1094.32/291.53	WORST_CASE(Omega(n^1), ?)
1097.00/292.14	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1097.00/292.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1097.00/292.14	
1097.00/292.14	(0) CpxTRS
1097.00/292.14	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1097.00/292.14	(2) CpxTRS
1097.00/292.14	(3) SlicingProof [LOWER BOUND(ID), 0 ms]
1097.00/292.14	(4) CpxTRS
1097.00/292.14	(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1097.00/292.14	(6) typed CpxTrs
1097.00/292.14	(7) OrderProof [LOWER BOUND(ID), 0 ms]
1097.00/292.14	(8) typed CpxTrs
1097.00/292.14	(9) RewriteLemmaProof [LOWER BOUND(ID), 296 ms]
1097.00/292.14	(10) BEST
1097.00/292.14	    (11) proven lower bound
1097.00/292.14	        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
1097.00/292.14	        (13) BOUNDS(n^1, INF)
1097.00/292.14	    (14) typed CpxTrs
1097.00/292.14	        (15) RewriteLemmaProof [LOWER BOUND(ID), 386 ms]
1097.00/292.14	        (16) BOUNDS(1, INF)
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(0)
1097.00/292.14	Obligation:
1097.00/292.14	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	The TRS R consists of the following rules:
1097.00/292.14	
1097.00/292.14	   numbers -> d(0)
1097.00/292.14	   d(x) -> if(le(x, nr), x)
1097.00/292.14	   if(true, x) -> cons(x, d(s(x)))
1097.00/292.14	   if(false, x) -> nil
1097.00/292.14	   le(0, y) -> true
1097.00/292.14	   le(s(x), 0) -> false
1097.00/292.14	   le(s(x), s(y)) -> le(x, y)
1097.00/292.14	   nr -> ack(s(s(s(s(s(s(0)))))), 0)
1097.00/292.14	   ack(0, x) -> s(x)
1097.00/292.14	   ack(s(x), 0) -> ack(x, s(0))
1097.00/292.14	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1097.00/292.14	
1097.00/292.14	S is empty.
1097.00/292.14	Rewrite Strategy: INNERMOST
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1097.00/292.14	Renamed function symbols to avoid clashes with predefined symbol.
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(2)
1097.00/292.14	Obligation:
1097.00/292.14	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	The TRS R consists of the following rules:
1097.00/292.14	
1097.00/292.14	   numbers -> d(0')
1097.00/292.14	   d(x) -> if(le(x, nr), x)
1097.00/292.14	   if(true, x) -> cons(x, d(s(x)))
1097.00/292.14	   if(false, x) -> nil
1097.00/292.14	   le(0', y) -> true
1097.00/292.14	   le(s(x), 0') -> false
1097.00/292.14	   le(s(x), s(y)) -> le(x, y)
1097.00/292.14	   nr -> ack(s(s(s(s(s(s(0')))))), 0')
1097.00/292.14	   ack(0', x) -> s(x)
1097.00/292.14	   ack(s(x), 0') -> ack(x, s(0'))
1097.00/292.14	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1097.00/292.14	
1097.00/292.14	S is empty.
1097.00/292.14	Rewrite Strategy: INNERMOST
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(3) SlicingProof (LOWER BOUND(ID))
1097.00/292.14	Sliced the following arguments:
1097.00/292.14	cons/0
1097.00/292.14	
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(4)
1097.00/292.14	Obligation:
1097.00/292.14	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	The TRS R consists of the following rules:
1097.00/292.14	
1097.00/292.14	   numbers -> d(0')
1097.00/292.14	   d(x) -> if(le(x, nr), x)
1097.00/292.14	   if(true, x) -> cons(d(s(x)))
1097.00/292.14	   if(false, x) -> nil
1097.00/292.14	   le(0', y) -> true
1097.00/292.14	   le(s(x), 0') -> false
1097.00/292.14	   le(s(x), s(y)) -> le(x, y)
1097.00/292.14	   nr -> ack(s(s(s(s(s(s(0')))))), 0')
1097.00/292.14	   ack(0', x) -> s(x)
1097.00/292.14	   ack(s(x), 0') -> ack(x, s(0'))
1097.00/292.14	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1097.00/292.14	
1097.00/292.14	S is empty.
1097.00/292.14	Rewrite Strategy: INNERMOST
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1097.00/292.14	Infered types.
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(6)
1097.00/292.14	Obligation:
1097.00/292.14	Innermost TRS:
1097.00/292.14	Rules:
1097.00/292.14	numbers -> d(0')
1097.00/292.14	d(x) -> if(le(x, nr), x)
1097.00/292.14	if(true, x) -> cons(d(s(x)))
1097.00/292.14	if(false, x) -> nil
1097.00/292.14	le(0', y) -> true
1097.00/292.14	le(s(x), 0') -> false
1097.00/292.14	le(s(x), s(y)) -> le(x, y)
1097.00/292.14	nr -> ack(s(s(s(s(s(s(0')))))), 0')
1097.00/292.14	ack(0', x) -> s(x)
1097.00/292.14	ack(s(x), 0') -> ack(x, s(0'))
1097.00/292.14	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1097.00/292.14	
1097.00/292.14	Types:
1097.00/292.14	numbers :: cons:nil
1097.00/292.14	d :: 0':s -> cons:nil
1097.00/292.14	0' :: 0':s
1097.00/292.14	if :: true:false -> 0':s -> cons:nil
1097.00/292.14	le :: 0':s -> 0':s -> true:false
1097.00/292.14	nr :: 0':s
1097.00/292.14	true :: true:false
1097.00/292.14	cons :: cons:nil -> cons:nil
1097.00/292.14	s :: 0':s -> 0':s
1097.00/292.14	false :: true:false
1097.00/292.14	nil :: cons:nil
1097.00/292.14	ack :: 0':s -> 0':s -> 0':s
1097.00/292.14	hole_cons:nil1_0 :: cons:nil
1097.00/292.14	hole_0':s2_0 :: 0':s
1097.00/292.14	hole_true:false3_0 :: true:false
1097.00/292.14	gen_cons:nil4_0 :: Nat -> cons:nil
1097.00/292.14	gen_0':s5_0 :: Nat -> 0':s
1097.00/292.14	
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(7) OrderProof (LOWER BOUND(ID))
1097.00/292.14	Heuristically decided to analyse the following defined symbols:
1097.00/292.14	d, le, ack
1097.00/292.14	
1097.00/292.14	They will be analysed ascendingly in the following order:
1097.00/292.14	le < d
1097.00/292.14	
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(8)
1097.00/292.14	Obligation:
1097.00/292.14	Innermost TRS:
1097.00/292.14	Rules:
1097.00/292.14	numbers -> d(0')
1097.00/292.14	d(x) -> if(le(x, nr), x)
1097.00/292.14	if(true, x) -> cons(d(s(x)))
1097.00/292.14	if(false, x) -> nil
1097.00/292.14	le(0', y) -> true
1097.00/292.14	le(s(x), 0') -> false
1097.00/292.14	le(s(x), s(y)) -> le(x, y)
1097.00/292.14	nr -> ack(s(s(s(s(s(s(0')))))), 0')
1097.00/292.14	ack(0', x) -> s(x)
1097.00/292.14	ack(s(x), 0') -> ack(x, s(0'))
1097.00/292.14	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1097.00/292.14	
1097.00/292.14	Types:
1097.00/292.14	numbers :: cons:nil
1097.00/292.14	d :: 0':s -> cons:nil
1097.00/292.14	0' :: 0':s
1097.00/292.14	if :: true:false -> 0':s -> cons:nil
1097.00/292.14	le :: 0':s -> 0':s -> true:false
1097.00/292.14	nr :: 0':s
1097.00/292.14	true :: true:false
1097.00/292.14	cons :: cons:nil -> cons:nil
1097.00/292.14	s :: 0':s -> 0':s
1097.00/292.14	false :: true:false
1097.00/292.14	nil :: cons:nil
1097.00/292.14	ack :: 0':s -> 0':s -> 0':s
1097.00/292.14	hole_cons:nil1_0 :: cons:nil
1097.00/292.14	hole_0':s2_0 :: 0':s
1097.00/292.14	hole_true:false3_0 :: true:false
1097.00/292.14	gen_cons:nil4_0 :: Nat -> cons:nil
1097.00/292.14	gen_0':s5_0 :: Nat -> 0':s
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	Generator Equations:
1097.00/292.14	gen_cons:nil4_0(0) <=> nil
1097.00/292.14	gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
1097.00/292.14	gen_0':s5_0(0) <=> 0'
1097.00/292.14	gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	The following defined symbols remain to be analysed:
1097.00/292.14	le, d, ack
1097.00/292.14	
1097.00/292.14	They will be analysed ascendingly in the following order:
1097.00/292.14	le < d
1097.00/292.14	
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(9) RewriteLemmaProof (LOWER BOUND(ID))
1097.00/292.14	Proved the following rewrite lemma:
1097.00/292.14	le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_0)) -> true, rt in Omega(1 + n7_0)
1097.00/292.14	
1097.00/292.14	Induction Base:
1097.00/292.14	le(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1)
1097.00/292.14	true
1097.00/292.14	
1097.00/292.14	Induction Step:
1097.00/292.14	le(gen_0':s5_0(+(n7_0, 1)), gen_0':s5_0(+(n7_0, 1))) ->_R^Omega(1)
1097.00/292.14	le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_0)) ->_IH
1097.00/292.14	true
1097.00/292.14	
1097.00/292.14	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(10)
1097.00/292.14	Complex Obligation (BEST)
1097.00/292.14	
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(11)
1097.00/292.14	Obligation:
1097.00/292.14	Proved the lower bound n^1 for the following obligation:
1097.00/292.14	
1097.00/292.14	Innermost TRS:
1097.00/292.14	Rules:
1097.00/292.14	numbers -> d(0')
1097.00/292.14	d(x) -> if(le(x, nr), x)
1097.00/292.14	if(true, x) -> cons(d(s(x)))
1097.00/292.14	if(false, x) -> nil
1097.00/292.14	le(0', y) -> true
1097.00/292.14	le(s(x), 0') -> false
1097.00/292.14	le(s(x), s(y)) -> le(x, y)
1097.00/292.14	nr -> ack(s(s(s(s(s(s(0')))))), 0')
1097.00/292.14	ack(0', x) -> s(x)
1097.00/292.14	ack(s(x), 0') -> ack(x, s(0'))
1097.00/292.14	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1097.00/292.14	
1097.00/292.14	Types:
1097.00/292.14	numbers :: cons:nil
1097.00/292.14	d :: 0':s -> cons:nil
1097.00/292.14	0' :: 0':s
1097.00/292.14	if :: true:false -> 0':s -> cons:nil
1097.00/292.14	le :: 0':s -> 0':s -> true:false
1097.00/292.14	nr :: 0':s
1097.00/292.14	true :: true:false
1097.00/292.14	cons :: cons:nil -> cons:nil
1097.00/292.14	s :: 0':s -> 0':s
1097.00/292.14	false :: true:false
1097.00/292.14	nil :: cons:nil
1097.00/292.14	ack :: 0':s -> 0':s -> 0':s
1097.00/292.14	hole_cons:nil1_0 :: cons:nil
1097.00/292.14	hole_0':s2_0 :: 0':s
1097.00/292.14	hole_true:false3_0 :: true:false
1097.00/292.14	gen_cons:nil4_0 :: Nat -> cons:nil
1097.00/292.14	gen_0':s5_0 :: Nat -> 0':s
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	Generator Equations:
1097.00/292.14	gen_cons:nil4_0(0) <=> nil
1097.00/292.14	gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
1097.00/292.14	gen_0':s5_0(0) <=> 0'
1097.00/292.14	gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	The following defined symbols remain to be analysed:
1097.00/292.14	le, d, ack
1097.00/292.14	
1097.00/292.14	They will be analysed ascendingly in the following order:
1097.00/292.14	le < d
1097.00/292.14	
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(12) LowerBoundPropagationProof (FINISHED)
1097.00/292.14	Propagated lower bound.
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(13)
1097.00/292.14	BOUNDS(n^1, INF)
1097.00/292.14	
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(14)
1097.00/292.14	Obligation:
1097.00/292.14	Innermost TRS:
1097.00/292.14	Rules:
1097.00/292.14	numbers -> d(0')
1097.00/292.14	d(x) -> if(le(x, nr), x)
1097.00/292.14	if(true, x) -> cons(d(s(x)))
1097.00/292.14	if(false, x) -> nil
1097.00/292.14	le(0', y) -> true
1097.00/292.14	le(s(x), 0') -> false
1097.00/292.14	le(s(x), s(y)) -> le(x, y)
1097.00/292.14	nr -> ack(s(s(s(s(s(s(0')))))), 0')
1097.00/292.14	ack(0', x) -> s(x)
1097.00/292.14	ack(s(x), 0') -> ack(x, s(0'))
1097.00/292.14	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1097.00/292.14	
1097.00/292.14	Types:
1097.00/292.14	numbers :: cons:nil
1097.00/292.14	d :: 0':s -> cons:nil
1097.00/292.14	0' :: 0':s
1097.00/292.14	if :: true:false -> 0':s -> cons:nil
1097.00/292.14	le :: 0':s -> 0':s -> true:false
1097.00/292.14	nr :: 0':s
1097.00/292.14	true :: true:false
1097.00/292.14	cons :: cons:nil -> cons:nil
1097.00/292.14	s :: 0':s -> 0':s
1097.00/292.14	false :: true:false
1097.00/292.14	nil :: cons:nil
1097.00/292.14	ack :: 0':s -> 0':s -> 0':s
1097.00/292.14	hole_cons:nil1_0 :: cons:nil
1097.00/292.14	hole_0':s2_0 :: 0':s
1097.00/292.14	hole_true:false3_0 :: true:false
1097.00/292.14	gen_cons:nil4_0 :: Nat -> cons:nil
1097.00/292.14	gen_0':s5_0 :: Nat -> 0':s
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	Lemmas:
1097.00/292.14	le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_0)) -> true, rt in Omega(1 + n7_0)
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	Generator Equations:
1097.00/292.14	gen_cons:nil4_0(0) <=> nil
1097.00/292.14	gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
1097.00/292.14	gen_0':s5_0(0) <=> 0'
1097.00/292.14	gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
1097.00/292.14	
1097.00/292.14	
1097.00/292.14	The following defined symbols remain to be analysed:
1097.00/292.14	d, ack
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(15) RewriteLemmaProof (LOWER BOUND(ID))
1097.00/292.14	Proved the following rewrite lemma:
1097.00/292.14	ack(gen_0':s5_0(1), gen_0':s5_0(+(1, n934_0))) -> *6_0, rt in Omega(n934_0)
1097.00/292.14	
1097.00/292.14	Induction Base:
1097.00/292.14	ack(gen_0':s5_0(1), gen_0':s5_0(+(1, 0)))
1097.00/292.14	
1097.00/292.14	Induction Step:
1097.00/292.14	ack(gen_0':s5_0(1), gen_0':s5_0(+(1, +(n934_0, 1)))) ->_R^Omega(1)
1097.00/292.14	ack(gen_0':s5_0(0), ack(s(gen_0':s5_0(0)), gen_0':s5_0(+(1, n934_0)))) ->_IH
1097.00/292.14	ack(gen_0':s5_0(0), *6_0)
1097.00/292.14	
1097.00/292.14	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1097.00/292.14	----------------------------------------
1097.00/292.14	
1097.00/292.14	(16)
1097.00/292.14	BOUNDS(1, INF)
1097.20/292.25	EOF