1122.38/291.48	WORST_CASE(Omega(n^4), ?)
1122.38/291.52	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1122.38/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^4, INF).
1122.38/291.52	
1122.38/291.52	(0) CpxTRS
1122.38/291.52	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1122.38/291.52	(2) CpxTRS
1122.38/291.52	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1122.38/291.52	(4) typed CpxTrs
1122.38/291.52	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1122.38/291.52	(6) typed CpxTrs
1122.38/291.52	(7) RewriteLemmaProof [LOWER BOUND(ID), 289 ms]
1122.38/291.52	(8) BEST
1122.38/291.52	    (9) proven lower bound
1122.38/291.52	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1122.38/291.52	        (11) BOUNDS(n^1, INF)
1122.38/291.52	    (12) typed CpxTrs
1122.38/291.52	        (13) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
1122.38/291.52	        (14) BEST
1122.38/291.52	            (15) proven lower bound
1122.38/291.52	                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
1122.38/291.52	                (17) BOUNDS(n^2, INF)
1122.38/291.52	            (18) typed CpxTrs
1122.38/291.52	                (19) RewriteLemmaProof [LOWER BOUND(ID), 38 ms]
1122.38/291.52	                (20) BEST
1122.38/291.52	                    (21) proven lower bound
1122.38/291.52	                        (22) LowerBoundPropagationProof [FINISHED, 0 ms]
1122.38/291.52	                        (23) BOUNDS(n^3, INF)
1122.38/291.52	                    (24) typed CpxTrs
1122.38/291.52	                        (25) RewriteLemmaProof [LOWER BOUND(ID), 1326 ms]
1122.38/291.52	                        (26) proven lower bound
1122.38/291.52	                        (27) LowerBoundPropagationProof [FINISHED, 0 ms]
1122.38/291.52	                        (28) BOUNDS(n^4, INF)
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(0)
1122.38/291.52	Obligation:
1122.38/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^4, INF).
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	The TRS R consists of the following rules:
1122.38/291.52	
1122.38/291.52	   plus(0, x) -> x
1122.38/291.52	   plus(s(x), y) -> s(plus(p(s(x)), y))
1122.38/291.52	   times(0, y) -> 0
1122.38/291.52	   times(s(x), y) -> plus(y, times(p(s(x)), y))
1122.38/291.52	   p(s(0)) -> 0
1122.38/291.52	   p(s(s(x))) -> s(p(s(x)))
1122.38/291.52	   fac(0, x) -> x
1122.38/291.52	   fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
1122.38/291.52	   factorial(x) -> fac(x, s(0))
1122.38/291.52	
1122.38/291.52	S is empty.
1122.38/291.52	Rewrite Strategy: INNERMOST
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1122.38/291.52	Renamed function symbols to avoid clashes with predefined symbol.
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(2)
1122.38/291.52	Obligation:
1122.38/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^4, INF).
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	The TRS R consists of the following rules:
1122.38/291.52	
1122.38/291.52	   plus(0', x) -> x
1122.38/291.52	   plus(s(x), y) -> s(plus(p(s(x)), y))
1122.38/291.52	   times(0', y) -> 0'
1122.38/291.52	   times(s(x), y) -> plus(y, times(p(s(x)), y))
1122.38/291.52	   p(s(0')) -> 0'
1122.38/291.52	   p(s(s(x))) -> s(p(s(x)))
1122.38/291.52	   fac(0', x) -> x
1122.38/291.52	   fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
1122.38/291.52	   factorial(x) -> fac(x, s(0'))
1122.38/291.52	
1122.38/291.52	S is empty.
1122.38/291.52	Rewrite Strategy: INNERMOST
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1122.38/291.52	Infered types.
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(4)
1122.38/291.52	Obligation:
1122.38/291.52	Innermost TRS:
1122.38/291.52	Rules:
1122.38/291.52	plus(0', x) -> x
1122.38/291.52	plus(s(x), y) -> s(plus(p(s(x)), y))
1122.38/291.52	times(0', y) -> 0'
1122.38/291.52	times(s(x), y) -> plus(y, times(p(s(x)), y))
1122.38/291.52	p(s(0')) -> 0'
1122.38/291.52	p(s(s(x))) -> s(p(s(x)))
1122.38/291.52	fac(0', x) -> x
1122.38/291.52	fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
1122.38/291.52	factorial(x) -> fac(x, s(0'))
1122.38/291.52	
1122.38/291.52	Types:
1122.38/291.52	plus :: 0':s -> 0':s -> 0':s
1122.38/291.52	0' :: 0':s
1122.38/291.52	s :: 0':s -> 0':s
1122.38/291.52	p :: 0':s -> 0':s
1122.38/291.52	times :: 0':s -> 0':s -> 0':s
1122.38/291.52	fac :: 0':s -> 0':s -> 0':s
1122.38/291.52	factorial :: 0':s -> 0':s
1122.38/291.52	hole_0':s1_0 :: 0':s
1122.38/291.52	gen_0':s2_0 :: Nat -> 0':s
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(5) OrderProof (LOWER BOUND(ID))
1122.38/291.52	Heuristically decided to analyse the following defined symbols:
1122.38/291.52	plus, p, times, fac
1122.38/291.52	
1122.38/291.52	They will be analysed ascendingly in the following order:
1122.38/291.52	p < plus
1122.38/291.52	plus < times
1122.38/291.52	p < times
1122.38/291.52	p < fac
1122.38/291.52	times < fac
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(6)
1122.38/291.52	Obligation:
1122.38/291.52	Innermost TRS:
1122.38/291.52	Rules:
1122.38/291.52	plus(0', x) -> x
1122.38/291.52	plus(s(x), y) -> s(plus(p(s(x)), y))
1122.38/291.52	times(0', y) -> 0'
1122.38/291.52	times(s(x), y) -> plus(y, times(p(s(x)), y))
1122.38/291.52	p(s(0')) -> 0'
1122.38/291.52	p(s(s(x))) -> s(p(s(x)))
1122.38/291.52	fac(0', x) -> x
1122.38/291.52	fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
1122.38/291.52	factorial(x) -> fac(x, s(0'))
1122.38/291.52	
1122.38/291.52	Types:
1122.38/291.52	plus :: 0':s -> 0':s -> 0':s
1122.38/291.52	0' :: 0':s
1122.38/291.52	s :: 0':s -> 0':s
1122.38/291.52	p :: 0':s -> 0':s
1122.38/291.52	times :: 0':s -> 0':s -> 0':s
1122.38/291.52	fac :: 0':s -> 0':s -> 0':s
1122.38/291.52	factorial :: 0':s -> 0':s
1122.38/291.52	hole_0':s1_0 :: 0':s
1122.38/291.52	gen_0':s2_0 :: Nat -> 0':s
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Generator Equations:
1122.38/291.52	gen_0':s2_0(0) <=> 0'
1122.38/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	The following defined symbols remain to be analysed:
1122.38/291.52	p, plus, times, fac
1122.38/291.52	
1122.38/291.52	They will be analysed ascendingly in the following order:
1122.38/291.52	p < plus
1122.38/291.52	plus < times
1122.38/291.52	p < times
1122.38/291.52	p < fac
1122.38/291.52	times < fac
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(7) RewriteLemmaProof (LOWER BOUND(ID))
1122.38/291.52	Proved the following rewrite lemma:
1122.38/291.52	p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
1122.38/291.52	
1122.38/291.52	Induction Base:
1122.38/291.52	p(gen_0':s2_0(+(1, 0))) ->_R^Omega(1)
1122.38/291.52	0'
1122.38/291.52	
1122.38/291.52	Induction Step:
1122.38/291.52	p(gen_0':s2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
1122.38/291.52	s(p(s(gen_0':s2_0(n4_0)))) ->_IH
1122.38/291.52	s(gen_0':s2_0(c5_0))
1122.38/291.52	
1122.38/291.52	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(8)
1122.38/291.52	Complex Obligation (BEST)
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(9)
1122.38/291.52	Obligation:
1122.38/291.52	Proved the lower bound n^1 for the following obligation:
1122.38/291.52	
1122.38/291.52	Innermost TRS:
1122.38/291.52	Rules:
1122.38/291.52	plus(0', x) -> x
1122.38/291.52	plus(s(x), y) -> s(plus(p(s(x)), y))
1122.38/291.52	times(0', y) -> 0'
1122.38/291.52	times(s(x), y) -> plus(y, times(p(s(x)), y))
1122.38/291.52	p(s(0')) -> 0'
1122.38/291.52	p(s(s(x))) -> s(p(s(x)))
1122.38/291.52	fac(0', x) -> x
1122.38/291.52	fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
1122.38/291.52	factorial(x) -> fac(x, s(0'))
1122.38/291.52	
1122.38/291.52	Types:
1122.38/291.52	plus :: 0':s -> 0':s -> 0':s
1122.38/291.52	0' :: 0':s
1122.38/291.52	s :: 0':s -> 0':s
1122.38/291.52	p :: 0':s -> 0':s
1122.38/291.52	times :: 0':s -> 0':s -> 0':s
1122.38/291.52	fac :: 0':s -> 0':s -> 0':s
1122.38/291.52	factorial :: 0':s -> 0':s
1122.38/291.52	hole_0':s1_0 :: 0':s
1122.38/291.52	gen_0':s2_0 :: Nat -> 0':s
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Generator Equations:
1122.38/291.52	gen_0':s2_0(0) <=> 0'
1122.38/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	The following defined symbols remain to be analysed:
1122.38/291.52	p, plus, times, fac
1122.38/291.52	
1122.38/291.52	They will be analysed ascendingly in the following order:
1122.38/291.52	p < plus
1122.38/291.52	plus < times
1122.38/291.52	p < times
1122.38/291.52	p < fac
1122.38/291.52	times < fac
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(10) LowerBoundPropagationProof (FINISHED)
1122.38/291.52	Propagated lower bound.
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(11)
1122.38/291.52	BOUNDS(n^1, INF)
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(12)
1122.38/291.52	Obligation:
1122.38/291.52	Innermost TRS:
1122.38/291.52	Rules:
1122.38/291.52	plus(0', x) -> x
1122.38/291.52	plus(s(x), y) -> s(plus(p(s(x)), y))
1122.38/291.52	times(0', y) -> 0'
1122.38/291.52	times(s(x), y) -> plus(y, times(p(s(x)), y))
1122.38/291.52	p(s(0')) -> 0'
1122.38/291.52	p(s(s(x))) -> s(p(s(x)))
1122.38/291.52	fac(0', x) -> x
1122.38/291.52	fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
1122.38/291.52	factorial(x) -> fac(x, s(0'))
1122.38/291.52	
1122.38/291.52	Types:
1122.38/291.52	plus :: 0':s -> 0':s -> 0':s
1122.38/291.52	0' :: 0':s
1122.38/291.52	s :: 0':s -> 0':s
1122.38/291.52	p :: 0':s -> 0':s
1122.38/291.52	times :: 0':s -> 0':s -> 0':s
1122.38/291.52	fac :: 0':s -> 0':s -> 0':s
1122.38/291.52	factorial :: 0':s -> 0':s
1122.38/291.52	hole_0':s1_0 :: 0':s
1122.38/291.52	gen_0':s2_0 :: Nat -> 0':s
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Lemmas:
1122.38/291.52	p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Generator Equations:
1122.38/291.52	gen_0':s2_0(0) <=> 0'
1122.38/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	The following defined symbols remain to be analysed:
1122.38/291.52	plus, times, fac
1122.38/291.52	
1122.38/291.52	They will be analysed ascendingly in the following order:
1122.38/291.52	plus < times
1122.38/291.52	times < fac
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(13) RewriteLemmaProof (LOWER BOUND(ID))
1122.38/291.52	Proved the following rewrite lemma:
1122.38/291.52	plus(gen_0':s2_0(n229_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n229_0, b)), rt in Omega(1 + n229_0 + n229_0^2)
1122.38/291.52	
1122.38/291.52	Induction Base:
1122.38/291.52	plus(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
1122.38/291.52	gen_0':s2_0(b)
1122.38/291.52	
1122.38/291.52	Induction Step:
1122.38/291.52	plus(gen_0':s2_0(+(n229_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
1122.38/291.52	s(plus(p(s(gen_0':s2_0(n229_0))), gen_0':s2_0(b))) ->_L^Omega(1 + n229_0)
1122.38/291.52	s(plus(gen_0':s2_0(n229_0), gen_0':s2_0(b))) ->_IH
1122.38/291.52	s(gen_0':s2_0(+(b, c230_0)))
1122.38/291.52	
1122.38/291.52	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(14)
1122.38/291.52	Complex Obligation (BEST)
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(15)
1122.38/291.52	Obligation:
1122.38/291.52	Proved the lower bound n^2 for the following obligation:
1122.38/291.52	
1122.38/291.52	Innermost TRS:
1122.38/291.52	Rules:
1122.38/291.52	plus(0', x) -> x
1122.38/291.52	plus(s(x), y) -> s(plus(p(s(x)), y))
1122.38/291.52	times(0', y) -> 0'
1122.38/291.52	times(s(x), y) -> plus(y, times(p(s(x)), y))
1122.38/291.52	p(s(0')) -> 0'
1122.38/291.52	p(s(s(x))) -> s(p(s(x)))
1122.38/291.52	fac(0', x) -> x
1122.38/291.52	fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
1122.38/291.52	factorial(x) -> fac(x, s(0'))
1122.38/291.52	
1122.38/291.52	Types:
1122.38/291.52	plus :: 0':s -> 0':s -> 0':s
1122.38/291.52	0' :: 0':s
1122.38/291.52	s :: 0':s -> 0':s
1122.38/291.52	p :: 0':s -> 0':s
1122.38/291.52	times :: 0':s -> 0':s -> 0':s
1122.38/291.52	fac :: 0':s -> 0':s -> 0':s
1122.38/291.52	factorial :: 0':s -> 0':s
1122.38/291.52	hole_0':s1_0 :: 0':s
1122.38/291.52	gen_0':s2_0 :: Nat -> 0':s
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Lemmas:
1122.38/291.52	p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Generator Equations:
1122.38/291.52	gen_0':s2_0(0) <=> 0'
1122.38/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	The following defined symbols remain to be analysed:
1122.38/291.52	plus, times, fac
1122.38/291.52	
1122.38/291.52	They will be analysed ascendingly in the following order:
1122.38/291.52	plus < times
1122.38/291.52	times < fac
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(16) LowerBoundPropagationProof (FINISHED)
1122.38/291.52	Propagated lower bound.
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(17)
1122.38/291.52	BOUNDS(n^2, INF)
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(18)
1122.38/291.52	Obligation:
1122.38/291.52	Innermost TRS:
1122.38/291.52	Rules:
1122.38/291.52	plus(0', x) -> x
1122.38/291.52	plus(s(x), y) -> s(plus(p(s(x)), y))
1122.38/291.52	times(0', y) -> 0'
1122.38/291.52	times(s(x), y) -> plus(y, times(p(s(x)), y))
1122.38/291.52	p(s(0')) -> 0'
1122.38/291.52	p(s(s(x))) -> s(p(s(x)))
1122.38/291.52	fac(0', x) -> x
1122.38/291.52	fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
1122.38/291.52	factorial(x) -> fac(x, s(0'))
1122.38/291.52	
1122.38/291.52	Types:
1122.38/291.52	plus :: 0':s -> 0':s -> 0':s
1122.38/291.52	0' :: 0':s
1122.38/291.52	s :: 0':s -> 0':s
1122.38/291.52	p :: 0':s -> 0':s
1122.38/291.52	times :: 0':s -> 0':s -> 0':s
1122.38/291.52	fac :: 0':s -> 0':s -> 0':s
1122.38/291.52	factorial :: 0':s -> 0':s
1122.38/291.52	hole_0':s1_0 :: 0':s
1122.38/291.52	gen_0':s2_0 :: Nat -> 0':s
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Lemmas:
1122.38/291.52	p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
1122.38/291.52	plus(gen_0':s2_0(n229_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n229_0, b)), rt in Omega(1 + n229_0 + n229_0^2)
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Generator Equations:
1122.38/291.52	gen_0':s2_0(0) <=> 0'
1122.38/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	The following defined symbols remain to be analysed:
1122.38/291.52	times, fac
1122.38/291.52	
1122.38/291.52	They will be analysed ascendingly in the following order:
1122.38/291.52	times < fac
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(19) RewriteLemmaProof (LOWER BOUND(ID))
1122.38/291.52	Proved the following rewrite lemma:
1122.38/291.52	times(gen_0':s2_0(n695_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n695_0, b)), rt in Omega(1 + b*n695_0 + b^2*n695_0 + n695_0 + n695_0^2)
1122.38/291.52	
1122.38/291.52	Induction Base:
1122.38/291.52	times(gen_0':s2_0(0), gen_0':s2_0(b)) ->_R^Omega(1)
1122.38/291.52	0'
1122.38/291.52	
1122.38/291.52	Induction Step:
1122.38/291.52	times(gen_0':s2_0(+(n695_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
1122.38/291.52	plus(gen_0':s2_0(b), times(p(s(gen_0':s2_0(n695_0))), gen_0':s2_0(b))) ->_L^Omega(1 + n695_0)
1122.38/291.52	plus(gen_0':s2_0(b), times(gen_0':s2_0(n695_0), gen_0':s2_0(b))) ->_IH
1122.38/291.52	plus(gen_0':s2_0(b), gen_0':s2_0(*(c696_0, b))) ->_L^Omega(1 + b + b^2)
1122.38/291.52	gen_0':s2_0(+(b, *(n695_0, b)))
1122.38/291.52	
1122.38/291.52	We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(20)
1122.38/291.52	Complex Obligation (BEST)
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(21)
1122.38/291.52	Obligation:
1122.38/291.52	Proved the lower bound n^3 for the following obligation:
1122.38/291.52	
1122.38/291.52	Innermost TRS:
1122.38/291.52	Rules:
1122.38/291.52	plus(0', x) -> x
1122.38/291.52	plus(s(x), y) -> s(plus(p(s(x)), y))
1122.38/291.52	times(0', y) -> 0'
1122.38/291.52	times(s(x), y) -> plus(y, times(p(s(x)), y))
1122.38/291.52	p(s(0')) -> 0'
1122.38/291.52	p(s(s(x))) -> s(p(s(x)))
1122.38/291.52	fac(0', x) -> x
1122.38/291.52	fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
1122.38/291.52	factorial(x) -> fac(x, s(0'))
1122.38/291.52	
1122.38/291.52	Types:
1122.38/291.52	plus :: 0':s -> 0':s -> 0':s
1122.38/291.52	0' :: 0':s
1122.38/291.52	s :: 0':s -> 0':s
1122.38/291.52	p :: 0':s -> 0':s
1122.38/291.52	times :: 0':s -> 0':s -> 0':s
1122.38/291.52	fac :: 0':s -> 0':s -> 0':s
1122.38/291.52	factorial :: 0':s -> 0':s
1122.38/291.52	hole_0':s1_0 :: 0':s
1122.38/291.52	gen_0':s2_0 :: Nat -> 0':s
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Lemmas:
1122.38/291.52	p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
1122.38/291.52	plus(gen_0':s2_0(n229_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n229_0, b)), rt in Omega(1 + n229_0 + n229_0^2)
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Generator Equations:
1122.38/291.52	gen_0':s2_0(0) <=> 0'
1122.38/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	The following defined symbols remain to be analysed:
1122.38/291.52	times, fac
1122.38/291.52	
1122.38/291.52	They will be analysed ascendingly in the following order:
1122.38/291.52	times < fac
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(22) LowerBoundPropagationProof (FINISHED)
1122.38/291.52	Propagated lower bound.
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(23)
1122.38/291.52	BOUNDS(n^3, INF)
1122.38/291.52	
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(24)
1122.38/291.52	Obligation:
1122.38/291.52	Innermost TRS:
1122.38/291.52	Rules:
1122.38/291.52	plus(0', x) -> x
1122.38/291.52	plus(s(x), y) -> s(plus(p(s(x)), y))
1122.38/291.52	times(0', y) -> 0'
1122.38/291.52	times(s(x), y) -> plus(y, times(p(s(x)), y))
1122.38/291.52	p(s(0')) -> 0'
1122.38/291.52	p(s(s(x))) -> s(p(s(x)))
1122.38/291.52	fac(0', x) -> x
1122.38/291.52	fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
1122.38/291.52	factorial(x) -> fac(x, s(0'))
1122.38/291.52	
1122.38/291.52	Types:
1122.38/291.52	plus :: 0':s -> 0':s -> 0':s
1122.38/291.52	0' :: 0':s
1122.38/291.52	s :: 0':s -> 0':s
1122.38/291.52	p :: 0':s -> 0':s
1122.38/291.52	times :: 0':s -> 0':s -> 0':s
1122.38/291.52	fac :: 0':s -> 0':s -> 0':s
1122.38/291.52	factorial :: 0':s -> 0':s
1122.38/291.52	hole_0':s1_0 :: 0':s
1122.38/291.52	gen_0':s2_0 :: Nat -> 0':s
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Lemmas:
1122.38/291.52	p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
1122.38/291.52	plus(gen_0':s2_0(n229_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n229_0, b)), rt in Omega(1 + n229_0 + n229_0^2)
1122.38/291.52	times(gen_0':s2_0(n695_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n695_0, b)), rt in Omega(1 + b*n695_0 + b^2*n695_0 + n695_0 + n695_0^2)
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Generator Equations:
1122.38/291.52	gen_0':s2_0(0) <=> 0'
1122.38/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	The following defined symbols remain to be analysed:
1122.38/291.52	fac
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(25) RewriteLemmaProof (LOWER BOUND(ID))
1122.38/291.52	Proved the following rewrite lemma:
1122.38/291.52	fac(gen_0':s2_0(n1347_0), gen_0':s2_0(b)) -> *3_0, rt in Omega(b*n1347_0 + b*n1347_0^2 + b^2*n1347_0 + b^2*n1347_0^2 + n1347_0 + n1347_0^2 + n1347_0^3)
1122.38/291.52	
1122.38/291.52	Induction Base:
1122.38/291.52	fac(gen_0':s2_0(0), gen_0':s2_0(b))
1122.38/291.52	
1122.38/291.52	Induction Step:
1122.38/291.52	fac(gen_0':s2_0(+(n1347_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
1122.38/291.52	fac(p(s(gen_0':s2_0(n1347_0))), times(s(gen_0':s2_0(n1347_0)), gen_0':s2_0(b))) ->_L^Omega(1 + n1347_0)
1122.38/291.52	fac(gen_0':s2_0(n1347_0), times(s(gen_0':s2_0(n1347_0)), gen_0':s2_0(b))) ->_L^Omega(3 + b + b*n1347_0 + b^2 + b^2*n1347_0 + 3*n1347_0 + n1347_0^2)
1122.38/291.52	fac(gen_0':s2_0(n1347_0), gen_0':s2_0(*(+(n1347_0, 1), b))) ->_IH
1122.38/291.52	*3_0
1122.38/291.52	
1122.38/291.52	We have rt in Omega(n^4) and sz in O(n). Thus, we have irc_R in Omega(n^4).
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(26)
1122.38/291.52	Obligation:
1122.38/291.52	Proved the lower bound n^4 for the following obligation:
1122.38/291.52	
1122.38/291.52	Innermost TRS:
1122.38/291.52	Rules:
1122.38/291.52	plus(0', x) -> x
1122.38/291.52	plus(s(x), y) -> s(plus(p(s(x)), y))
1122.38/291.52	times(0', y) -> 0'
1122.38/291.52	times(s(x), y) -> plus(y, times(p(s(x)), y))
1122.38/291.52	p(s(0')) -> 0'
1122.38/291.52	p(s(s(x))) -> s(p(s(x)))
1122.38/291.52	fac(0', x) -> x
1122.38/291.52	fac(s(x), y) -> fac(p(s(x)), times(s(x), y))
1122.38/291.52	factorial(x) -> fac(x, s(0'))
1122.38/291.52	
1122.38/291.52	Types:
1122.38/291.52	plus :: 0':s -> 0':s -> 0':s
1122.38/291.52	0' :: 0':s
1122.38/291.52	s :: 0':s -> 0':s
1122.38/291.52	p :: 0':s -> 0':s
1122.38/291.52	times :: 0':s -> 0':s -> 0':s
1122.38/291.52	fac :: 0':s -> 0':s -> 0':s
1122.38/291.52	factorial :: 0':s -> 0':s
1122.38/291.52	hole_0':s1_0 :: 0':s
1122.38/291.52	gen_0':s2_0 :: Nat -> 0':s
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Lemmas:
1122.38/291.52	p(gen_0':s2_0(+(1, n4_0))) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
1122.38/291.52	plus(gen_0':s2_0(n229_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n229_0, b)), rt in Omega(1 + n229_0 + n229_0^2)
1122.38/291.52	times(gen_0':s2_0(n695_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n695_0, b)), rt in Omega(1 + b*n695_0 + b^2*n695_0 + n695_0 + n695_0^2)
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	Generator Equations:
1122.38/291.52	gen_0':s2_0(0) <=> 0'
1122.38/291.52	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1122.38/291.52	
1122.38/291.52	
1122.38/291.52	The following defined symbols remain to be analysed:
1122.38/291.52	fac
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(27) LowerBoundPropagationProof (FINISHED)
1122.38/291.52	Propagated lower bound.
1122.38/291.52	----------------------------------------
1122.38/291.52	
1122.38/291.52	(28)
1122.38/291.52	BOUNDS(n^4, INF)
1122.80/291.59	EOF