1119.38/291.50	WORST_CASE(Omega(n^2), ?)
1119.38/291.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1119.38/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1119.38/291.51	
1119.38/291.51	(0) CpxTRS
1119.38/291.51	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1119.38/291.51	(2) CpxTRS
1119.38/291.51	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1119.38/291.51	(4) typed CpxTrs
1119.38/291.51	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1119.38/291.51	(6) typed CpxTrs
1119.38/291.51	(7) RewriteLemmaProof [LOWER BOUND(ID), 274 ms]
1119.38/291.51	(8) BEST
1119.38/291.51	    (9) proven lower bound
1119.38/291.51	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1119.38/291.51	        (11) BOUNDS(n^1, INF)
1119.38/291.51	    (12) typed CpxTrs
1119.38/291.51	        (13) RewriteLemmaProof [LOWER BOUND(ID), 98 ms]
1119.38/291.51	        (14) BEST
1119.38/291.51	            (15) proven lower bound
1119.38/291.51	                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
1119.38/291.51	                (17) BOUNDS(n^2, INF)
1119.38/291.51	            (18) typed CpxTrs
1119.38/291.51	                (19) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
1119.38/291.51	                (20) typed CpxTrs
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(0)
1119.38/291.51	Obligation:
1119.38/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	The TRS R consists of the following rules:
1119.38/291.51	
1119.38/291.51	   a__fib(N) -> a__sel(mark(N), a__fib1(s(0), s(0)))
1119.38/291.51	   a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
1119.38/291.51	   a__add(0, X) -> mark(X)
1119.38/291.51	   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
1119.38/291.51	   a__sel(0, cons(X, XS)) -> mark(X)
1119.38/291.51	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
1119.38/291.51	   mark(fib(X)) -> a__fib(mark(X))
1119.38/291.51	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1119.38/291.51	   mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
1119.38/291.51	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1119.38/291.51	   mark(s(X)) -> s(mark(X))
1119.38/291.51	   mark(0) -> 0
1119.38/291.51	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1119.38/291.51	   a__fib(X) -> fib(X)
1119.38/291.51	   a__sel(X1, X2) -> sel(X1, X2)
1119.38/291.51	   a__fib1(X1, X2) -> fib1(X1, X2)
1119.38/291.51	   a__add(X1, X2) -> add(X1, X2)
1119.38/291.51	
1119.38/291.51	S is empty.
1119.38/291.51	Rewrite Strategy: INNERMOST
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1119.38/291.51	Renamed function symbols to avoid clashes with predefined symbol.
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(2)
1119.38/291.51	Obligation:
1119.38/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	The TRS R consists of the following rules:
1119.38/291.51	
1119.38/291.51	   a__fib(N) -> a__sel(mark(N), a__fib1(s(0'), s(0')))
1119.38/291.51	   a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
1119.38/291.51	   a__add(0', X) -> mark(X)
1119.38/291.51	   a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
1119.38/291.51	   a__sel(0', cons(X, XS)) -> mark(X)
1119.38/291.51	   a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
1119.38/291.51	   mark(fib(X)) -> a__fib(mark(X))
1119.38/291.51	   mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1119.38/291.51	   mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
1119.38/291.51	   mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1119.38/291.51	   mark(s(X)) -> s(mark(X))
1119.38/291.51	   mark(0') -> 0'
1119.38/291.51	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1119.38/291.51	   a__fib(X) -> fib(X)
1119.38/291.51	   a__sel(X1, X2) -> sel(X1, X2)
1119.38/291.51	   a__fib1(X1, X2) -> fib1(X1, X2)
1119.38/291.51	   a__add(X1, X2) -> add(X1, X2)
1119.38/291.51	
1119.38/291.51	S is empty.
1119.38/291.51	Rewrite Strategy: INNERMOST
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1119.38/291.51	Infered types.
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(4)
1119.38/291.51	Obligation:
1119.38/291.51	Innermost TRS:
1119.38/291.51	Rules:
1119.38/291.51	a__fib(N) -> a__sel(mark(N), a__fib1(s(0'), s(0')))
1119.38/291.51	a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
1119.38/291.51	a__add(0', X) -> mark(X)
1119.38/291.51	a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
1119.38/291.51	a__sel(0', cons(X, XS)) -> mark(X)
1119.38/291.51	a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
1119.38/291.51	mark(fib(X)) -> a__fib(mark(X))
1119.38/291.51	mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1119.38/291.51	mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
1119.38/291.51	mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1119.38/291.51	mark(s(X)) -> s(mark(X))
1119.38/291.51	mark(0') -> 0'
1119.38/291.51	mark(cons(X1, X2)) -> cons(mark(X1), X2)
1119.38/291.51	a__fib(X) -> fib(X)
1119.38/291.51	a__sel(X1, X2) -> sel(X1, X2)
1119.38/291.51	a__fib1(X1, X2) -> fib1(X1, X2)
1119.38/291.51	a__add(X1, X2) -> add(X1, X2)
1119.38/291.51	
1119.38/291.51	Types:
1119.38/291.51	a__fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	mark :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	s :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	0' :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	cons :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	hole_0':s:add:fib1:cons:fib:sel1_0 :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0 :: Nat -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(5) OrderProof (LOWER BOUND(ID))
1119.38/291.51	Heuristically decided to analyse the following defined symbols:
1119.38/291.51	a__fib, a__sel, mark, a__fib1, a__add
1119.38/291.51	
1119.38/291.51	They will be analysed ascendingly in the following order:
1119.38/291.51	a__fib = a__sel
1119.38/291.51	a__fib = mark
1119.38/291.51	a__fib = a__fib1
1119.38/291.51	a__fib = a__add
1119.38/291.51	a__sel = mark
1119.38/291.51	a__sel = a__fib1
1119.38/291.51	a__sel = a__add
1119.38/291.51	mark = a__fib1
1119.38/291.51	mark = a__add
1119.38/291.51	a__fib1 = a__add
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(6)
1119.38/291.51	Obligation:
1119.38/291.51	Innermost TRS:
1119.38/291.51	Rules:
1119.38/291.51	a__fib(N) -> a__sel(mark(N), a__fib1(s(0'), s(0')))
1119.38/291.51	a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
1119.38/291.51	a__add(0', X) -> mark(X)
1119.38/291.51	a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
1119.38/291.51	a__sel(0', cons(X, XS)) -> mark(X)
1119.38/291.51	a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
1119.38/291.51	mark(fib(X)) -> a__fib(mark(X))
1119.38/291.51	mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1119.38/291.51	mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
1119.38/291.51	mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1119.38/291.51	mark(s(X)) -> s(mark(X))
1119.38/291.51	mark(0') -> 0'
1119.38/291.51	mark(cons(X1, X2)) -> cons(mark(X1), X2)
1119.38/291.51	a__fib(X) -> fib(X)
1119.38/291.51	a__sel(X1, X2) -> sel(X1, X2)
1119.38/291.51	a__fib1(X1, X2) -> fib1(X1, X2)
1119.38/291.51	a__add(X1, X2) -> add(X1, X2)
1119.38/291.51	
1119.38/291.51	Types:
1119.38/291.51	a__fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	mark :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	s :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	0' :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	cons :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	hole_0':s:add:fib1:cons:fib:sel1_0 :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0 :: Nat -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	Generator Equations:
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(0) <=> 0'
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(+(x, 1)) <=> s(gen_0':s:add:fib1:cons:fib:sel2_0(x))
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	The following defined symbols remain to be analysed:
1119.38/291.51	a__sel, a__fib, mark, a__fib1, a__add
1119.38/291.51	
1119.38/291.51	They will be analysed ascendingly in the following order:
1119.38/291.51	a__fib = a__sel
1119.38/291.51	a__fib = mark
1119.38/291.51	a__fib = a__fib1
1119.38/291.51	a__fib = a__add
1119.38/291.51	a__sel = mark
1119.38/291.51	a__sel = a__fib1
1119.38/291.51	a__sel = a__add
1119.38/291.51	mark = a__fib1
1119.38/291.51	mark = a__add
1119.38/291.51	a__fib1 = a__add
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(7) RewriteLemmaProof (LOWER BOUND(ID))
1119.38/291.51	Proved the following rewrite lemma:
1119.38/291.51	mark(gen_0':s:add:fib1:cons:fib:sel2_0(n26_0)) -> gen_0':s:add:fib1:cons:fib:sel2_0(n26_0), rt in Omega(1 + n26_0)
1119.38/291.51	
1119.38/291.51	Induction Base:
1119.38/291.51	mark(gen_0':s:add:fib1:cons:fib:sel2_0(0)) ->_R^Omega(1)
1119.38/291.51	0'
1119.38/291.51	
1119.38/291.51	Induction Step:
1119.38/291.51	mark(gen_0':s:add:fib1:cons:fib:sel2_0(+(n26_0, 1))) ->_R^Omega(1)
1119.38/291.51	s(mark(gen_0':s:add:fib1:cons:fib:sel2_0(n26_0))) ->_IH
1119.38/291.51	s(gen_0':s:add:fib1:cons:fib:sel2_0(c27_0))
1119.38/291.51	
1119.38/291.51	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(8)
1119.38/291.51	Complex Obligation (BEST)
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(9)
1119.38/291.51	Obligation:
1119.38/291.51	Proved the lower bound n^1 for the following obligation:
1119.38/291.51	
1119.38/291.51	Innermost TRS:
1119.38/291.51	Rules:
1119.38/291.51	a__fib(N) -> a__sel(mark(N), a__fib1(s(0'), s(0')))
1119.38/291.51	a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
1119.38/291.51	a__add(0', X) -> mark(X)
1119.38/291.51	a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
1119.38/291.51	a__sel(0', cons(X, XS)) -> mark(X)
1119.38/291.51	a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
1119.38/291.51	mark(fib(X)) -> a__fib(mark(X))
1119.38/291.51	mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1119.38/291.51	mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
1119.38/291.51	mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1119.38/291.51	mark(s(X)) -> s(mark(X))
1119.38/291.51	mark(0') -> 0'
1119.38/291.51	mark(cons(X1, X2)) -> cons(mark(X1), X2)
1119.38/291.51	a__fib(X) -> fib(X)
1119.38/291.51	a__sel(X1, X2) -> sel(X1, X2)
1119.38/291.51	a__fib1(X1, X2) -> fib1(X1, X2)
1119.38/291.51	a__add(X1, X2) -> add(X1, X2)
1119.38/291.51	
1119.38/291.51	Types:
1119.38/291.51	a__fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	mark :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	s :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	0' :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	cons :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	hole_0':s:add:fib1:cons:fib:sel1_0 :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0 :: Nat -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	Generator Equations:
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(0) <=> 0'
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(+(x, 1)) <=> s(gen_0':s:add:fib1:cons:fib:sel2_0(x))
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	The following defined symbols remain to be analysed:
1119.38/291.51	mark, a__fib, a__fib1, a__add
1119.38/291.51	
1119.38/291.51	They will be analysed ascendingly in the following order:
1119.38/291.51	a__fib = a__sel
1119.38/291.51	a__fib = mark
1119.38/291.51	a__fib = a__fib1
1119.38/291.51	a__fib = a__add
1119.38/291.51	a__sel = mark
1119.38/291.51	a__sel = a__fib1
1119.38/291.51	a__sel = a__add
1119.38/291.51	mark = a__fib1
1119.38/291.51	mark = a__add
1119.38/291.51	a__fib1 = a__add
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(10) LowerBoundPropagationProof (FINISHED)
1119.38/291.51	Propagated lower bound.
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(11)
1119.38/291.51	BOUNDS(n^1, INF)
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(12)
1119.38/291.51	Obligation:
1119.38/291.51	Innermost TRS:
1119.38/291.51	Rules:
1119.38/291.51	a__fib(N) -> a__sel(mark(N), a__fib1(s(0'), s(0')))
1119.38/291.51	a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
1119.38/291.51	a__add(0', X) -> mark(X)
1119.38/291.51	a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
1119.38/291.51	a__sel(0', cons(X, XS)) -> mark(X)
1119.38/291.51	a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
1119.38/291.51	mark(fib(X)) -> a__fib(mark(X))
1119.38/291.51	mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1119.38/291.51	mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
1119.38/291.51	mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1119.38/291.51	mark(s(X)) -> s(mark(X))
1119.38/291.51	mark(0') -> 0'
1119.38/291.51	mark(cons(X1, X2)) -> cons(mark(X1), X2)
1119.38/291.51	a__fib(X) -> fib(X)
1119.38/291.51	a__sel(X1, X2) -> sel(X1, X2)
1119.38/291.51	a__fib1(X1, X2) -> fib1(X1, X2)
1119.38/291.51	a__add(X1, X2) -> add(X1, X2)
1119.38/291.51	
1119.38/291.51	Types:
1119.38/291.51	a__fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	mark :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	s :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	0' :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	cons :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	hole_0':s:add:fib1:cons:fib:sel1_0 :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0 :: Nat -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	Lemmas:
1119.38/291.51	mark(gen_0':s:add:fib1:cons:fib:sel2_0(n26_0)) -> gen_0':s:add:fib1:cons:fib:sel2_0(n26_0), rt in Omega(1 + n26_0)
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	Generator Equations:
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(0) <=> 0'
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(+(x, 1)) <=> s(gen_0':s:add:fib1:cons:fib:sel2_0(x))
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	The following defined symbols remain to be analysed:
1119.38/291.51	a__fib, a__sel, a__fib1, a__add
1119.38/291.51	
1119.38/291.51	They will be analysed ascendingly in the following order:
1119.38/291.51	a__fib = a__sel
1119.38/291.51	a__fib = mark
1119.38/291.51	a__fib = a__fib1
1119.38/291.51	a__fib = a__add
1119.38/291.51	a__sel = mark
1119.38/291.51	a__sel = a__fib1
1119.38/291.51	a__sel = a__add
1119.38/291.51	mark = a__fib1
1119.38/291.51	mark = a__add
1119.38/291.51	a__fib1 = a__add
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(13) RewriteLemmaProof (LOWER BOUND(ID))
1119.38/291.51	Proved the following rewrite lemma:
1119.38/291.51	a__add(gen_0':s:add:fib1:cons:fib:sel2_0(n4941_0), gen_0':s:add:fib1:cons:fib:sel2_0(b)) -> gen_0':s:add:fib1:cons:fib:sel2_0(+(n4941_0, b)), rt in Omega(1 + b + b*n4941_0 + n4941_0 + n4941_0^2)
1119.38/291.51	
1119.38/291.51	Induction Base:
1119.38/291.51	a__add(gen_0':s:add:fib1:cons:fib:sel2_0(0), gen_0':s:add:fib1:cons:fib:sel2_0(b)) ->_R^Omega(1)
1119.38/291.51	mark(gen_0':s:add:fib1:cons:fib:sel2_0(b)) ->_L^Omega(1 + b)
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(b)
1119.38/291.51	
1119.38/291.51	Induction Step:
1119.38/291.51	a__add(gen_0':s:add:fib1:cons:fib:sel2_0(+(n4941_0, 1)), gen_0':s:add:fib1:cons:fib:sel2_0(b)) ->_R^Omega(1)
1119.38/291.51	s(a__add(mark(gen_0':s:add:fib1:cons:fib:sel2_0(n4941_0)), mark(gen_0':s:add:fib1:cons:fib:sel2_0(b)))) ->_L^Omega(1 + n4941_0)
1119.38/291.51	s(a__add(gen_0':s:add:fib1:cons:fib:sel2_0(n4941_0), mark(gen_0':s:add:fib1:cons:fib:sel2_0(b)))) ->_L^Omega(1 + b)
1119.38/291.51	s(a__add(gen_0':s:add:fib1:cons:fib:sel2_0(n4941_0), gen_0':s:add:fib1:cons:fib:sel2_0(b))) ->_IH
1119.38/291.51	s(gen_0':s:add:fib1:cons:fib:sel2_0(+(b, c4942_0)))
1119.38/291.51	
1119.38/291.51	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(14)
1119.38/291.51	Complex Obligation (BEST)
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(15)
1119.38/291.51	Obligation:
1119.38/291.51	Proved the lower bound n^2 for the following obligation:
1119.38/291.51	
1119.38/291.51	Innermost TRS:
1119.38/291.51	Rules:
1119.38/291.51	a__fib(N) -> a__sel(mark(N), a__fib1(s(0'), s(0')))
1119.38/291.51	a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
1119.38/291.51	a__add(0', X) -> mark(X)
1119.38/291.51	a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
1119.38/291.51	a__sel(0', cons(X, XS)) -> mark(X)
1119.38/291.51	a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
1119.38/291.51	mark(fib(X)) -> a__fib(mark(X))
1119.38/291.51	mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1119.38/291.51	mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
1119.38/291.51	mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1119.38/291.51	mark(s(X)) -> s(mark(X))
1119.38/291.51	mark(0') -> 0'
1119.38/291.51	mark(cons(X1, X2)) -> cons(mark(X1), X2)
1119.38/291.51	a__fib(X) -> fib(X)
1119.38/291.51	a__sel(X1, X2) -> sel(X1, X2)
1119.38/291.51	a__fib1(X1, X2) -> fib1(X1, X2)
1119.38/291.51	a__add(X1, X2) -> add(X1, X2)
1119.38/291.51	
1119.38/291.51	Types:
1119.38/291.51	a__fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	mark :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	s :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	0' :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	cons :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	hole_0':s:add:fib1:cons:fib:sel1_0 :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0 :: Nat -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	Lemmas:
1119.38/291.51	mark(gen_0':s:add:fib1:cons:fib:sel2_0(n26_0)) -> gen_0':s:add:fib1:cons:fib:sel2_0(n26_0), rt in Omega(1 + n26_0)
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	Generator Equations:
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(0) <=> 0'
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(+(x, 1)) <=> s(gen_0':s:add:fib1:cons:fib:sel2_0(x))
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	The following defined symbols remain to be analysed:
1119.38/291.51	a__add, a__sel
1119.38/291.51	
1119.38/291.51	They will be analysed ascendingly in the following order:
1119.38/291.51	a__fib = a__sel
1119.38/291.51	a__fib = mark
1119.38/291.51	a__fib = a__fib1
1119.38/291.51	a__fib = a__add
1119.38/291.51	a__sel = mark
1119.38/291.51	a__sel = a__fib1
1119.38/291.51	a__sel = a__add
1119.38/291.51	mark = a__fib1
1119.38/291.51	mark = a__add
1119.38/291.51	a__fib1 = a__add
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(16) LowerBoundPropagationProof (FINISHED)
1119.38/291.51	Propagated lower bound.
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(17)
1119.38/291.51	BOUNDS(n^2, INF)
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(18)
1119.38/291.51	Obligation:
1119.38/291.51	Innermost TRS:
1119.38/291.51	Rules:
1119.38/291.51	a__fib(N) -> a__sel(mark(N), a__fib1(s(0'), s(0')))
1119.38/291.51	a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
1119.38/291.51	a__add(0', X) -> mark(X)
1119.38/291.51	a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
1119.38/291.51	a__sel(0', cons(X, XS)) -> mark(X)
1119.38/291.51	a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
1119.38/291.51	mark(fib(X)) -> a__fib(mark(X))
1119.38/291.51	mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1119.38/291.51	mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
1119.38/291.51	mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1119.38/291.51	mark(s(X)) -> s(mark(X))
1119.38/291.51	mark(0') -> 0'
1119.38/291.51	mark(cons(X1, X2)) -> cons(mark(X1), X2)
1119.38/291.51	a__fib(X) -> fib(X)
1119.38/291.51	a__sel(X1, X2) -> sel(X1, X2)
1119.38/291.51	a__fib1(X1, X2) -> fib1(X1, X2)
1119.38/291.51	a__add(X1, X2) -> add(X1, X2)
1119.38/291.51	
1119.38/291.51	Types:
1119.38/291.51	a__fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	mark :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	s :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	0' :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	cons :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	hole_0':s:add:fib1:cons:fib:sel1_0 :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0 :: Nat -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	Lemmas:
1119.38/291.51	mark(gen_0':s:add:fib1:cons:fib:sel2_0(n26_0)) -> gen_0':s:add:fib1:cons:fib:sel2_0(n26_0), rt in Omega(1 + n26_0)
1119.38/291.51	a__add(gen_0':s:add:fib1:cons:fib:sel2_0(n4941_0), gen_0':s:add:fib1:cons:fib:sel2_0(b)) -> gen_0':s:add:fib1:cons:fib:sel2_0(+(n4941_0, b)), rt in Omega(1 + b + b*n4941_0 + n4941_0 + n4941_0^2)
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	Generator Equations:
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(0) <=> 0'
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(+(x, 1)) <=> s(gen_0':s:add:fib1:cons:fib:sel2_0(x))
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	The following defined symbols remain to be analysed:
1119.38/291.51	a__sel, a__fib, mark, a__fib1
1119.38/291.51	
1119.38/291.51	They will be analysed ascendingly in the following order:
1119.38/291.51	a__fib = a__sel
1119.38/291.51	a__fib = mark
1119.38/291.51	a__fib = a__fib1
1119.38/291.51	a__fib = a__add
1119.38/291.51	a__sel = mark
1119.38/291.51	a__sel = a__fib1
1119.38/291.51	a__sel = a__add
1119.38/291.51	mark = a__fib1
1119.38/291.51	mark = a__add
1119.38/291.51	a__fib1 = a__add
1119.38/291.51	
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(19) RewriteLemmaProof (LOWER BOUND(ID))
1119.38/291.51	Proved the following rewrite lemma:
1119.38/291.51	mark(gen_0':s:add:fib1:cons:fib:sel2_0(n6310_0)) -> gen_0':s:add:fib1:cons:fib:sel2_0(n6310_0), rt in Omega(1 + n6310_0)
1119.38/291.51	
1119.38/291.51	Induction Base:
1119.38/291.51	mark(gen_0':s:add:fib1:cons:fib:sel2_0(0)) ->_R^Omega(1)
1119.38/291.51	0'
1119.38/291.51	
1119.38/291.51	Induction Step:
1119.38/291.51	mark(gen_0':s:add:fib1:cons:fib:sel2_0(+(n6310_0, 1))) ->_R^Omega(1)
1119.38/291.51	s(mark(gen_0':s:add:fib1:cons:fib:sel2_0(n6310_0))) ->_IH
1119.38/291.51	s(gen_0':s:add:fib1:cons:fib:sel2_0(c6311_0))
1119.38/291.51	
1119.38/291.51	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1119.38/291.51	----------------------------------------
1119.38/291.51	
1119.38/291.51	(20)
1119.38/291.51	Obligation:
1119.38/291.51	Innermost TRS:
1119.38/291.51	Rules:
1119.38/291.51	a__fib(N) -> a__sel(mark(N), a__fib1(s(0'), s(0')))
1119.38/291.51	a__fib1(X, Y) -> cons(mark(X), fib1(Y, add(X, Y)))
1119.38/291.51	a__add(0', X) -> mark(X)
1119.38/291.51	a__add(s(X), Y) -> s(a__add(mark(X), mark(Y)))
1119.38/291.51	a__sel(0', cons(X, XS)) -> mark(X)
1119.38/291.51	a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS))
1119.38/291.51	mark(fib(X)) -> a__fib(mark(X))
1119.38/291.51	mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2))
1119.38/291.51	mark(fib1(X1, X2)) -> a__fib1(mark(X1), mark(X2))
1119.38/291.51	mark(add(X1, X2)) -> a__add(mark(X1), mark(X2))
1119.38/291.51	mark(s(X)) -> s(mark(X))
1119.38/291.51	mark(0') -> 0'
1119.38/291.51	mark(cons(X1, X2)) -> cons(mark(X1), X2)
1119.38/291.51	a__fib(X) -> fib(X)
1119.38/291.51	a__sel(X1, X2) -> sel(X1, X2)
1119.38/291.51	a__fib1(X1, X2) -> fib1(X1, X2)
1119.38/291.51	a__add(X1, X2) -> add(X1, X2)
1119.38/291.51	
1119.38/291.51	Types:
1119.38/291.51	a__fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	mark :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	s :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	0' :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	cons :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib1 :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	a__add :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	fib :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	sel :: 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	hole_0':s:add:fib1:cons:fib:sel1_0 :: 0':s:add:fib1:cons:fib:sel
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0 :: Nat -> 0':s:add:fib1:cons:fib:sel
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	Lemmas:
1119.38/291.51	mark(gen_0':s:add:fib1:cons:fib:sel2_0(n6310_0)) -> gen_0':s:add:fib1:cons:fib:sel2_0(n6310_0), rt in Omega(1 + n6310_0)
1119.38/291.51	a__add(gen_0':s:add:fib1:cons:fib:sel2_0(n4941_0), gen_0':s:add:fib1:cons:fib:sel2_0(b)) -> gen_0':s:add:fib1:cons:fib:sel2_0(+(n4941_0, b)), rt in Omega(1 + b + b*n4941_0 + n4941_0 + n4941_0^2)
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	Generator Equations:
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(0) <=> 0'
1119.38/291.51	gen_0':s:add:fib1:cons:fib:sel2_0(+(x, 1)) <=> s(gen_0':s:add:fib1:cons:fib:sel2_0(x))
1119.38/291.51	
1119.38/291.51	
1119.38/291.51	The following defined symbols remain to be analysed:
1119.38/291.51	a__fib, a__fib1
1119.38/291.51	
1119.38/291.51	They will be analysed ascendingly in the following order:
1119.38/291.51	a__fib = a__sel
1119.38/291.51	a__fib = mark
1119.38/291.51	a__fib = a__fib1
1119.38/291.51	a__fib = a__add
1119.38/291.51	a__sel = mark
1119.38/291.51	a__sel = a__fib1
1119.38/291.51	a__sel = a__add
1119.38/291.51	mark = a__fib1
1119.38/291.51	mark = a__add
1119.38/291.51	a__fib1 = a__add
1119.57/291.57	EOF