951.64/291.51	WORST_CASE(Omega(n^3), ?)
951.64/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
951.64/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
951.64/291.52	
951.64/291.52	
951.64/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).
951.64/291.52	
951.64/291.52	(0) CpxTRS
951.64/291.52	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
951.64/291.52	(2) CpxTRS
951.64/291.52	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
951.64/291.52	(4) typed CpxTrs
951.64/291.52	(5) OrderProof [LOWER BOUND(ID), 0 ms]
951.64/291.52	(6) typed CpxTrs
951.64/291.52	(7) RewriteLemmaProof [LOWER BOUND(ID), 295 ms]
951.64/291.52	(8) BEST
951.64/291.52	    (9) proven lower bound
951.64/291.52	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
951.64/291.52	        (11) BOUNDS(n^1, INF)
951.64/291.52	    (12) typed CpxTrs
951.64/291.52	        (13) RewriteLemmaProof [LOWER BOUND(ID), 32 ms]
951.64/291.52	        (14) BEST
951.64/291.52	            (15) proven lower bound
951.64/291.52	                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
951.64/291.52	                (17) BOUNDS(n^2, INF)
951.64/291.52	            (18) typed CpxTrs
951.64/291.52	                (19) RewriteLemmaProof [LOWER BOUND(ID), 728 ms]
951.64/291.52	                (20) typed CpxTrs
951.64/291.52	                (21) RewriteLemmaProof [LOWER BOUND(ID), 5854 ms]
951.64/291.52	                (22) proven lower bound
951.64/291.52	                (23) LowerBoundPropagationProof [FINISHED, 0 ms]
951.64/291.52	                (24) BOUNDS(n^3, INF)
951.64/291.52	
951.64/291.52	
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(0)
951.64/291.52	Obligation:
951.64/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).
951.64/291.52	
951.64/291.52	
951.64/291.52	The TRS R consists of the following rules:
951.64/291.52	
951.64/291.52	   fac(0) -> 1
951.64/291.52	   fac(s(x)) -> *(s(x), fac(x))
951.64/291.52	   floop(0, y) -> y
951.64/291.52	   floop(s(x), y) -> floop(x, *(s(x), y))
951.64/291.52	   *(x, 0) -> 0
951.64/291.52	   *(x, s(y)) -> +(*(x, y), x)
951.64/291.52	   +(x, 0) -> x
951.64/291.52	   +(x, s(y)) -> s(+(x, y))
951.64/291.52	   1 -> s(0)
951.64/291.52	   fac(0) -> s(0)
951.64/291.52	
951.64/291.52	S is empty.
951.64/291.52	Rewrite Strategy: FULL
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(1) RenamingProof (BOTH BOUNDS(ID, ID))
951.64/291.52	Renamed function symbols to avoid clashes with predefined symbol.
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(2)
951.64/291.52	Obligation:
951.64/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).
951.64/291.52	
951.64/291.52	
951.64/291.52	The TRS R consists of the following rules:
951.64/291.52	
951.64/291.52	   fac(0') -> 1'
951.64/291.52	   fac(s(x)) -> *'(s(x), fac(x))
951.64/291.52	   floop(0', y) -> y
951.64/291.52	   floop(s(x), y) -> floop(x, *'(s(x), y))
951.64/291.52	   *'(x, 0') -> 0'
951.64/291.52	   *'(x, s(y)) -> +'(*'(x, y), x)
951.64/291.52	   +'(x, 0') -> x
951.64/291.52	   +'(x, s(y)) -> s(+'(x, y))
951.64/291.52	   1' -> s(0')
951.64/291.52	   fac(0') -> s(0')
951.64/291.52	
951.64/291.52	S is empty.
951.64/291.52	Rewrite Strategy: FULL
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
951.64/291.52	Infered types.
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(4)
951.64/291.52	Obligation:
951.64/291.52	TRS:
951.64/291.52	Rules:
951.64/291.52	fac(0') -> 1'
951.64/291.52	fac(s(x)) -> *'(s(x), fac(x))
951.64/291.52	floop(0', y) -> y
951.64/291.52	floop(s(x), y) -> floop(x, *'(s(x), y))
951.64/291.52	*'(x, 0') -> 0'
951.64/291.52	*'(x, s(y)) -> +'(*'(x, y), x)
951.64/291.52	+'(x, 0') -> x
951.64/291.52	+'(x, s(y)) -> s(+'(x, y))
951.64/291.52	1' -> s(0')
951.64/291.52	fac(0') -> s(0')
951.64/291.52	
951.64/291.52	Types:
951.64/291.52	fac :: 0':s -> 0':s
951.64/291.52	0' :: 0':s
951.64/291.52	1' :: 0':s
951.64/291.52	s :: 0':s -> 0':s
951.64/291.52	*' :: 0':s -> 0':s -> 0':s
951.64/291.52	floop :: 0':s -> 0':s -> 0':s
951.64/291.52	+' :: 0':s -> 0':s -> 0':s
951.64/291.52	hole_0':s1_0 :: 0':s
951.64/291.52	gen_0':s2_0 :: Nat -> 0':s
951.64/291.52	
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(5) OrderProof (LOWER BOUND(ID))
951.64/291.52	Heuristically decided to analyse the following defined symbols:
951.64/291.52	fac, *', floop, +'
951.64/291.52	
951.64/291.52	They will be analysed ascendingly in the following order:
951.64/291.52	*' < fac
951.64/291.52	*' < floop
951.64/291.52	+' < *'
951.64/291.52	
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(6)
951.64/291.52	Obligation:
951.64/291.52	TRS:
951.64/291.52	Rules:
951.64/291.52	fac(0') -> 1'
951.64/291.52	fac(s(x)) -> *'(s(x), fac(x))
951.64/291.52	floop(0', y) -> y
951.64/291.52	floop(s(x), y) -> floop(x, *'(s(x), y))
951.64/291.52	*'(x, 0') -> 0'
951.64/291.52	*'(x, s(y)) -> +'(*'(x, y), x)
951.64/291.52	+'(x, 0') -> x
951.64/291.52	+'(x, s(y)) -> s(+'(x, y))
951.64/291.52	1' -> s(0')
951.64/291.52	fac(0') -> s(0')
951.64/291.52	
951.64/291.52	Types:
951.64/291.52	fac :: 0':s -> 0':s
951.64/291.52	0' :: 0':s
951.64/291.52	1' :: 0':s
951.64/291.52	s :: 0':s -> 0':s
951.64/291.52	*' :: 0':s -> 0':s -> 0':s
951.64/291.52	floop :: 0':s -> 0':s -> 0':s
951.64/291.52	+' :: 0':s -> 0':s -> 0':s
951.64/291.52	hole_0':s1_0 :: 0':s
951.64/291.52	gen_0':s2_0 :: Nat -> 0':s
951.64/291.52	
951.64/291.52	
951.64/291.52	Generator Equations:
951.64/291.52	gen_0':s2_0(0) <=> 0'
951.64/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
951.64/291.52	
951.64/291.52	
951.64/291.52	The following defined symbols remain to be analysed:
951.64/291.52	+', fac, *', floop
951.64/291.52	
951.64/291.52	They will be analysed ascendingly in the following order:
951.64/291.52	*' < fac
951.64/291.52	*' < floop
951.64/291.52	+' < *'
951.64/291.52	
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(7) RewriteLemmaProof (LOWER BOUND(ID))
951.64/291.52	Proved the following rewrite lemma:
951.64/291.52	+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
951.64/291.52	
951.64/291.52	Induction Base:
951.64/291.52	+'(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1)
951.64/291.52	gen_0':s2_0(a)
951.64/291.52	
951.64/291.52	Induction Step:
951.64/291.52	+'(gen_0':s2_0(a), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
951.64/291.52	s(+'(gen_0':s2_0(a), gen_0':s2_0(n4_0))) ->_IH
951.64/291.52	s(gen_0':s2_0(+(a, c5_0)))
951.64/291.52	
951.64/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(8)
951.64/291.52	Complex Obligation (BEST)
951.64/291.52	
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(9)
951.64/291.52	Obligation:
951.64/291.52	Proved the lower bound n^1 for the following obligation:
951.64/291.52	
951.64/291.52	TRS:
951.64/291.52	Rules:
951.64/291.52	fac(0') -> 1'
951.64/291.52	fac(s(x)) -> *'(s(x), fac(x))
951.64/291.52	floop(0', y) -> y
951.64/291.52	floop(s(x), y) -> floop(x, *'(s(x), y))
951.64/291.52	*'(x, 0') -> 0'
951.64/291.52	*'(x, s(y)) -> +'(*'(x, y), x)
951.64/291.52	+'(x, 0') -> x
951.64/291.52	+'(x, s(y)) -> s(+'(x, y))
951.64/291.52	1' -> s(0')
951.64/291.52	fac(0') -> s(0')
951.64/291.52	
951.64/291.52	Types:
951.64/291.52	fac :: 0':s -> 0':s
951.64/291.52	0' :: 0':s
951.64/291.52	1' :: 0':s
951.64/291.52	s :: 0':s -> 0':s
951.64/291.52	*' :: 0':s -> 0':s -> 0':s
951.64/291.52	floop :: 0':s -> 0':s -> 0':s
951.64/291.52	+' :: 0':s -> 0':s -> 0':s
951.64/291.52	hole_0':s1_0 :: 0':s
951.64/291.52	gen_0':s2_0 :: Nat -> 0':s
951.64/291.52	
951.64/291.52	
951.64/291.52	Generator Equations:
951.64/291.52	gen_0':s2_0(0) <=> 0'
951.64/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
951.64/291.52	
951.64/291.52	
951.64/291.52	The following defined symbols remain to be analysed:
951.64/291.52	+', fac, *', floop
951.64/291.52	
951.64/291.52	They will be analysed ascendingly in the following order:
951.64/291.52	*' < fac
951.64/291.52	*' < floop
951.64/291.52	+' < *'
951.64/291.52	
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(10) LowerBoundPropagationProof (FINISHED)
951.64/291.52	Propagated lower bound.
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(11)
951.64/291.52	BOUNDS(n^1, INF)
951.64/291.52	
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(12)
951.64/291.52	Obligation:
951.64/291.52	TRS:
951.64/291.52	Rules:
951.64/291.52	fac(0') -> 1'
951.64/291.52	fac(s(x)) -> *'(s(x), fac(x))
951.64/291.52	floop(0', y) -> y
951.64/291.52	floop(s(x), y) -> floop(x, *'(s(x), y))
951.64/291.52	*'(x, 0') -> 0'
951.64/291.52	*'(x, s(y)) -> +'(*'(x, y), x)
951.64/291.52	+'(x, 0') -> x
951.64/291.52	+'(x, s(y)) -> s(+'(x, y))
951.64/291.52	1' -> s(0')
951.64/291.52	fac(0') -> s(0')
951.64/291.52	
951.64/291.52	Types:
951.64/291.52	fac :: 0':s -> 0':s
951.64/291.52	0' :: 0':s
951.64/291.52	1' :: 0':s
951.64/291.52	s :: 0':s -> 0':s
951.64/291.52	*' :: 0':s -> 0':s -> 0':s
951.64/291.52	floop :: 0':s -> 0':s -> 0':s
951.64/291.52	+' :: 0':s -> 0':s -> 0':s
951.64/291.52	hole_0':s1_0 :: 0':s
951.64/291.52	gen_0':s2_0 :: Nat -> 0':s
951.64/291.52	
951.64/291.52	
951.64/291.52	Lemmas:
951.64/291.52	+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
951.64/291.52	
951.64/291.52	
951.64/291.52	Generator Equations:
951.64/291.52	gen_0':s2_0(0) <=> 0'
951.64/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
951.64/291.52	
951.64/291.52	
951.64/291.52	The following defined symbols remain to be analysed:
951.64/291.52	*', fac, floop
951.64/291.52	
951.64/291.52	They will be analysed ascendingly in the following order:
951.64/291.52	*' < fac
951.64/291.52	*' < floop
951.64/291.52	
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(13) RewriteLemmaProof (LOWER BOUND(ID))
951.64/291.52	Proved the following rewrite lemma:
951.64/291.52	*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) -> gen_0':s2_0(*(n511_0, a)), rt in Omega(1 + a*n511_0 + n511_0)
951.64/291.52	
951.64/291.52	Induction Base:
951.64/291.52	*'(gen_0':s2_0(a), gen_0':s2_0(0)) ->_R^Omega(1)
951.64/291.52	0'
951.64/291.52	
951.64/291.52	Induction Step:
951.64/291.52	*'(gen_0':s2_0(a), gen_0':s2_0(+(n511_0, 1))) ->_R^Omega(1)
951.64/291.52	+'(*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)), gen_0':s2_0(a)) ->_IH
951.64/291.52	+'(gen_0':s2_0(*(c512_0, a)), gen_0':s2_0(a)) ->_L^Omega(1 + a)
951.64/291.52	gen_0':s2_0(+(a, *(n511_0, a)))
951.64/291.52	
951.64/291.52	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(14)
951.64/291.52	Complex Obligation (BEST)
951.64/291.52	
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(15)
951.64/291.52	Obligation:
951.64/291.52	Proved the lower bound n^2 for the following obligation:
951.64/291.52	
951.64/291.52	TRS:
951.64/291.52	Rules:
951.64/291.52	fac(0') -> 1'
951.64/291.52	fac(s(x)) -> *'(s(x), fac(x))
951.64/291.52	floop(0', y) -> y
951.64/291.52	floop(s(x), y) -> floop(x, *'(s(x), y))
951.64/291.52	*'(x, 0') -> 0'
951.64/291.52	*'(x, s(y)) -> +'(*'(x, y), x)
951.64/291.52	+'(x, 0') -> x
951.64/291.52	+'(x, s(y)) -> s(+'(x, y))
951.64/291.52	1' -> s(0')
951.64/291.52	fac(0') -> s(0')
951.64/291.52	
951.64/291.52	Types:
951.64/291.52	fac :: 0':s -> 0':s
951.64/291.52	0' :: 0':s
951.64/291.52	1' :: 0':s
951.64/291.52	s :: 0':s -> 0':s
951.64/291.52	*' :: 0':s -> 0':s -> 0':s
951.64/291.52	floop :: 0':s -> 0':s -> 0':s
951.64/291.52	+' :: 0':s -> 0':s -> 0':s
951.64/291.52	hole_0':s1_0 :: 0':s
951.64/291.52	gen_0':s2_0 :: Nat -> 0':s
951.64/291.52	
951.64/291.52	
951.64/291.52	Lemmas:
951.64/291.52	+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
951.64/291.52	
951.64/291.52	
951.64/291.52	Generator Equations:
951.64/291.52	gen_0':s2_0(0) <=> 0'
951.64/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
951.64/291.52	
951.64/291.52	
951.64/291.52	The following defined symbols remain to be analysed:
951.64/291.52	*', fac, floop
951.64/291.52	
951.64/291.52	They will be analysed ascendingly in the following order:
951.64/291.52	*' < fac
951.64/291.52	*' < floop
951.64/291.52	
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(16) LowerBoundPropagationProof (FINISHED)
951.64/291.52	Propagated lower bound.
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(17)
951.64/291.52	BOUNDS(n^2, INF)
951.64/291.52	
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(18)
951.64/291.52	Obligation:
951.64/291.52	TRS:
951.64/291.52	Rules:
951.64/291.52	fac(0') -> 1'
951.64/291.52	fac(s(x)) -> *'(s(x), fac(x))
951.64/291.52	floop(0', y) -> y
951.64/291.52	floop(s(x), y) -> floop(x, *'(s(x), y))
951.64/291.52	*'(x, 0') -> 0'
951.64/291.52	*'(x, s(y)) -> +'(*'(x, y), x)
951.64/291.52	+'(x, 0') -> x
951.64/291.52	+'(x, s(y)) -> s(+'(x, y))
951.64/291.52	1' -> s(0')
951.64/291.52	fac(0') -> s(0')
951.64/291.52	
951.64/291.52	Types:
951.64/291.52	fac :: 0':s -> 0':s
951.64/291.52	0' :: 0':s
951.64/291.52	1' :: 0':s
951.64/291.52	s :: 0':s -> 0':s
951.64/291.52	*' :: 0':s -> 0':s -> 0':s
951.64/291.52	floop :: 0':s -> 0':s -> 0':s
951.64/291.52	+' :: 0':s -> 0':s -> 0':s
951.64/291.52	hole_0':s1_0 :: 0':s
951.64/291.52	gen_0':s2_0 :: Nat -> 0':s
951.64/291.52	
951.64/291.52	
951.64/291.52	Lemmas:
951.64/291.52	+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
951.64/291.52	*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) -> gen_0':s2_0(*(n511_0, a)), rt in Omega(1 + a*n511_0 + n511_0)
951.64/291.52	
951.64/291.52	
951.64/291.52	Generator Equations:
951.64/291.52	gen_0':s2_0(0) <=> 0'
951.64/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
951.64/291.52	
951.64/291.52	
951.64/291.52	The following defined symbols remain to be analysed:
951.64/291.52	fac, floop
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(19) RewriteLemmaProof (LOWER BOUND(ID))
951.64/291.52	Proved the following rewrite lemma:
951.64/291.52	fac(gen_0':s2_0(+(1, n1127_0))) -> *3_0, rt in Omega(n1127_0)
951.64/291.52	
951.64/291.52	Induction Base:
951.64/291.52	fac(gen_0':s2_0(+(1, 0)))
951.64/291.52	
951.64/291.52	Induction Step:
951.64/291.52	fac(gen_0':s2_0(+(1, +(n1127_0, 1)))) ->_R^Omega(1)
951.64/291.52	*'(s(gen_0':s2_0(+(1, n1127_0))), fac(gen_0':s2_0(+(1, n1127_0)))) ->_IH
951.64/291.52	*'(s(gen_0':s2_0(+(1, n1127_0))), *3_0)
951.64/291.52	
951.64/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(20)
951.64/291.52	Obligation:
951.64/291.52	TRS:
951.64/291.52	Rules:
951.64/291.52	fac(0') -> 1'
951.64/291.52	fac(s(x)) -> *'(s(x), fac(x))
951.64/291.52	floop(0', y) -> y
951.64/291.52	floop(s(x), y) -> floop(x, *'(s(x), y))
951.64/291.52	*'(x, 0') -> 0'
951.64/291.52	*'(x, s(y)) -> +'(*'(x, y), x)
951.64/291.52	+'(x, 0') -> x
951.64/291.52	+'(x, s(y)) -> s(+'(x, y))
951.64/291.52	1' -> s(0')
951.64/291.52	fac(0') -> s(0')
951.64/291.52	
951.64/291.52	Types:
951.64/291.52	fac :: 0':s -> 0':s
951.64/291.52	0' :: 0':s
951.64/291.52	1' :: 0':s
951.64/291.52	s :: 0':s -> 0':s
951.64/291.52	*' :: 0':s -> 0':s -> 0':s
951.64/291.52	floop :: 0':s -> 0':s -> 0':s
951.64/291.52	+' :: 0':s -> 0':s -> 0':s
951.64/291.52	hole_0':s1_0 :: 0':s
951.64/291.52	gen_0':s2_0 :: Nat -> 0':s
951.64/291.52	
951.64/291.52	
951.64/291.52	Lemmas:
951.64/291.52	+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
951.64/291.52	*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) -> gen_0':s2_0(*(n511_0, a)), rt in Omega(1 + a*n511_0 + n511_0)
951.64/291.52	fac(gen_0':s2_0(+(1, n1127_0))) -> *3_0, rt in Omega(n1127_0)
951.64/291.52	
951.64/291.52	
951.64/291.52	Generator Equations:
951.64/291.52	gen_0':s2_0(0) <=> 0'
951.64/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
951.64/291.52	
951.64/291.52	
951.64/291.52	The following defined symbols remain to be analysed:
951.64/291.52	floop
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(21) RewriteLemmaProof (LOWER BOUND(ID))
951.64/291.52	Proved the following rewrite lemma:
951.64/291.52	floop(gen_0':s2_0(n2194_0), gen_0':s2_0(b)) -> *3_0, rt in Omega(b*n2194_0 + b*n2194_0^2 + n2194_0)
951.64/291.52	
951.64/291.52	Induction Base:
951.64/291.52	floop(gen_0':s2_0(0), gen_0':s2_0(b))
951.64/291.52	
951.64/291.52	Induction Step:
951.64/291.52	floop(gen_0':s2_0(+(n2194_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
951.64/291.52	floop(gen_0':s2_0(n2194_0), *'(s(gen_0':s2_0(n2194_0)), gen_0':s2_0(b))) ->_L^Omega(1 + 2*b + b*n2194_0)
951.64/291.52	floop(gen_0':s2_0(n2194_0), gen_0':s2_0(*(b, +(n2194_0, 1)))) ->_IH
951.64/291.52	*3_0
951.64/291.52	
951.64/291.52	We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(22)
951.64/291.52	Obligation:
951.64/291.52	Proved the lower bound n^3 for the following obligation:
951.64/291.52	
951.64/291.52	TRS:
951.64/291.52	Rules:
951.64/291.52	fac(0') -> 1'
951.64/291.52	fac(s(x)) -> *'(s(x), fac(x))
951.64/291.52	floop(0', y) -> y
951.64/291.52	floop(s(x), y) -> floop(x, *'(s(x), y))
951.64/291.52	*'(x, 0') -> 0'
951.64/291.52	*'(x, s(y)) -> +'(*'(x, y), x)
951.64/291.52	+'(x, 0') -> x
951.64/291.52	+'(x, s(y)) -> s(+'(x, y))
951.64/291.52	1' -> s(0')
951.64/291.52	fac(0') -> s(0')
951.64/291.52	
951.64/291.52	Types:
951.64/291.52	fac :: 0':s -> 0':s
951.64/291.52	0' :: 0':s
951.64/291.52	1' :: 0':s
951.64/291.52	s :: 0':s -> 0':s
951.64/291.52	*' :: 0':s -> 0':s -> 0':s
951.64/291.52	floop :: 0':s -> 0':s -> 0':s
951.64/291.52	+' :: 0':s -> 0':s -> 0':s
951.64/291.52	hole_0':s1_0 :: 0':s
951.64/291.52	gen_0':s2_0 :: Nat -> 0':s
951.64/291.52	
951.64/291.52	
951.64/291.52	Lemmas:
951.64/291.52	+'(gen_0':s2_0(a), gen_0':s2_0(n4_0)) -> gen_0':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)
951.64/291.52	*'(gen_0':s2_0(a), gen_0':s2_0(n511_0)) -> gen_0':s2_0(*(n511_0, a)), rt in Omega(1 + a*n511_0 + n511_0)
951.64/291.52	fac(gen_0':s2_0(+(1, n1127_0))) -> *3_0, rt in Omega(n1127_0)
951.64/291.52	
951.64/291.52	
951.64/291.52	Generator Equations:
951.64/291.52	gen_0':s2_0(0) <=> 0'
951.64/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
951.64/291.52	
951.64/291.52	
951.64/291.52	The following defined symbols remain to be analysed:
951.64/291.52	floop
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(23) LowerBoundPropagationProof (FINISHED)
951.64/291.52	Propagated lower bound.
951.64/291.52	----------------------------------------
951.64/291.52	
951.64/291.52	(24)
951.64/291.52	BOUNDS(n^3, INF)
951.93/291.56	EOF