1126.85/291.50	WORST_CASE(Omega(n^3), ?)
1126.85/291.52	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1126.85/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).
1126.85/291.52	
1126.85/291.52	(0) CpxTRS
1126.85/291.52	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1126.85/291.52	(2) CpxTRS
1126.85/291.52	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1126.85/291.52	(4) typed CpxTrs
1126.85/291.52	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1126.85/291.52	(6) typed CpxTrs
1126.85/291.52	(7) RewriteLemmaProof [LOWER BOUND(ID), 311 ms]
1126.85/291.52	(8) BEST
1126.85/291.52	    (9) proven lower bound
1126.85/291.52	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1126.85/291.52	        (11) BOUNDS(n^1, INF)
1126.85/291.52	    (12) typed CpxTrs
1126.85/291.52	        (13) RewriteLemmaProof [LOWER BOUND(ID), 85 ms]
1126.85/291.52	        (14) BEST
1126.85/291.52	            (15) proven lower bound
1126.85/291.52	                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
1126.85/291.52	                (17) BOUNDS(n^2, INF)
1126.85/291.52	            (18) typed CpxTrs
1126.85/291.52	                (19) RewriteLemmaProof [LOWER BOUND(ID), 609 ms]
1126.85/291.52	                (20) typed CpxTrs
1126.85/291.52	                (21) RewriteLemmaProof [LOWER BOUND(ID), 266 ms]
1126.85/291.52	                (22) proven lower bound
1126.85/291.52	                (23) LowerBoundPropagationProof [FINISHED, 0 ms]
1126.85/291.52	                (24) BOUNDS(n^3, INF)
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(0)
1126.85/291.52	Obligation:
1126.85/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	The TRS R consists of the following rules:
1126.85/291.52	
1126.85/291.52	   plus(0, x) -> x
1126.85/291.52	   plus(s(x), y) -> s(plus(x, y))
1126.85/291.52	   times(0, y) -> 0
1126.85/291.52	   times(s(x), y) -> plus(y, times(x, y))
1126.85/291.52	   p(s(x)) -> x
1126.85/291.52	   p(0) -> 0
1126.85/291.52	   minus(x, 0) -> x
1126.85/291.52	   minus(0, x) -> 0
1126.85/291.52	   minus(x, s(y)) -> p(minus(x, y))
1126.85/291.52	   isZero(0) -> true
1126.85/291.52	   isZero(s(x)) -> false
1126.85/291.52	   facIter(x, y) -> if(isZero(x), minus(x, s(0)), y, times(y, x))
1126.85/291.52	   if(true, x, y, z) -> y
1126.85/291.52	   if(false, x, y, z) -> facIter(x, z)
1126.85/291.52	   factorial(x) -> facIter(x, s(0))
1126.85/291.52	
1126.85/291.52	S is empty.
1126.85/291.52	Rewrite Strategy: INNERMOST
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1126.85/291.52	Renamed function symbols to avoid clashes with predefined symbol.
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(2)
1126.85/291.52	Obligation:
1126.85/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	The TRS R consists of the following rules:
1126.85/291.52	
1126.85/291.52	   plus(0', x) -> x
1126.85/291.52	   plus(s(x), y) -> s(plus(x, y))
1126.85/291.52	   times(0', y) -> 0'
1126.85/291.52	   times(s(x), y) -> plus(y, times(x, y))
1126.85/291.52	   p(s(x)) -> x
1126.85/291.52	   p(0') -> 0'
1126.85/291.52	   minus(x, 0') -> x
1126.85/291.52	   minus(0', x) -> 0'
1126.85/291.52	   minus(x, s(y)) -> p(minus(x, y))
1126.85/291.52	   isZero(0') -> true
1126.85/291.52	   isZero(s(x)) -> false
1126.85/291.52	   facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
1126.85/291.52	   if(true, x, y, z) -> y
1126.85/291.52	   if(false, x, y, z) -> facIter(x, z)
1126.85/291.52	   factorial(x) -> facIter(x, s(0'))
1126.85/291.52	
1126.85/291.52	S is empty.
1126.85/291.52	Rewrite Strategy: INNERMOST
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1126.85/291.52	Infered types.
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(4)
1126.85/291.52	Obligation:
1126.85/291.52	Innermost TRS:
1126.85/291.52	Rules:
1126.85/291.52	plus(0', x) -> x
1126.85/291.52	plus(s(x), y) -> s(plus(x, y))
1126.85/291.52	times(0', y) -> 0'
1126.85/291.52	times(s(x), y) -> plus(y, times(x, y))
1126.85/291.52	p(s(x)) -> x
1126.85/291.52	p(0') -> 0'
1126.85/291.52	minus(x, 0') -> x
1126.85/291.52	minus(0', x) -> 0'
1126.85/291.52	minus(x, s(y)) -> p(minus(x, y))
1126.85/291.52	isZero(0') -> true
1126.85/291.52	isZero(s(x)) -> false
1126.85/291.52	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
1126.85/291.52	if(true, x, y, z) -> y
1126.85/291.52	if(false, x, y, z) -> facIter(x, z)
1126.85/291.52	factorial(x) -> facIter(x, s(0'))
1126.85/291.52	
1126.85/291.52	Types:
1126.85/291.52	plus :: 0':s -> 0':s -> 0':s
1126.85/291.52	0' :: 0':s
1126.85/291.52	s :: 0':s -> 0':s
1126.85/291.52	times :: 0':s -> 0':s -> 0':s
1126.85/291.52	p :: 0':s -> 0':s
1126.85/291.52	minus :: 0':s -> 0':s -> 0':s
1126.85/291.52	isZero :: 0':s -> true:false
1126.85/291.52	true :: true:false
1126.85/291.52	false :: true:false
1126.85/291.52	facIter :: 0':s -> 0':s -> 0':s
1126.85/291.52	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1126.85/291.52	factorial :: 0':s -> 0':s
1126.85/291.52	hole_0':s1_0 :: 0':s
1126.85/291.52	hole_true:false2_0 :: true:false
1126.85/291.52	gen_0':s3_0 :: Nat -> 0':s
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(5) OrderProof (LOWER BOUND(ID))
1126.85/291.52	Heuristically decided to analyse the following defined symbols:
1126.85/291.52	plus, times, minus, facIter
1126.85/291.52	
1126.85/291.52	They will be analysed ascendingly in the following order:
1126.85/291.52	plus < times
1126.85/291.52	times < facIter
1126.85/291.52	minus < facIter
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(6)
1126.85/291.52	Obligation:
1126.85/291.52	Innermost TRS:
1126.85/291.52	Rules:
1126.85/291.52	plus(0', x) -> x
1126.85/291.52	plus(s(x), y) -> s(plus(x, y))
1126.85/291.52	times(0', y) -> 0'
1126.85/291.52	times(s(x), y) -> plus(y, times(x, y))
1126.85/291.52	p(s(x)) -> x
1126.85/291.52	p(0') -> 0'
1126.85/291.52	minus(x, 0') -> x
1126.85/291.52	minus(0', x) -> 0'
1126.85/291.52	minus(x, s(y)) -> p(minus(x, y))
1126.85/291.52	isZero(0') -> true
1126.85/291.52	isZero(s(x)) -> false
1126.85/291.52	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
1126.85/291.52	if(true, x, y, z) -> y
1126.85/291.52	if(false, x, y, z) -> facIter(x, z)
1126.85/291.52	factorial(x) -> facIter(x, s(0'))
1126.85/291.52	
1126.85/291.52	Types:
1126.85/291.52	plus :: 0':s -> 0':s -> 0':s
1126.85/291.52	0' :: 0':s
1126.85/291.52	s :: 0':s -> 0':s
1126.85/291.52	times :: 0':s -> 0':s -> 0':s
1126.85/291.52	p :: 0':s -> 0':s
1126.85/291.52	minus :: 0':s -> 0':s -> 0':s
1126.85/291.52	isZero :: 0':s -> true:false
1126.85/291.52	true :: true:false
1126.85/291.52	false :: true:false
1126.85/291.52	facIter :: 0':s -> 0':s -> 0':s
1126.85/291.52	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1126.85/291.52	factorial :: 0':s -> 0':s
1126.85/291.52	hole_0':s1_0 :: 0':s
1126.85/291.52	hole_true:false2_0 :: true:false
1126.85/291.52	gen_0':s3_0 :: Nat -> 0':s
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Generator Equations:
1126.85/291.52	gen_0':s3_0(0) <=> 0'
1126.85/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	The following defined symbols remain to be analysed:
1126.85/291.52	plus, times, minus, facIter
1126.85/291.52	
1126.85/291.52	They will be analysed ascendingly in the following order:
1126.85/291.52	plus < times
1126.85/291.52	times < facIter
1126.85/291.52	minus < facIter
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(7) RewriteLemmaProof (LOWER BOUND(ID))
1126.85/291.52	Proved the following rewrite lemma:
1126.85/291.52	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)
1126.85/291.52	
1126.85/291.52	Induction Base:
1126.85/291.52	plus(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
1126.85/291.52	gen_0':s3_0(b)
1126.85/291.52	
1126.85/291.52	Induction Step:
1126.85/291.52	plus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
1126.85/291.52	s(plus(gen_0':s3_0(n5_0), gen_0':s3_0(b))) ->_IH
1126.85/291.52	s(gen_0':s3_0(+(b, c6_0)))
1126.85/291.52	
1126.85/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(8)
1126.85/291.52	Complex Obligation (BEST)
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(9)
1126.85/291.52	Obligation:
1126.85/291.52	Proved the lower bound n^1 for the following obligation:
1126.85/291.52	
1126.85/291.52	Innermost TRS:
1126.85/291.52	Rules:
1126.85/291.52	plus(0', x) -> x
1126.85/291.52	plus(s(x), y) -> s(plus(x, y))
1126.85/291.52	times(0', y) -> 0'
1126.85/291.52	times(s(x), y) -> plus(y, times(x, y))
1126.85/291.52	p(s(x)) -> x
1126.85/291.52	p(0') -> 0'
1126.85/291.52	minus(x, 0') -> x
1126.85/291.52	minus(0', x) -> 0'
1126.85/291.52	minus(x, s(y)) -> p(minus(x, y))
1126.85/291.52	isZero(0') -> true
1126.85/291.52	isZero(s(x)) -> false
1126.85/291.52	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
1126.85/291.52	if(true, x, y, z) -> y
1126.85/291.52	if(false, x, y, z) -> facIter(x, z)
1126.85/291.52	factorial(x) -> facIter(x, s(0'))
1126.85/291.52	
1126.85/291.52	Types:
1126.85/291.52	plus :: 0':s -> 0':s -> 0':s
1126.85/291.52	0' :: 0':s
1126.85/291.52	s :: 0':s -> 0':s
1126.85/291.52	times :: 0':s -> 0':s -> 0':s
1126.85/291.52	p :: 0':s -> 0':s
1126.85/291.52	minus :: 0':s -> 0':s -> 0':s
1126.85/291.52	isZero :: 0':s -> true:false
1126.85/291.52	true :: true:false
1126.85/291.52	false :: true:false
1126.85/291.52	facIter :: 0':s -> 0':s -> 0':s
1126.85/291.52	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1126.85/291.52	factorial :: 0':s -> 0':s
1126.85/291.52	hole_0':s1_0 :: 0':s
1126.85/291.52	hole_true:false2_0 :: true:false
1126.85/291.52	gen_0':s3_0 :: Nat -> 0':s
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Generator Equations:
1126.85/291.52	gen_0':s3_0(0) <=> 0'
1126.85/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	The following defined symbols remain to be analysed:
1126.85/291.52	plus, times, minus, facIter
1126.85/291.52	
1126.85/291.52	They will be analysed ascendingly in the following order:
1126.85/291.52	plus < times
1126.85/291.52	times < facIter
1126.85/291.52	minus < facIter
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(10) LowerBoundPropagationProof (FINISHED)
1126.85/291.52	Propagated lower bound.
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(11)
1126.85/291.52	BOUNDS(n^1, INF)
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(12)
1126.85/291.52	Obligation:
1126.85/291.52	Innermost TRS:
1126.85/291.52	Rules:
1126.85/291.52	plus(0', x) -> x
1126.85/291.52	plus(s(x), y) -> s(plus(x, y))
1126.85/291.52	times(0', y) -> 0'
1126.85/291.52	times(s(x), y) -> plus(y, times(x, y))
1126.85/291.52	p(s(x)) -> x
1126.85/291.52	p(0') -> 0'
1126.85/291.52	minus(x, 0') -> x
1126.85/291.52	minus(0', x) -> 0'
1126.85/291.52	minus(x, s(y)) -> p(minus(x, y))
1126.85/291.52	isZero(0') -> true
1126.85/291.52	isZero(s(x)) -> false
1126.85/291.52	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
1126.85/291.52	if(true, x, y, z) -> y
1126.85/291.52	if(false, x, y, z) -> facIter(x, z)
1126.85/291.52	factorial(x) -> facIter(x, s(0'))
1126.85/291.52	
1126.85/291.52	Types:
1126.85/291.52	plus :: 0':s -> 0':s -> 0':s
1126.85/291.52	0' :: 0':s
1126.85/291.52	s :: 0':s -> 0':s
1126.85/291.52	times :: 0':s -> 0':s -> 0':s
1126.85/291.52	p :: 0':s -> 0':s
1126.85/291.52	minus :: 0':s -> 0':s -> 0':s
1126.85/291.52	isZero :: 0':s -> true:false
1126.85/291.52	true :: true:false
1126.85/291.52	false :: true:false
1126.85/291.52	facIter :: 0':s -> 0':s -> 0':s
1126.85/291.52	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1126.85/291.52	factorial :: 0':s -> 0':s
1126.85/291.52	hole_0':s1_0 :: 0':s
1126.85/291.52	hole_true:false2_0 :: true:false
1126.85/291.52	gen_0':s3_0 :: Nat -> 0':s
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Lemmas:
1126.85/291.52	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)
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Generator Equations:
1126.85/291.52	gen_0':s3_0(0) <=> 0'
1126.85/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	The following defined symbols remain to be analysed:
1126.85/291.52	times, minus, facIter
1126.85/291.52	
1126.85/291.52	They will be analysed ascendingly in the following order:
1126.85/291.52	times < facIter
1126.85/291.52	minus < facIter
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(13) RewriteLemmaProof (LOWER BOUND(ID))
1126.85/291.52	Proved the following rewrite lemma:
1126.85/291.52	times(gen_0':s3_0(n706_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n706_0, b)), rt in Omega(1 + b*n706_0 + n706_0)
1126.85/291.52	
1126.85/291.52	Induction Base:
1126.85/291.52	times(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
1126.85/291.52	0'
1126.85/291.52	
1126.85/291.52	Induction Step:
1126.85/291.52	times(gen_0':s3_0(+(n706_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
1126.85/291.52	plus(gen_0':s3_0(b), times(gen_0':s3_0(n706_0), gen_0':s3_0(b))) ->_IH
1126.85/291.52	plus(gen_0':s3_0(b), gen_0':s3_0(*(c707_0, b))) ->_L^Omega(1 + b)
1126.85/291.52	gen_0':s3_0(+(b, *(n706_0, b)))
1126.85/291.52	
1126.85/291.52	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(14)
1126.85/291.52	Complex Obligation (BEST)
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(15)
1126.85/291.52	Obligation:
1126.85/291.52	Proved the lower bound n^2 for the following obligation:
1126.85/291.52	
1126.85/291.52	Innermost TRS:
1126.85/291.52	Rules:
1126.85/291.52	plus(0', x) -> x
1126.85/291.52	plus(s(x), y) -> s(plus(x, y))
1126.85/291.52	times(0', y) -> 0'
1126.85/291.52	times(s(x), y) -> plus(y, times(x, y))
1126.85/291.52	p(s(x)) -> x
1126.85/291.52	p(0') -> 0'
1126.85/291.52	minus(x, 0') -> x
1126.85/291.52	minus(0', x) -> 0'
1126.85/291.52	minus(x, s(y)) -> p(minus(x, y))
1126.85/291.52	isZero(0') -> true
1126.85/291.52	isZero(s(x)) -> false
1126.85/291.52	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
1126.85/291.52	if(true, x, y, z) -> y
1126.85/291.52	if(false, x, y, z) -> facIter(x, z)
1126.85/291.52	factorial(x) -> facIter(x, s(0'))
1126.85/291.52	
1126.85/291.52	Types:
1126.85/291.52	plus :: 0':s -> 0':s -> 0':s
1126.85/291.52	0' :: 0':s
1126.85/291.52	s :: 0':s -> 0':s
1126.85/291.52	times :: 0':s -> 0':s -> 0':s
1126.85/291.52	p :: 0':s -> 0':s
1126.85/291.52	minus :: 0':s -> 0':s -> 0':s
1126.85/291.52	isZero :: 0':s -> true:false
1126.85/291.52	true :: true:false
1126.85/291.52	false :: true:false
1126.85/291.52	facIter :: 0':s -> 0':s -> 0':s
1126.85/291.52	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1126.85/291.52	factorial :: 0':s -> 0':s
1126.85/291.52	hole_0':s1_0 :: 0':s
1126.85/291.52	hole_true:false2_0 :: true:false
1126.85/291.52	gen_0':s3_0 :: Nat -> 0':s
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Lemmas:
1126.85/291.52	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)
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Generator Equations:
1126.85/291.52	gen_0':s3_0(0) <=> 0'
1126.85/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	The following defined symbols remain to be analysed:
1126.85/291.52	times, minus, facIter
1126.85/291.52	
1126.85/291.52	They will be analysed ascendingly in the following order:
1126.85/291.52	times < facIter
1126.85/291.52	minus < facIter
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(16) LowerBoundPropagationProof (FINISHED)
1126.85/291.52	Propagated lower bound.
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(17)
1126.85/291.52	BOUNDS(n^2, INF)
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(18)
1126.85/291.52	Obligation:
1126.85/291.52	Innermost TRS:
1126.85/291.52	Rules:
1126.85/291.52	plus(0', x) -> x
1126.85/291.52	plus(s(x), y) -> s(plus(x, y))
1126.85/291.52	times(0', y) -> 0'
1126.85/291.52	times(s(x), y) -> plus(y, times(x, y))
1126.85/291.52	p(s(x)) -> x
1126.85/291.52	p(0') -> 0'
1126.85/291.52	minus(x, 0') -> x
1126.85/291.52	minus(0', x) -> 0'
1126.85/291.52	minus(x, s(y)) -> p(minus(x, y))
1126.85/291.52	isZero(0') -> true
1126.85/291.52	isZero(s(x)) -> false
1126.85/291.52	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
1126.85/291.52	if(true, x, y, z) -> y
1126.85/291.52	if(false, x, y, z) -> facIter(x, z)
1126.85/291.52	factorial(x) -> facIter(x, s(0'))
1126.85/291.52	
1126.85/291.52	Types:
1126.85/291.52	plus :: 0':s -> 0':s -> 0':s
1126.85/291.52	0' :: 0':s
1126.85/291.52	s :: 0':s -> 0':s
1126.85/291.52	times :: 0':s -> 0':s -> 0':s
1126.85/291.52	p :: 0':s -> 0':s
1126.85/291.52	minus :: 0':s -> 0':s -> 0':s
1126.85/291.52	isZero :: 0':s -> true:false
1126.85/291.52	true :: true:false
1126.85/291.52	false :: true:false
1126.85/291.52	facIter :: 0':s -> 0':s -> 0':s
1126.85/291.52	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1126.85/291.52	factorial :: 0':s -> 0':s
1126.85/291.52	hole_0':s1_0 :: 0':s
1126.85/291.52	hole_true:false2_0 :: true:false
1126.85/291.52	gen_0':s3_0 :: Nat -> 0':s
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Lemmas:
1126.85/291.52	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)
1126.85/291.52	times(gen_0':s3_0(n706_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n706_0, b)), rt in Omega(1 + b*n706_0 + n706_0)
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Generator Equations:
1126.85/291.52	gen_0':s3_0(0) <=> 0'
1126.85/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	The following defined symbols remain to be analysed:
1126.85/291.52	minus, facIter
1126.85/291.52	
1126.85/291.52	They will be analysed ascendingly in the following order:
1126.85/291.52	minus < facIter
1126.85/291.52	
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(19) RewriteLemmaProof (LOWER BOUND(ID))
1126.85/291.52	Proved the following rewrite lemma:
1126.85/291.52	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1596_0))) -> *4_0, rt in Omega(n1596_0)
1126.85/291.52	
1126.85/291.52	Induction Base:
1126.85/291.52	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, 0)))
1126.85/291.52	
1126.85/291.52	Induction Step:
1126.85/291.52	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, +(n1596_0, 1)))) ->_R^Omega(1)
1126.85/291.52	p(minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1596_0)))) ->_IH
1126.85/291.52	p(*4_0)
1126.85/291.52	
1126.85/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(20)
1126.85/291.52	Obligation:
1126.85/291.52	Innermost TRS:
1126.85/291.52	Rules:
1126.85/291.52	plus(0', x) -> x
1126.85/291.52	plus(s(x), y) -> s(plus(x, y))
1126.85/291.52	times(0', y) -> 0'
1126.85/291.52	times(s(x), y) -> plus(y, times(x, y))
1126.85/291.52	p(s(x)) -> x
1126.85/291.52	p(0') -> 0'
1126.85/291.52	minus(x, 0') -> x
1126.85/291.52	minus(0', x) -> 0'
1126.85/291.52	minus(x, s(y)) -> p(minus(x, y))
1126.85/291.52	isZero(0') -> true
1126.85/291.52	isZero(s(x)) -> false
1126.85/291.52	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
1126.85/291.52	if(true, x, y, z) -> y
1126.85/291.52	if(false, x, y, z) -> facIter(x, z)
1126.85/291.52	factorial(x) -> facIter(x, s(0'))
1126.85/291.52	
1126.85/291.52	Types:
1126.85/291.52	plus :: 0':s -> 0':s -> 0':s
1126.85/291.52	0' :: 0':s
1126.85/291.52	s :: 0':s -> 0':s
1126.85/291.52	times :: 0':s -> 0':s -> 0':s
1126.85/291.52	p :: 0':s -> 0':s
1126.85/291.52	minus :: 0':s -> 0':s -> 0':s
1126.85/291.52	isZero :: 0':s -> true:false
1126.85/291.52	true :: true:false
1126.85/291.52	false :: true:false
1126.85/291.52	facIter :: 0':s -> 0':s -> 0':s
1126.85/291.52	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1126.85/291.52	factorial :: 0':s -> 0':s
1126.85/291.52	hole_0':s1_0 :: 0':s
1126.85/291.52	hole_true:false2_0 :: true:false
1126.85/291.52	gen_0':s3_0 :: Nat -> 0':s
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Lemmas:
1126.85/291.52	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)
1126.85/291.52	times(gen_0':s3_0(n706_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n706_0, b)), rt in Omega(1 + b*n706_0 + n706_0)
1126.85/291.52	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1596_0))) -> *4_0, rt in Omega(n1596_0)
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Generator Equations:
1126.85/291.52	gen_0':s3_0(0) <=> 0'
1126.85/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	The following defined symbols remain to be analysed:
1126.85/291.52	facIter
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(21) RewriteLemmaProof (LOWER BOUND(ID))
1126.85/291.52	Proved the following rewrite lemma:
1126.85/291.52	facIter(gen_0':s3_0(n5565_0), gen_0':s3_0(b)) -> *4_0, rt in Omega(n5565_0 + n5565_0^2 + n5565_0^3)
1126.85/291.52	
1126.85/291.52	Induction Base:
1126.85/291.52	facIter(gen_0':s3_0(0), gen_0':s3_0(b))
1126.85/291.52	
1126.85/291.52	Induction Step:
1126.85/291.52	facIter(gen_0':s3_0(+(n5565_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
1126.85/291.52	if(isZero(gen_0':s3_0(+(n5565_0, 1))), minus(gen_0':s3_0(+(n5565_0, 1)), s(0')), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(n5565_0, 1)))) ->_R^Omega(1)
1126.85/291.52	if(false, minus(gen_0':s3_0(+(1, n5565_0)), s(0')), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n5565_0)))) ->_R^Omega(1)
1126.85/291.52	if(false, p(minus(gen_0':s3_0(+(1, n5565_0)), 0')), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n5565_0)))) ->_R^Omega(1)
1126.85/291.52	if(false, p(gen_0':s3_0(+(1, n5565_0))), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n5565_0)))) ->_R^Omega(1)
1126.85/291.52	if(false, gen_0':s3_0(n5565_0), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n5565_0)))) ->_L^Omega(3 + 3*n5565_0 + n5565_0^2)
1126.85/291.52	if(false, gen_0':s3_0(n5565_0), gen_0':s3_0(+(1, n5565_0)), gen_0':s3_0(*(b, +(1, n5565_0)))) ->_R^Omega(1)
1126.85/291.52	facIter(gen_0':s3_0(n5565_0), gen_0':s3_0(+(b, *(b, n5565_0)))) ->_IH
1126.85/291.52	*4_0
1126.85/291.52	
1126.85/291.52	We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(22)
1126.85/291.52	Obligation:
1126.85/291.52	Proved the lower bound n^3 for the following obligation:
1126.85/291.52	
1126.85/291.52	Innermost TRS:
1126.85/291.52	Rules:
1126.85/291.52	plus(0', x) -> x
1126.85/291.52	plus(s(x), y) -> s(plus(x, y))
1126.85/291.52	times(0', y) -> 0'
1126.85/291.52	times(s(x), y) -> plus(y, times(x, y))
1126.85/291.52	p(s(x)) -> x
1126.85/291.52	p(0') -> 0'
1126.85/291.52	minus(x, 0') -> x
1126.85/291.52	minus(0', x) -> 0'
1126.85/291.52	minus(x, s(y)) -> p(minus(x, y))
1126.85/291.52	isZero(0') -> true
1126.85/291.52	isZero(s(x)) -> false
1126.85/291.52	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
1126.85/291.52	if(true, x, y, z) -> y
1126.85/291.52	if(false, x, y, z) -> facIter(x, z)
1126.85/291.52	factorial(x) -> facIter(x, s(0'))
1126.85/291.52	
1126.85/291.52	Types:
1126.85/291.52	plus :: 0':s -> 0':s -> 0':s
1126.85/291.52	0' :: 0':s
1126.85/291.52	s :: 0':s -> 0':s
1126.85/291.52	times :: 0':s -> 0':s -> 0':s
1126.85/291.52	p :: 0':s -> 0':s
1126.85/291.52	minus :: 0':s -> 0':s -> 0':s
1126.85/291.52	isZero :: 0':s -> true:false
1126.85/291.52	true :: true:false
1126.85/291.52	false :: true:false
1126.85/291.52	facIter :: 0':s -> 0':s -> 0':s
1126.85/291.52	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
1126.85/291.52	factorial :: 0':s -> 0':s
1126.85/291.52	hole_0':s1_0 :: 0':s
1126.85/291.52	hole_true:false2_0 :: true:false
1126.85/291.52	gen_0':s3_0 :: Nat -> 0':s
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Lemmas:
1126.85/291.52	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)
1126.85/291.52	times(gen_0':s3_0(n706_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n706_0, b)), rt in Omega(1 + b*n706_0 + n706_0)
1126.85/291.52	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1596_0))) -> *4_0, rt in Omega(n1596_0)
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	Generator Equations:
1126.85/291.52	gen_0':s3_0(0) <=> 0'
1126.85/291.52	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
1126.85/291.52	
1126.85/291.52	
1126.85/291.52	The following defined symbols remain to be analysed:
1126.85/291.52	facIter
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(23) LowerBoundPropagationProof (FINISHED)
1126.85/291.52	Propagated lower bound.
1126.85/291.52	----------------------------------------
1126.85/291.52	
1126.85/291.52	(24)
1126.85/291.52	BOUNDS(n^3, INF)
1127.12/291.59	EOF