958.18/291.58	WORST_CASE(Omega(n^1), ?)
958.18/291.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
958.18/291.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
958.18/291.59	
958.18/291.59	
958.18/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
958.18/291.59	
958.18/291.59	(0) CpxTRS
958.18/291.59	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
958.18/291.59	(2) CpxTRS
958.18/291.59	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
958.18/291.59	(4) typed CpxTrs
958.18/291.59	(5) OrderProof [LOWER BOUND(ID), 0 ms]
958.18/291.59	(6) typed CpxTrs
958.18/291.59	(7) RewriteLemmaProof [LOWER BOUND(ID), 301 ms]
958.18/291.59	(8) BEST
958.18/291.59	    (9) proven lower bound
958.18/291.59	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
958.18/291.59	        (11) BOUNDS(n^1, INF)
958.18/291.59	    (12) typed CpxTrs
958.18/291.59	        (13) RewriteLemmaProof [LOWER BOUND(ID), 78 ms]
958.18/291.59	        (14) typed CpxTrs
958.18/291.59	
958.18/291.59	
958.18/291.59	----------------------------------------
958.18/291.59	
958.18/291.59	(0)
958.18/291.59	Obligation:
958.18/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
958.18/291.59	
958.18/291.59	
958.18/291.59	The TRS R consists of the following rules:
958.18/291.59	
958.18/291.59	   plus(0, y) -> y
958.18/291.59	   plus(s(x), y) -> s(plus(x, y))
958.18/291.59	   lt(0, s(y)) -> true
958.18/291.59	   lt(x, 0) -> false
958.18/291.59	   lt(s(x), s(y)) -> lt(x, y)
958.18/291.59	   fib(x) -> fibiter(x, 0, 0, s(0))
958.18/291.59	   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
958.18/291.59	   if(false, b, c, x, y) -> x
958.18/291.59	   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))
958.18/291.59	
958.18/291.59	S is empty.
958.18/291.59	Rewrite Strategy: INNERMOST
958.18/291.59	----------------------------------------
958.18/291.59	
958.18/291.59	(1) RenamingProof (BOTH BOUNDS(ID, ID))
958.18/291.59	Renamed function symbols to avoid clashes with predefined symbol.
958.18/291.59	----------------------------------------
958.18/291.59	
958.18/291.59	(2)
958.18/291.59	Obligation:
958.18/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
958.18/291.59	
958.18/291.59	
958.18/291.59	The TRS R consists of the following rules:
958.18/291.59	
958.18/291.59	   plus(0', y) -> y
958.18/291.59	   plus(s(x), y) -> s(plus(x, y))
958.18/291.59	   lt(0', s(y)) -> true
958.18/291.59	   lt(x, 0') -> false
958.18/291.59	   lt(s(x), s(y)) -> lt(x, y)
958.18/291.59	   fib(x) -> fibiter(x, 0', 0', s(0'))
958.18/291.59	   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
958.18/291.59	   if(false, b, c, x, y) -> x
958.18/291.59	   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))
958.18/291.59	
958.18/291.59	S is empty.
958.18/291.59	Rewrite Strategy: INNERMOST
958.18/291.59	----------------------------------------
958.18/291.59	
958.18/291.59	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
958.18/291.60	Infered types.
958.18/291.60	----------------------------------------
958.18/291.60	
958.18/291.60	(4)
958.18/291.60	Obligation:
958.18/291.60	Innermost TRS:
958.18/291.60	Rules:
958.18/291.60	plus(0', y) -> y
958.18/291.60	plus(s(x), y) -> s(plus(x, y))
958.18/291.60	lt(0', s(y)) -> true
958.18/291.60	lt(x, 0') -> false
958.18/291.60	lt(s(x), s(y)) -> lt(x, y)
958.18/291.60	fib(x) -> fibiter(x, 0', 0', s(0'))
958.18/291.60	fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
958.18/291.60	if(false, b, c, x, y) -> x
958.18/291.60	if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))
958.18/291.60	
958.18/291.60	Types:
958.18/291.60	plus :: 0':s -> 0':s -> 0':s
958.18/291.60	0' :: 0':s
958.18/291.60	s :: 0':s -> 0':s
958.18/291.60	lt :: 0':s -> 0':s -> true:false
958.18/291.60	true :: true:false
958.18/291.60	false :: true:false
958.18/291.60	fib :: 0':s -> 0':s
958.18/291.60	fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
958.18/291.60	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
958.18/291.60	hole_0':s1_0 :: 0':s
958.18/291.60	hole_true:false2_0 :: true:false
958.18/291.60	gen_0':s3_0 :: Nat -> 0':s
958.18/291.60	
958.18/291.60	----------------------------------------
958.18/291.60	
958.18/291.60	(5) OrderProof (LOWER BOUND(ID))
958.18/291.60	Heuristically decided to analyse the following defined symbols:
958.18/291.60	plus, lt, fibiter
958.18/291.60	
958.18/291.60	They will be analysed ascendingly in the following order:
958.18/291.60	plus < fibiter
958.18/291.60	lt < fibiter
958.18/291.60	
958.18/291.60	----------------------------------------
958.18/291.60	
958.18/291.60	(6)
958.18/291.60	Obligation:
958.18/291.60	Innermost TRS:
958.18/291.60	Rules:
958.18/291.60	plus(0', y) -> y
958.18/291.60	plus(s(x), y) -> s(plus(x, y))
958.18/291.60	lt(0', s(y)) -> true
958.18/291.60	lt(x, 0') -> false
958.18/291.60	lt(s(x), s(y)) -> lt(x, y)
958.18/291.60	fib(x) -> fibiter(x, 0', 0', s(0'))
958.18/291.60	fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
958.18/291.60	if(false, b, c, x, y) -> x
958.18/291.60	if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))
958.18/291.60	
958.18/291.60	Types:
958.18/291.60	plus :: 0':s -> 0':s -> 0':s
958.18/291.60	0' :: 0':s
958.18/291.60	s :: 0':s -> 0':s
958.18/291.60	lt :: 0':s -> 0':s -> true:false
958.18/291.60	true :: true:false
958.18/291.60	false :: true:false
958.18/291.60	fib :: 0':s -> 0':s
958.18/291.60	fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
958.18/291.60	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
958.18/291.60	hole_0':s1_0 :: 0':s
958.18/291.60	hole_true:false2_0 :: true:false
958.18/291.60	gen_0':s3_0 :: Nat -> 0':s
958.18/291.60	
958.18/291.60	
958.18/291.60	Generator Equations:
958.18/291.60	gen_0':s3_0(0) <=> 0'
958.18/291.60	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
958.18/291.60	
958.18/291.60	
958.18/291.60	The following defined symbols remain to be analysed:
958.18/291.60	plus, lt, fibiter
958.18/291.60	
958.18/291.60	They will be analysed ascendingly in the following order:
958.18/291.60	plus < fibiter
958.18/291.60	lt < fibiter
958.18/291.60	
958.18/291.60	----------------------------------------
958.18/291.60	
958.18/291.60	(7) RewriteLemmaProof (LOWER BOUND(ID))
958.18/291.60	Proved the following rewrite lemma:
958.18/291.60	plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
958.18/291.60	
958.18/291.60	Induction Base:
958.18/291.60	plus(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
958.18/291.60	gen_0':s3_0(b)
958.18/291.60	
958.18/291.60	Induction Step:
958.18/291.60	plus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
958.18/291.60	s(plus(gen_0':s3_0(n5_0), gen_0':s3_0(b))) ->_IH
958.18/291.60	s(gen_0':s3_0(+(b, c6_0)))
958.18/291.60	
958.18/291.60	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
958.18/291.60	----------------------------------------
958.18/291.60	
958.18/291.60	(8)
958.18/291.60	Complex Obligation (BEST)
958.18/291.60	
958.18/291.60	----------------------------------------
958.18/291.60	
958.18/291.60	(9)
958.18/291.60	Obligation:
958.18/291.60	Proved the lower bound n^1 for the following obligation:
958.18/291.60	
958.18/291.60	Innermost TRS:
958.18/291.60	Rules:
958.18/291.60	plus(0', y) -> y
958.18/291.60	plus(s(x), y) -> s(plus(x, y))
958.18/291.60	lt(0', s(y)) -> true
958.18/291.60	lt(x, 0') -> false
958.18/291.60	lt(s(x), s(y)) -> lt(x, y)
958.18/291.60	fib(x) -> fibiter(x, 0', 0', s(0'))
958.18/291.60	fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
958.18/291.60	if(false, b, c, x, y) -> x
958.18/291.60	if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))
958.18/291.60	
958.18/291.60	Types:
958.18/291.60	plus :: 0':s -> 0':s -> 0':s
958.18/291.60	0' :: 0':s
958.18/291.60	s :: 0':s -> 0':s
958.18/291.60	lt :: 0':s -> 0':s -> true:false
958.18/291.60	true :: true:false
958.18/291.60	false :: true:false
958.18/291.60	fib :: 0':s -> 0':s
958.18/291.60	fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
958.18/291.60	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
958.18/291.60	hole_0':s1_0 :: 0':s
958.18/291.60	hole_true:false2_0 :: true:false
958.18/291.60	gen_0':s3_0 :: Nat -> 0':s
958.18/291.60	
958.18/291.60	
958.18/291.60	Generator Equations:
958.18/291.60	gen_0':s3_0(0) <=> 0'
958.18/291.60	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
958.18/291.60	
958.18/291.60	
958.18/291.60	The following defined symbols remain to be analysed:
958.18/291.60	plus, lt, fibiter
958.18/291.60	
958.18/291.60	They will be analysed ascendingly in the following order:
958.18/291.60	plus < fibiter
958.18/291.60	lt < fibiter
958.18/291.60	
958.18/291.60	----------------------------------------
958.18/291.60	
958.18/291.60	(10) LowerBoundPropagationProof (FINISHED)
958.18/291.60	Propagated lower bound.
958.18/291.60	----------------------------------------
958.18/291.60	
958.18/291.60	(11)
958.18/291.60	BOUNDS(n^1, INF)
958.18/291.60	
958.18/291.60	----------------------------------------
958.18/291.60	
958.18/291.60	(12)
958.18/291.60	Obligation:
958.18/291.60	Innermost TRS:
958.18/291.60	Rules:
958.18/291.60	plus(0', y) -> y
958.18/291.60	plus(s(x), y) -> s(plus(x, y))
958.18/291.60	lt(0', s(y)) -> true
958.18/291.60	lt(x, 0') -> false
958.18/291.60	lt(s(x), s(y)) -> lt(x, y)
958.18/291.60	fib(x) -> fibiter(x, 0', 0', s(0'))
958.18/291.60	fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
958.18/291.60	if(false, b, c, x, y) -> x
958.18/291.60	if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))
958.18/291.60	
958.18/291.60	Types:
958.18/291.60	plus :: 0':s -> 0':s -> 0':s
958.18/291.60	0' :: 0':s
958.18/291.60	s :: 0':s -> 0':s
958.18/291.60	lt :: 0':s -> 0':s -> true:false
958.18/291.60	true :: true:false
958.18/291.60	false :: true:false
958.18/291.60	fib :: 0':s -> 0':s
958.18/291.60	fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
958.18/291.60	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
958.18/291.60	hole_0':s1_0 :: 0':s
958.18/291.60	hole_true:false2_0 :: true:false
958.18/291.60	gen_0':s3_0 :: Nat -> 0':s
958.18/291.60	
958.18/291.60	
958.18/291.60	Lemmas:
958.18/291.60	plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
958.18/291.60	
958.18/291.60	
958.18/291.60	Generator Equations:
958.18/291.60	gen_0':s3_0(0) <=> 0'
958.18/291.60	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
958.18/291.60	
958.18/291.60	
958.18/291.60	The following defined symbols remain to be analysed:
958.18/291.60	lt, fibiter
958.18/291.60	
958.18/291.60	They will be analysed ascendingly in the following order:
958.18/291.60	lt < fibiter
958.18/291.60	
958.18/291.60	----------------------------------------
958.18/291.60	
958.18/291.60	(13) RewriteLemmaProof (LOWER BOUND(ID))
958.18/291.60	Proved the following rewrite lemma:
958.18/291.60	lt(gen_0':s3_0(n542_0), gen_0':s3_0(+(1, n542_0))) -> true, rt in Omega(1 + n542_0)
958.18/291.60	
958.18/291.60	Induction Base:
958.18/291.60	lt(gen_0':s3_0(0), gen_0':s3_0(+(1, 0))) ->_R^Omega(1)
958.18/291.60	true
958.18/291.60	
958.18/291.60	Induction Step:
958.18/291.60	lt(gen_0':s3_0(+(n542_0, 1)), gen_0':s3_0(+(1, +(n542_0, 1)))) ->_R^Omega(1)
958.18/291.60	lt(gen_0':s3_0(n542_0), gen_0':s3_0(+(1, n542_0))) ->_IH
958.18/291.60	true
958.18/291.60	
958.18/291.60	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
958.18/291.60	----------------------------------------
958.18/291.60	
958.18/291.60	(14)
958.18/291.60	Obligation:
958.18/291.60	Innermost TRS:
958.18/291.60	Rules:
958.18/291.60	plus(0', y) -> y
958.18/291.60	plus(s(x), y) -> s(plus(x, y))
958.18/291.60	lt(0', s(y)) -> true
958.18/291.60	lt(x, 0') -> false
958.18/291.60	lt(s(x), s(y)) -> lt(x, y)
958.18/291.60	fib(x) -> fibiter(x, 0', 0', s(0'))
958.18/291.60	fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
958.18/291.60	if(false, b, c, x, y) -> x
958.18/291.60	if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))
958.18/291.60	
958.18/291.60	Types:
958.18/291.60	plus :: 0':s -> 0':s -> 0':s
958.18/291.60	0' :: 0':s
958.18/291.60	s :: 0':s -> 0':s
958.18/291.60	lt :: 0':s -> 0':s -> true:false
958.18/291.60	true :: true:false
958.18/291.60	false :: true:false
958.18/291.60	fib :: 0':s -> 0':s
958.18/291.60	fibiter :: 0':s -> 0':s -> 0':s -> 0':s -> 0':s
958.18/291.60	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s -> 0':s
958.18/291.60	hole_0':s1_0 :: 0':s
958.18/291.60	hole_true:false2_0 :: true:false
958.18/291.60	gen_0':s3_0 :: Nat -> 0':s
958.18/291.60	
958.18/291.60	
958.18/291.60	Lemmas:
958.18/291.60	plus(gen_0':s3_0(n5_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n5_0, b)), rt in Omega(1 + n5_0)
958.18/291.60	lt(gen_0':s3_0(n542_0), gen_0':s3_0(+(1, n542_0))) -> true, rt in Omega(1 + n542_0)
958.18/291.60	
958.18/291.60	
958.18/291.60	Generator Equations:
958.18/291.60	gen_0':s3_0(0) <=> 0'
958.18/291.60	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
958.18/291.60	
958.18/291.60	
958.18/291.60	The following defined symbols remain to be analysed:
958.18/291.60	fibiter
958.33/291.64	EOF