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	top(mark(X)) -> top(proper(X))
25.81/9.52	top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	Types:
25.81/9.52	active :: mark:0':ok -> mark:0':ok
25.81/9.52	f :: mark:0':ok -> mark:0':ok
25.81/9.52	mark :: mark:0':ok -> mark:0':ok
25.81/9.52	cons :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	g :: mark:0':ok -> mark:0':ok
25.81/9.52	0' :: mark:0':ok
25.81/9.52	s :: mark:0':ok -> mark:0':ok
25.81/9.52	sel :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	proper :: mark:0':ok -> mark:0':ok
25.81/9.52	ok :: mark:0':ok -> mark:0':ok
25.81/9.52	top :: mark:0':ok -> top
25.81/9.52	hole_mark:0':ok1_0 :: mark:0':ok
25.81/9.52	hole_top2_0 :: top
25.81/9.52	gen_mark:0':ok3_0 :: Nat -> mark:0':ok
25.81/9.52	
25.81/9.52	
25.81/9.52	Lemmas:
25.81/9.52	cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
25.81/9.52	f(gen_mark:0':ok3_0(+(1, n860_0))) -> *4_0, rt in Omega(n860_0)
25.81/9.52	
25.81/9.52	
25.81/9.52	Generator Equations:
25.81/9.52	gen_mark:0':ok3_0(0) <=> 0'
25.81/9.52	gen_mark:0':ok3_0(+(x, 1)) <=> mark(gen_mark:0':ok3_0(x))
25.81/9.52	
25.81/9.52	
25.81/9.52	The following defined symbols remain to be analysed:
25.81/9.52	g, active, s, sel, proper, top
25.81/9.52	
25.81/9.52	They will be analysed ascendingly in the following order:
25.81/9.52	g < active
25.81/9.52	s < active
25.81/9.52	sel < active
25.81/9.52	active < top
25.81/9.52	g < proper
25.81/9.52	s < proper
25.81/9.52	sel < proper
25.81/9.52	proper < top
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(21) RewriteLemmaProof (LOWER BOUND(ID))
25.81/9.52	Proved the following rewrite lemma:
25.81/9.52	g(gen_mark:0':ok3_0(+(1, n1367_0))) -> *4_0, rt in Omega(n1367_0)
25.81/9.52	
25.81/9.52	Induction Base:
25.81/9.52	g(gen_mark:0':ok3_0(+(1, 0)))
25.81/9.52	
25.81/9.52	Induction Step:
25.81/9.52	g(gen_mark:0':ok3_0(+(1, +(n1367_0, 1)))) ->_R^Omega(1)
25.81/9.52	mark(g(gen_mark:0':ok3_0(+(1, n1367_0)))) ->_IH
25.81/9.52	mark(*4_0)
25.81/9.52	
25.81/9.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(22)
25.81/9.52	Obligation:
25.81/9.52	TRS:
25.81/9.52	Rules:
25.81/9.52	active(f(X)) -> mark(cons(X, f(g(X))))
25.81/9.52	active(g(0')) -> mark(s(0'))
25.81/9.52	active(g(s(X))) -> mark(s(s(g(X))))
25.81/9.52	active(sel(0', cons(X, Y))) -> mark(X)
25.81/9.52	active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
25.81/9.52	active(f(X)) -> f(active(X))
25.81/9.52	active(cons(X1, X2)) -> cons(active(X1), X2)
25.81/9.52	active(g(X)) -> g(active(X))
25.81/9.52	active(s(X)) -> s(active(X))
25.81/9.52	active(sel(X1, X2)) -> sel(active(X1), X2)
25.81/9.52	active(sel(X1, X2)) -> sel(X1, active(X2))
25.81/9.52	f(mark(X)) -> mark(f(X))
25.81/9.52	cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	g(mark(X)) -> mark(g(X))
25.81/9.52	s(mark(X)) -> mark(s(X))
25.81/9.52	sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	proper(f(X)) -> f(proper(X))
25.81/9.52	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
25.81/9.52	proper(g(X)) -> g(proper(X))
25.81/9.52	proper(0') -> ok(0')
25.81/9.52	proper(s(X)) -> s(proper(X))
25.81/9.52	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
25.81/9.52	f(ok(X)) -> ok(f(X))
25.81/9.52	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	g(ok(X)) -> ok(g(X))
25.81/9.52	s(ok(X)) -> ok(s(X))
25.81/9.52	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	top(mark(X)) -> top(proper(X))
25.81/9.52	top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	Types:
25.81/9.52	active :: mark:0':ok -> mark:0':ok
25.81/9.52	f :: mark:0':ok -> mark:0':ok
25.81/9.52	mark :: mark:0':ok -> mark:0':ok
25.81/9.52	cons :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	g :: mark:0':ok -> mark:0':ok
25.81/9.52	0' :: mark:0':ok
25.81/9.52	s :: mark:0':ok -> mark:0':ok
25.81/9.52	sel :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	proper :: mark:0':ok -> mark:0':ok
25.81/9.52	ok :: mark:0':ok -> mark:0':ok
25.81/9.52	top :: mark:0':ok -> top
25.81/9.52	hole_mark:0':ok1_0 :: mark:0':ok
25.81/9.52	hole_top2_0 :: top
25.81/9.52	gen_mark:0':ok3_0 :: Nat -> mark:0':ok
25.81/9.52	
25.81/9.52	
25.81/9.52	Lemmas:
25.81/9.52	cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
25.81/9.52	f(gen_mark:0':ok3_0(+(1, n860_0))) -> *4_0, rt in Omega(n860_0)
25.81/9.52	g(gen_mark:0':ok3_0(+(1, n1367_0))) -> *4_0, rt in Omega(n1367_0)
25.81/9.52	
25.81/9.52	
25.81/9.52	Generator Equations:
25.81/9.52	gen_mark:0':ok3_0(0) <=> 0'
25.81/9.52	gen_mark:0':ok3_0(+(x, 1)) <=> mark(gen_mark:0':ok3_0(x))
25.81/9.52	
25.81/9.52	
25.81/9.52	The following defined symbols remain to be analysed:
25.81/9.52	s, active, sel, proper, top
25.81/9.52	
25.81/9.52	They will be analysed ascendingly in the following order:
25.81/9.52	s < active
25.81/9.52	sel < active
25.81/9.52	active < top
25.81/9.52	s < proper
25.81/9.52	sel < proper
25.81/9.52	proper < top
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(23) RewriteLemmaProof (LOWER BOUND(ID))
25.81/9.52	Proved the following rewrite lemma:
25.81/9.52	s(gen_mark:0':ok3_0(+(1, n1975_0))) -> *4_0, rt in Omega(n1975_0)
25.81/9.52	
25.81/9.52	Induction Base:
25.81/9.52	s(gen_mark:0':ok3_0(+(1, 0)))
25.81/9.52	
25.81/9.52	Induction Step:
25.81/9.52	s(gen_mark:0':ok3_0(+(1, +(n1975_0, 1)))) ->_R^Omega(1)
25.81/9.52	mark(s(gen_mark:0':ok3_0(+(1, n1975_0)))) ->_IH
25.81/9.52	mark(*4_0)
25.81/9.52	
25.81/9.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(24)
25.81/9.52	Obligation:
25.81/9.52	TRS:
25.81/9.52	Rules:
25.81/9.52	active(f(X)) -> mark(cons(X, f(g(X))))
25.81/9.52	active(g(0')) -> mark(s(0'))
25.81/9.52	active(g(s(X))) -> mark(s(s(g(X))))
25.81/9.52	active(sel(0', cons(X, Y))) -> mark(X)
25.81/9.52	active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
25.81/9.52	active(f(X)) -> f(active(X))
25.81/9.52	active(cons(X1, X2)) -> cons(active(X1), X2)
25.81/9.52	active(g(X)) -> g(active(X))
25.81/9.52	active(s(X)) -> s(active(X))
25.81/9.52	active(sel(X1, X2)) -> sel(active(X1), X2)
25.81/9.52	active(sel(X1, X2)) -> sel(X1, active(X2))
25.81/9.52	f(mark(X)) -> mark(f(X))
25.81/9.52	cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	g(mark(X)) -> mark(g(X))
25.81/9.52	s(mark(X)) -> mark(s(X))
25.81/9.52	sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	proper(f(X)) -> f(proper(X))
25.81/9.52	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
25.81/9.52	proper(g(X)) -> g(proper(X))
25.81/9.52	proper(0') -> ok(0')
25.81/9.52	proper(s(X)) -> s(proper(X))
25.81/9.52	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
25.81/9.52	f(ok(X)) -> ok(f(X))
25.81/9.52	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	g(ok(X)) -> ok(g(X))
25.81/9.52	s(ok(X)) -> ok(s(X))
25.81/9.52	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	top(mark(X)) -> top(proper(X))
25.81/9.52	top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	Types:
25.81/9.52	active :: mark:0':ok -> mark:0':ok
25.81/9.52	f :: mark:0':ok -> mark:0':ok
25.81/9.52	mark :: mark:0':ok -> mark:0':ok
25.81/9.52	cons :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	g :: mark:0':ok -> mark:0':ok
25.81/9.52	0' :: mark:0':ok
25.81/9.52	s :: mark:0':ok -> mark:0':ok
25.81/9.52	sel :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	proper :: mark:0':ok -> mark:0':ok
25.81/9.52	ok :: mark:0':ok -> mark:0':ok
25.81/9.52	top :: mark:0':ok -> top
25.81/9.52	hole_mark:0':ok1_0 :: mark:0':ok
25.81/9.52	hole_top2_0 :: top
25.81/9.52	gen_mark:0':ok3_0 :: Nat -> mark:0':ok
25.81/9.52	
25.81/9.52	
25.81/9.52	Lemmas:
25.81/9.52	cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
25.81/9.52	f(gen_mark:0':ok3_0(+(1, n860_0))) -> *4_0, rt in Omega(n860_0)
25.81/9.52	g(gen_mark:0':ok3_0(+(1, n1367_0))) -> *4_0, rt in Omega(n1367_0)
25.81/9.52	s(gen_mark:0':ok3_0(+(1, n1975_0))) -> *4_0, rt in Omega(n1975_0)
25.81/9.52	
25.81/9.52	
25.81/9.52	Generator Equations:
25.81/9.52	gen_mark:0':ok3_0(0) <=> 0'
25.81/9.52	gen_mark:0':ok3_0(+(x, 1)) <=> mark(gen_mark:0':ok3_0(x))
25.81/9.52	
25.81/9.52	
25.81/9.52	The following defined symbols remain to be analysed:
25.81/9.52	sel, active, proper, top
25.81/9.52	
25.81/9.52	They will be analysed ascendingly in the following order:
25.81/9.52	sel < active
25.81/9.52	active < top
25.81/9.52	sel < proper
25.81/9.52	proper < top
25.81/9.52	
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(25) RewriteLemmaProof (LOWER BOUND(ID))
25.81/9.52	Proved the following rewrite lemma:
25.81/9.52	sel(gen_mark:0':ok3_0(+(1, n2684_0)), gen_mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2684_0)
25.81/9.52	
25.81/9.52	Induction Base:
25.81/9.52	sel(gen_mark:0':ok3_0(+(1, 0)), gen_mark:0':ok3_0(b))
25.81/9.52	
25.81/9.52	Induction Step:
25.81/9.52	sel(gen_mark:0':ok3_0(+(1, +(n2684_0, 1))), gen_mark:0':ok3_0(b)) ->_R^Omega(1)
25.81/9.52	mark(sel(gen_mark:0':ok3_0(+(1, n2684_0)), gen_mark:0':ok3_0(b))) ->_IH
25.81/9.52	mark(*4_0)
25.81/9.52	
25.81/9.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
25.81/9.52	----------------------------------------
25.81/9.52	
25.81/9.52	(26)
25.81/9.52	Obligation:
25.81/9.52	TRS:
25.81/9.52	Rules:
25.81/9.52	active(f(X)) -> mark(cons(X, f(g(X))))
25.81/9.52	active(g(0')) -> mark(s(0'))
25.81/9.52	active(g(s(X))) -> mark(s(s(g(X))))
25.81/9.52	active(sel(0', cons(X, Y))) -> mark(X)
25.81/9.52	active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
25.81/9.52	active(f(X)) -> f(active(X))
25.81/9.52	active(cons(X1, X2)) -> cons(active(X1), X2)
25.81/9.52	active(g(X)) -> g(active(X))
25.81/9.52	active(s(X)) -> s(active(X))
25.81/9.52	active(sel(X1, X2)) -> sel(active(X1), X2)
25.81/9.52	active(sel(X1, X2)) -> sel(X1, active(X2))
25.81/9.52	f(mark(X)) -> mark(f(X))
25.81/9.52	cons(mark(X1), X2) -> mark(cons(X1, X2))
25.81/9.52	g(mark(X)) -> mark(g(X))
25.81/9.52	s(mark(X)) -> mark(s(X))
25.81/9.52	sel(mark(X1), X2) -> mark(sel(X1, X2))
25.81/9.52	sel(X1, mark(X2)) -> mark(sel(X1, X2))
25.81/9.52	proper(f(X)) -> f(proper(X))
25.81/9.52	proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
25.81/9.52	proper(g(X)) -> g(proper(X))
25.81/9.52	proper(0') -> ok(0')
25.81/9.52	proper(s(X)) -> s(proper(X))
25.81/9.52	proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
25.81/9.52	f(ok(X)) -> ok(f(X))
25.81/9.52	cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
25.81/9.52	g(ok(X)) -> ok(g(X))
25.81/9.52	s(ok(X)) -> ok(s(X))
25.81/9.52	sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
25.81/9.52	top(mark(X)) -> top(proper(X))
25.81/9.52	top(ok(X)) -> top(active(X))
25.81/9.52	
25.81/9.52	Types:
25.81/9.52	active :: mark:0':ok -> mark:0':ok
25.81/9.52	f :: mark:0':ok -> mark:0':ok
25.81/9.52	mark :: mark:0':ok -> mark:0':ok
25.81/9.52	cons :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	g :: mark:0':ok -> mark:0':ok
25.81/9.52	0' :: mark:0':ok
25.81/9.52	s :: mark:0':ok -> mark:0':ok
25.81/9.52	sel :: mark:0':ok -> mark:0':ok -> mark:0':ok
25.81/9.52	proper :: mark:0':ok -> mark:0':ok
25.81/9.52	ok :: mark:0':ok -> mark:0':ok
25.81/9.52	top :: mark:0':ok -> top
25.81/9.52	hole_mark:0':ok1_0 :: mark:0':ok
25.81/9.52	hole_top2_0 :: top
25.81/9.52	gen_mark:0':ok3_0 :: Nat -> mark:0':ok
25.81/9.52	
25.81/9.52	
25.81/9.52	Lemmas:
25.81/9.52	cons(gen_mark:0':ok3_0(+(1, n5_0)), gen_mark:0':ok3_0(b)) -> *4_0, rt in Omega(n5_0)
25.81/9.52	f(gen_mark:0':ok3_0(+(1, n860_0))) -> *4_0, rt in Omega(n860_0)
25.81/9.52	g(gen_mark:0':ok3_0(+(1, n1367_0))) -> *4_0, rt in Omega(n1367_0)
25.81/9.52	s(gen_mark:0':ok3_0(+(1, n1975_0))) -> *4_0, rt in Omega(n1975_0)
25.81/9.52	sel(gen_mark:0':ok3_0(+(1, n2684_0)), gen_mark:0':ok3_0(b)) -> *4_0, rt in Omega(n2684_0)
25.81/9.52	
25.81/9.52	
25.81/9.52	Generator Equations:
25.81/9.52	gen_mark:0':ok3_0(0) <=> 0'
25.81/9.52	gen_mark:0':ok3_0(+(x, 1)) <=> mark(gen_mark:0':ok3_0(x))
25.81/9.52	
25.81/9.52	
25.81/9.52	The following defined symbols remain to be analysed:
25.81/9.52	active, proper, top
25.81/9.52	
25.81/9.52	They will be analysed ascendingly in the following order:
25.81/9.52	active < top
25.81/9.52	proper < top
26.03/9.56	EOF