321.07/291.60	WORST_CASE(Omega(n^3), ?)
321.07/291.61	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
321.07/291.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
321.07/291.61	
321.07/291.61	
321.07/291.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).
321.07/291.61	
321.07/291.61	(0) CpxTRS
321.07/291.61	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
321.07/291.61	(2) CpxTRS
321.07/291.61	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
321.07/291.61	(4) typed CpxTrs
321.07/291.61	(5) OrderProof [LOWER BOUND(ID), 0 ms]
321.07/291.61	(6) typed CpxTrs
321.07/291.61	(7) RewriteLemmaProof [LOWER BOUND(ID), 301 ms]
321.07/291.61	(8) BEST
321.07/291.61	    (9) proven lower bound
321.07/291.61	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
321.07/291.61	        (11) BOUNDS(n^1, INF)
321.07/291.61	    (12) typed CpxTrs
321.07/291.61	        (13) RewriteLemmaProof [LOWER BOUND(ID), 52 ms]
321.07/291.61	        (14) BEST
321.07/291.61	            (15) proven lower bound
321.07/291.61	                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
321.07/291.61	                (17) BOUNDS(n^2, INF)
321.07/291.61	            (18) typed CpxTrs
321.07/291.61	                (19) RewriteLemmaProof [LOWER BOUND(ID), 478 ms]
321.07/291.61	                (20) typed CpxTrs
321.07/291.61	                (21) RewriteLemmaProof [LOWER BOUND(ID), 307 ms]
321.07/291.61	                (22) proven lower bound
321.07/291.61	                (23) LowerBoundPropagationProof [FINISHED, 0 ms]
321.07/291.61	                (24) BOUNDS(n^3, INF)
321.07/291.61	
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(0)
321.07/291.61	Obligation:
321.07/291.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).
321.07/291.61	
321.07/291.61	
321.07/291.61	The TRS R consists of the following rules:
321.07/291.61	
321.07/291.61	   plus(0, x) -> x
321.07/291.61	   plus(s(x), y) -> s(plus(x, y))
321.07/291.61	   times(0, y) -> 0
321.07/291.61	   times(s(x), y) -> plus(y, times(x, y))
321.07/291.61	   p(s(x)) -> x
321.07/291.61	   p(0) -> 0
321.07/291.61	   minus(x, 0) -> x
321.07/291.61	   minus(0, x) -> 0
321.07/291.61	   minus(x, s(y)) -> p(minus(x, y))
321.07/291.61	   isZero(0) -> true
321.07/291.61	   isZero(s(x)) -> false
321.07/291.61	   facIter(x, y) -> if(isZero(x), minus(x, s(0)), y, times(y, x))
321.07/291.61	   if(true, x, y, z) -> y
321.07/291.61	   if(false, x, y, z) -> facIter(x, z)
321.07/291.61	   factorial(x) -> facIter(x, s(0))
321.07/291.61	
321.07/291.61	S is empty.
321.07/291.61	Rewrite Strategy: FULL
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(1) RenamingProof (BOTH BOUNDS(ID, ID))
321.07/291.61	Renamed function symbols to avoid clashes with predefined symbol.
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(2)
321.07/291.61	Obligation:
321.07/291.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).
321.07/291.61	
321.07/291.61	
321.07/291.61	The TRS R consists of the following rules:
321.07/291.61	
321.07/291.61	   plus(0', x) -> x
321.07/291.61	   plus(s(x), y) -> s(plus(x, y))
321.07/291.61	   times(0', y) -> 0'
321.07/291.61	   times(s(x), y) -> plus(y, times(x, y))
321.07/291.61	   p(s(x)) -> x
321.07/291.61	   p(0') -> 0'
321.07/291.61	   minus(x, 0') -> x
321.07/291.61	   minus(0', x) -> 0'
321.07/291.61	   minus(x, s(y)) -> p(minus(x, y))
321.07/291.61	   isZero(0') -> true
321.07/291.61	   isZero(s(x)) -> false
321.07/291.61	   facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
321.07/291.61	   if(true, x, y, z) -> y
321.07/291.61	   if(false, x, y, z) -> facIter(x, z)
321.07/291.61	   factorial(x) -> facIter(x, s(0'))
321.07/291.61	
321.07/291.61	S is empty.
321.07/291.61	Rewrite Strategy: FULL
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
321.07/291.61	Infered types.
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(4)
321.07/291.61	Obligation:
321.07/291.61	TRS:
321.07/291.61	Rules:
321.07/291.61	plus(0', x) -> x
321.07/291.61	plus(s(x), y) -> s(plus(x, y))
321.07/291.61	times(0', y) -> 0'
321.07/291.61	times(s(x), y) -> plus(y, times(x, y))
321.07/291.61	p(s(x)) -> x
321.07/291.61	p(0') -> 0'
321.07/291.61	minus(x, 0') -> x
321.07/291.61	minus(0', x) -> 0'
321.07/291.61	minus(x, s(y)) -> p(minus(x, y))
321.07/291.61	isZero(0') -> true
321.07/291.61	isZero(s(x)) -> false
321.07/291.61	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
321.07/291.61	if(true, x, y, z) -> y
321.07/291.61	if(false, x, y, z) -> facIter(x, z)
321.07/291.61	factorial(x) -> facIter(x, s(0'))
321.07/291.61	
321.07/291.61	Types:
321.07/291.61	plus :: 0':s -> 0':s -> 0':s
321.07/291.61	0' :: 0':s
321.07/291.61	s :: 0':s -> 0':s
321.07/291.61	times :: 0':s -> 0':s -> 0':s
321.07/291.61	p :: 0':s -> 0':s
321.07/291.61	minus :: 0':s -> 0':s -> 0':s
321.07/291.61	isZero :: 0':s -> true:false
321.07/291.61	true :: true:false
321.07/291.61	false :: true:false
321.07/291.61	facIter :: 0':s -> 0':s -> 0':s
321.07/291.61	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
321.07/291.61	factorial :: 0':s -> 0':s
321.07/291.61	hole_0':s1_0 :: 0':s
321.07/291.61	hole_true:false2_0 :: true:false
321.07/291.61	gen_0':s3_0 :: Nat -> 0':s
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(5) OrderProof (LOWER BOUND(ID))
321.07/291.61	Heuristically decided to analyse the following defined symbols:
321.07/291.61	plus, times, minus, facIter
321.07/291.61	
321.07/291.61	They will be analysed ascendingly in the following order:
321.07/291.61	plus < times
321.07/291.61	times < facIter
321.07/291.61	minus < facIter
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(6)
321.07/291.61	Obligation:
321.07/291.61	TRS:
321.07/291.61	Rules:
321.07/291.61	plus(0', x) -> x
321.07/291.61	plus(s(x), y) -> s(plus(x, y))
321.07/291.61	times(0', y) -> 0'
321.07/291.61	times(s(x), y) -> plus(y, times(x, y))
321.07/291.61	p(s(x)) -> x
321.07/291.61	p(0') -> 0'
321.07/291.61	minus(x, 0') -> x
321.07/291.61	minus(0', x) -> 0'
321.07/291.61	minus(x, s(y)) -> p(minus(x, y))
321.07/291.61	isZero(0') -> true
321.07/291.61	isZero(s(x)) -> false
321.07/291.61	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
321.07/291.61	if(true, x, y, z) -> y
321.07/291.61	if(false, x, y, z) -> facIter(x, z)
321.07/291.61	factorial(x) -> facIter(x, s(0'))
321.07/291.61	
321.07/291.61	Types:
321.07/291.61	plus :: 0':s -> 0':s -> 0':s
321.07/291.61	0' :: 0':s
321.07/291.61	s :: 0':s -> 0':s
321.07/291.61	times :: 0':s -> 0':s -> 0':s
321.07/291.61	p :: 0':s -> 0':s
321.07/291.61	minus :: 0':s -> 0':s -> 0':s
321.07/291.61	isZero :: 0':s -> true:false
321.07/291.61	true :: true:false
321.07/291.61	false :: true:false
321.07/291.61	facIter :: 0':s -> 0':s -> 0':s
321.07/291.61	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
321.07/291.61	factorial :: 0':s -> 0':s
321.07/291.61	hole_0':s1_0 :: 0':s
321.07/291.61	hole_true:false2_0 :: true:false
321.07/291.61	gen_0':s3_0 :: Nat -> 0':s
321.07/291.61	
321.07/291.61	
321.07/291.61	Generator Equations:
321.07/291.61	gen_0':s3_0(0) <=> 0'
321.07/291.61	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
321.07/291.61	
321.07/291.61	
321.07/291.61	The following defined symbols remain to be analysed:
321.07/291.61	plus, times, minus, facIter
321.07/291.61	
321.07/291.61	They will be analysed ascendingly in the following order:
321.07/291.61	plus < times
321.07/291.61	times < facIter
321.07/291.61	minus < facIter
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(7) RewriteLemmaProof (LOWER BOUND(ID))
321.07/291.61	Proved the following rewrite lemma:
321.07/291.61	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)
321.07/291.61	
321.07/291.61	Induction Base:
321.07/291.61	plus(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
321.07/291.61	gen_0':s3_0(b)
321.07/291.61	
321.07/291.61	Induction Step:
321.07/291.61	plus(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
321.07/291.61	s(plus(gen_0':s3_0(n5_0), gen_0':s3_0(b))) ->_IH
321.07/291.61	s(gen_0':s3_0(+(b, c6_0)))
321.07/291.61	
321.07/291.61	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(8)
321.07/291.61	Complex Obligation (BEST)
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(9)
321.07/291.61	Obligation:
321.07/291.61	Proved the lower bound n^1 for the following obligation:
321.07/291.61	
321.07/291.61	TRS:
321.07/291.61	Rules:
321.07/291.61	plus(0', x) -> x
321.07/291.61	plus(s(x), y) -> s(plus(x, y))
321.07/291.61	times(0', y) -> 0'
321.07/291.61	times(s(x), y) -> plus(y, times(x, y))
321.07/291.61	p(s(x)) -> x
321.07/291.61	p(0') -> 0'
321.07/291.61	minus(x, 0') -> x
321.07/291.61	minus(0', x) -> 0'
321.07/291.61	minus(x, s(y)) -> p(minus(x, y))
321.07/291.61	isZero(0') -> true
321.07/291.61	isZero(s(x)) -> false
321.07/291.61	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
321.07/291.61	if(true, x, y, z) -> y
321.07/291.61	if(false, x, y, z) -> facIter(x, z)
321.07/291.61	factorial(x) -> facIter(x, s(0'))
321.07/291.61	
321.07/291.61	Types:
321.07/291.61	plus :: 0':s -> 0':s -> 0':s
321.07/291.61	0' :: 0':s
321.07/291.61	s :: 0':s -> 0':s
321.07/291.61	times :: 0':s -> 0':s -> 0':s
321.07/291.61	p :: 0':s -> 0':s
321.07/291.61	minus :: 0':s -> 0':s -> 0':s
321.07/291.61	isZero :: 0':s -> true:false
321.07/291.61	true :: true:false
321.07/291.61	false :: true:false
321.07/291.61	facIter :: 0':s -> 0':s -> 0':s
321.07/291.61	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
321.07/291.61	factorial :: 0':s -> 0':s
321.07/291.61	hole_0':s1_0 :: 0':s
321.07/291.61	hole_true:false2_0 :: true:false
321.07/291.61	gen_0':s3_0 :: Nat -> 0':s
321.07/291.61	
321.07/291.61	
321.07/291.61	Generator Equations:
321.07/291.61	gen_0':s3_0(0) <=> 0'
321.07/291.61	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
321.07/291.61	
321.07/291.61	
321.07/291.61	The following defined symbols remain to be analysed:
321.07/291.61	plus, times, minus, facIter
321.07/291.61	
321.07/291.61	They will be analysed ascendingly in the following order:
321.07/291.61	plus < times
321.07/291.61	times < facIter
321.07/291.61	minus < facIter
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(10) LowerBoundPropagationProof (FINISHED)
321.07/291.61	Propagated lower bound.
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(11)
321.07/291.61	BOUNDS(n^1, INF)
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(12)
321.07/291.61	Obligation:
321.07/291.61	TRS:
321.07/291.61	Rules:
321.07/291.61	plus(0', x) -> x
321.07/291.61	plus(s(x), y) -> s(plus(x, y))
321.07/291.61	times(0', y) -> 0'
321.07/291.61	times(s(x), y) -> plus(y, times(x, y))
321.07/291.61	p(s(x)) -> x
321.07/291.61	p(0') -> 0'
321.07/291.61	minus(x, 0') -> x
321.07/291.61	minus(0', x) -> 0'
321.07/291.61	minus(x, s(y)) -> p(minus(x, y))
321.07/291.61	isZero(0') -> true
321.07/291.61	isZero(s(x)) -> false
321.07/291.61	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
321.07/291.61	if(true, x, y, z) -> y
321.07/291.61	if(false, x, y, z) -> facIter(x, z)
321.07/291.61	factorial(x) -> facIter(x, s(0'))
321.07/291.61	
321.07/291.61	Types:
321.07/291.61	plus :: 0':s -> 0':s -> 0':s
321.07/291.61	0' :: 0':s
321.07/291.61	s :: 0':s -> 0':s
321.07/291.61	times :: 0':s -> 0':s -> 0':s
321.07/291.61	p :: 0':s -> 0':s
321.07/291.61	minus :: 0':s -> 0':s -> 0':s
321.07/291.61	isZero :: 0':s -> true:false
321.07/291.61	true :: true:false
321.07/291.61	false :: true:false
321.07/291.61	facIter :: 0':s -> 0':s -> 0':s
321.07/291.61	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
321.07/291.61	factorial :: 0':s -> 0':s
321.07/291.61	hole_0':s1_0 :: 0':s
321.07/291.61	hole_true:false2_0 :: true:false
321.07/291.61	gen_0':s3_0 :: Nat -> 0':s
321.07/291.61	
321.07/291.61	
321.07/291.61	Lemmas:
321.07/291.61	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)
321.07/291.61	
321.07/291.61	
321.07/291.61	Generator Equations:
321.07/291.61	gen_0':s3_0(0) <=> 0'
321.07/291.61	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
321.07/291.61	
321.07/291.61	
321.07/291.61	The following defined symbols remain to be analysed:
321.07/291.61	times, minus, facIter
321.07/291.61	
321.07/291.61	They will be analysed ascendingly in the following order:
321.07/291.61	times < facIter
321.07/291.61	minus < facIter
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(13) RewriteLemmaProof (LOWER BOUND(ID))
321.07/291.61	Proved the following rewrite lemma:
321.07/291.61	times(gen_0':s3_0(n602_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n602_0, b)), rt in Omega(1 + b*n602_0 + n602_0)
321.07/291.61	
321.07/291.61	Induction Base:
321.07/291.61	times(gen_0':s3_0(0), gen_0':s3_0(b)) ->_R^Omega(1)
321.07/291.61	0'
321.07/291.61	
321.07/291.61	Induction Step:
321.07/291.61	times(gen_0':s3_0(+(n602_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
321.07/291.61	plus(gen_0':s3_0(b), times(gen_0':s3_0(n602_0), gen_0':s3_0(b))) ->_IH
321.07/291.61	plus(gen_0':s3_0(b), gen_0':s3_0(*(c603_0, b))) ->_L^Omega(1 + b)
321.07/291.61	gen_0':s3_0(+(b, *(n602_0, b)))
321.07/291.61	
321.07/291.61	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(14)
321.07/291.61	Complex Obligation (BEST)
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(15)
321.07/291.61	Obligation:
321.07/291.61	Proved the lower bound n^2 for the following obligation:
321.07/291.61	
321.07/291.61	TRS:
321.07/291.61	Rules:
321.07/291.61	plus(0', x) -> x
321.07/291.61	plus(s(x), y) -> s(plus(x, y))
321.07/291.61	times(0', y) -> 0'
321.07/291.61	times(s(x), y) -> plus(y, times(x, y))
321.07/291.61	p(s(x)) -> x
321.07/291.61	p(0') -> 0'
321.07/291.61	minus(x, 0') -> x
321.07/291.61	minus(0', x) -> 0'
321.07/291.61	minus(x, s(y)) -> p(minus(x, y))
321.07/291.61	isZero(0') -> true
321.07/291.61	isZero(s(x)) -> false
321.07/291.61	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
321.07/291.61	if(true, x, y, z) -> y
321.07/291.61	if(false, x, y, z) -> facIter(x, z)
321.07/291.61	factorial(x) -> facIter(x, s(0'))
321.07/291.61	
321.07/291.61	Types:
321.07/291.61	plus :: 0':s -> 0':s -> 0':s
321.07/291.61	0' :: 0':s
321.07/291.61	s :: 0':s -> 0':s
321.07/291.61	times :: 0':s -> 0':s -> 0':s
321.07/291.61	p :: 0':s -> 0':s
321.07/291.61	minus :: 0':s -> 0':s -> 0':s
321.07/291.61	isZero :: 0':s -> true:false
321.07/291.61	true :: true:false
321.07/291.61	false :: true:false
321.07/291.61	facIter :: 0':s -> 0':s -> 0':s
321.07/291.61	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
321.07/291.61	factorial :: 0':s -> 0':s
321.07/291.61	hole_0':s1_0 :: 0':s
321.07/291.61	hole_true:false2_0 :: true:false
321.07/291.61	gen_0':s3_0 :: Nat -> 0':s
321.07/291.61	
321.07/291.61	
321.07/291.61	Lemmas:
321.07/291.61	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)
321.07/291.61	
321.07/291.61	
321.07/291.61	Generator Equations:
321.07/291.61	gen_0':s3_0(0) <=> 0'
321.07/291.61	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
321.07/291.61	
321.07/291.61	
321.07/291.61	The following defined symbols remain to be analysed:
321.07/291.61	times, minus, facIter
321.07/291.61	
321.07/291.61	They will be analysed ascendingly in the following order:
321.07/291.61	times < facIter
321.07/291.61	minus < facIter
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(16) LowerBoundPropagationProof (FINISHED)
321.07/291.61	Propagated lower bound.
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(17)
321.07/291.61	BOUNDS(n^2, INF)
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(18)
321.07/291.61	Obligation:
321.07/291.61	TRS:
321.07/291.61	Rules:
321.07/291.61	plus(0', x) -> x
321.07/291.61	plus(s(x), y) -> s(plus(x, y))
321.07/291.61	times(0', y) -> 0'
321.07/291.61	times(s(x), y) -> plus(y, times(x, y))
321.07/291.61	p(s(x)) -> x
321.07/291.61	p(0') -> 0'
321.07/291.61	minus(x, 0') -> x
321.07/291.61	minus(0', x) -> 0'
321.07/291.61	minus(x, s(y)) -> p(minus(x, y))
321.07/291.61	isZero(0') -> true
321.07/291.61	isZero(s(x)) -> false
321.07/291.61	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
321.07/291.61	if(true, x, y, z) -> y
321.07/291.61	if(false, x, y, z) -> facIter(x, z)
321.07/291.61	factorial(x) -> facIter(x, s(0'))
321.07/291.61	
321.07/291.61	Types:
321.07/291.61	plus :: 0':s -> 0':s -> 0':s
321.07/291.61	0' :: 0':s
321.07/291.61	s :: 0':s -> 0':s
321.07/291.61	times :: 0':s -> 0':s -> 0':s
321.07/291.61	p :: 0':s -> 0':s
321.07/291.61	minus :: 0':s -> 0':s -> 0':s
321.07/291.61	isZero :: 0':s -> true:false
321.07/291.61	true :: true:false
321.07/291.61	false :: true:false
321.07/291.61	facIter :: 0':s -> 0':s -> 0':s
321.07/291.61	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
321.07/291.61	factorial :: 0':s -> 0':s
321.07/291.61	hole_0':s1_0 :: 0':s
321.07/291.61	hole_true:false2_0 :: true:false
321.07/291.61	gen_0':s3_0 :: Nat -> 0':s
321.07/291.61	
321.07/291.61	
321.07/291.61	Lemmas:
321.07/291.61	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)
321.07/291.61	times(gen_0':s3_0(n602_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n602_0, b)), rt in Omega(1 + b*n602_0 + n602_0)
321.07/291.61	
321.07/291.61	
321.07/291.61	Generator Equations:
321.07/291.61	gen_0':s3_0(0) <=> 0'
321.07/291.61	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
321.07/291.61	
321.07/291.61	
321.07/291.61	The following defined symbols remain to be analysed:
321.07/291.61	minus, facIter
321.07/291.61	
321.07/291.61	They will be analysed ascendingly in the following order:
321.07/291.61	minus < facIter
321.07/291.61	
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(19) RewriteLemmaProof (LOWER BOUND(ID))
321.07/291.61	Proved the following rewrite lemma:
321.07/291.61	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1362_0))) -> *4_0, rt in Omega(n1362_0)
321.07/291.61	
321.07/291.61	Induction Base:
321.07/291.61	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, 0)))
321.07/291.61	
321.07/291.61	Induction Step:
321.07/291.61	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, +(n1362_0, 1)))) ->_R^Omega(1)
321.07/291.61	p(minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1362_0)))) ->_IH
321.07/291.61	p(*4_0)
321.07/291.61	
321.07/291.61	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(20)
321.07/291.61	Obligation:
321.07/291.61	TRS:
321.07/291.61	Rules:
321.07/291.61	plus(0', x) -> x
321.07/291.61	plus(s(x), y) -> s(plus(x, y))
321.07/291.61	times(0', y) -> 0'
321.07/291.61	times(s(x), y) -> plus(y, times(x, y))
321.07/291.61	p(s(x)) -> x
321.07/291.61	p(0') -> 0'
321.07/291.61	minus(x, 0') -> x
321.07/291.61	minus(0', x) -> 0'
321.07/291.61	minus(x, s(y)) -> p(minus(x, y))
321.07/291.61	isZero(0') -> true
321.07/291.61	isZero(s(x)) -> false
321.07/291.61	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
321.07/291.61	if(true, x, y, z) -> y
321.07/291.61	if(false, x, y, z) -> facIter(x, z)
321.07/291.61	factorial(x) -> facIter(x, s(0'))
321.07/291.61	
321.07/291.61	Types:
321.07/291.61	plus :: 0':s -> 0':s -> 0':s
321.07/291.61	0' :: 0':s
321.07/291.61	s :: 0':s -> 0':s
321.07/291.61	times :: 0':s -> 0':s -> 0':s
321.07/291.61	p :: 0':s -> 0':s
321.07/291.61	minus :: 0':s -> 0':s -> 0':s
321.07/291.61	isZero :: 0':s -> true:false
321.07/291.61	true :: true:false
321.07/291.61	false :: true:false
321.07/291.61	facIter :: 0':s -> 0':s -> 0':s
321.07/291.61	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
321.07/291.61	factorial :: 0':s -> 0':s
321.07/291.61	hole_0':s1_0 :: 0':s
321.07/291.61	hole_true:false2_0 :: true:false
321.07/291.61	gen_0':s3_0 :: Nat -> 0':s
321.07/291.61	
321.07/291.61	
321.07/291.61	Lemmas:
321.07/291.61	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)
321.07/291.61	times(gen_0':s3_0(n602_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n602_0, b)), rt in Omega(1 + b*n602_0 + n602_0)
321.07/291.61	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1362_0))) -> *4_0, rt in Omega(n1362_0)
321.07/291.61	
321.07/291.61	
321.07/291.61	Generator Equations:
321.07/291.61	gen_0':s3_0(0) <=> 0'
321.07/291.61	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
321.07/291.61	
321.07/291.61	
321.07/291.61	The following defined symbols remain to be analysed:
321.07/291.61	facIter
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(21) RewriteLemmaProof (LOWER BOUND(ID))
321.07/291.61	Proved the following rewrite lemma:
321.07/291.61	facIter(gen_0':s3_0(n4187_0), gen_0':s3_0(b)) -> *4_0, rt in Omega(n4187_0 + n4187_0^2 + n4187_0^3)
321.07/291.61	
321.07/291.61	Induction Base:
321.07/291.61	facIter(gen_0':s3_0(0), gen_0':s3_0(b))
321.07/291.61	
321.07/291.61	Induction Step:
321.07/291.61	facIter(gen_0':s3_0(+(n4187_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
321.07/291.61	if(isZero(gen_0':s3_0(+(n4187_0, 1))), minus(gen_0':s3_0(+(n4187_0, 1)), s(0')), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(n4187_0, 1)))) ->_R^Omega(1)
321.07/291.61	if(false, minus(gen_0':s3_0(+(1, n4187_0)), s(0')), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n4187_0)))) ->_R^Omega(1)
321.07/291.61	if(false, p(minus(gen_0':s3_0(+(1, n4187_0)), 0')), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n4187_0)))) ->_R^Omega(1)
321.07/291.61	if(false, p(gen_0':s3_0(+(1, n4187_0))), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n4187_0)))) ->_R^Omega(1)
321.07/291.61	if(false, gen_0':s3_0(n4187_0), gen_0':s3_0(b), times(gen_0':s3_0(b), gen_0':s3_0(+(1, n4187_0)))) ->_L^Omega(3 + 3*n4187_0 + n4187_0^2)
321.07/291.61	if(false, gen_0':s3_0(n4187_0), gen_0':s3_0(+(1, n4187_0)), gen_0':s3_0(*(b, +(1, n4187_0)))) ->_R^Omega(1)
321.07/291.61	facIter(gen_0':s3_0(n4187_0), gen_0':s3_0(+(b, *(b, n4187_0)))) ->_IH
321.07/291.61	*4_0
321.07/291.61	
321.07/291.61	We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(22)
321.07/291.61	Obligation:
321.07/291.61	Proved the lower bound n^3 for the following obligation:
321.07/291.61	
321.07/291.61	TRS:
321.07/291.61	Rules:
321.07/291.61	plus(0', x) -> x
321.07/291.61	plus(s(x), y) -> s(plus(x, y))
321.07/291.61	times(0', y) -> 0'
321.07/291.61	times(s(x), y) -> plus(y, times(x, y))
321.07/291.61	p(s(x)) -> x
321.07/291.61	p(0') -> 0'
321.07/291.61	minus(x, 0') -> x
321.07/291.61	minus(0', x) -> 0'
321.07/291.61	minus(x, s(y)) -> p(minus(x, y))
321.07/291.61	isZero(0') -> true
321.07/291.61	isZero(s(x)) -> false
321.07/291.61	facIter(x, y) -> if(isZero(x), minus(x, s(0')), y, times(y, x))
321.07/291.61	if(true, x, y, z) -> y
321.07/291.61	if(false, x, y, z) -> facIter(x, z)
321.07/291.61	factorial(x) -> facIter(x, s(0'))
321.07/291.61	
321.07/291.61	Types:
321.07/291.61	plus :: 0':s -> 0':s -> 0':s
321.07/291.61	0' :: 0':s
321.07/291.61	s :: 0':s -> 0':s
321.07/291.61	times :: 0':s -> 0':s -> 0':s
321.07/291.61	p :: 0':s -> 0':s
321.07/291.61	minus :: 0':s -> 0':s -> 0':s
321.07/291.61	isZero :: 0':s -> true:false
321.07/291.61	true :: true:false
321.07/291.61	false :: true:false
321.07/291.61	facIter :: 0':s -> 0':s -> 0':s
321.07/291.61	if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
321.07/291.61	factorial :: 0':s -> 0':s
321.07/291.61	hole_0':s1_0 :: 0':s
321.07/291.61	hole_true:false2_0 :: true:false
321.07/291.61	gen_0':s3_0 :: Nat -> 0':s
321.07/291.61	
321.07/291.61	
321.07/291.61	Lemmas:
321.07/291.61	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)
321.07/291.61	times(gen_0':s3_0(n602_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n602_0, b)), rt in Omega(1 + b*n602_0 + n602_0)
321.07/291.61	minus(gen_0':s3_0(a), gen_0':s3_0(+(1, n1362_0))) -> *4_0, rt in Omega(n1362_0)
321.07/291.61	
321.07/291.61	
321.07/291.61	Generator Equations:
321.07/291.61	gen_0':s3_0(0) <=> 0'
321.07/291.61	gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
321.07/291.61	
321.07/291.61	
321.07/291.61	The following defined symbols remain to be analysed:
321.07/291.61	facIter
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(23) LowerBoundPropagationProof (FINISHED)
321.07/291.61	Propagated lower bound.
321.07/291.61	----------------------------------------
321.07/291.61	
321.07/291.61	(24)
321.07/291.61	BOUNDS(n^3, INF)
321.07/291.64	EOF