1087.22/291.50	WORST_CASE(Omega(n^1), ?)
1087.42/291.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1087.42/291.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1087.42/291.55	
1087.42/291.55	(0) CpxTRS
1087.42/291.55	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1087.42/291.55	(2) CpxTRS
1087.42/291.55	(3) SlicingProof [LOWER BOUND(ID), 0 ms]
1087.42/291.55	(4) CpxTRS
1087.42/291.55	(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1087.42/291.55	(6) typed CpxTrs
1087.42/291.55	(7) OrderProof [LOWER BOUND(ID), 0 ms]
1087.42/291.55	(8) typed CpxTrs
1087.42/291.55	(9) RewriteLemmaProof [LOWER BOUND(ID), 270 ms]
1087.42/291.55	(10) BEST
1087.42/291.55	    (11) proven lower bound
1087.42/291.55	        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
1087.42/291.55	        (13) BOUNDS(n^1, INF)
1087.42/291.55	    (14) typed CpxTrs
1087.42/291.55	        (15) RewriteLemmaProof [LOWER BOUND(ID), 392 ms]
1087.42/291.55	        (16) BOUNDS(1, INF)
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(0)
1087.42/291.55	Obligation:
1087.42/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	The TRS R consists of the following rules:
1087.42/291.55	
1087.42/291.55	   numbers -> d(0)
1087.42/291.55	   d(x) -> if(le(x, nr), x)
1087.42/291.55	   if(true, x) -> cons(x, d(s(x)))
1087.42/291.55	   if(false, x) -> nil
1087.42/291.55	   le(0, y) -> true
1087.42/291.55	   le(s(x), 0) -> false
1087.42/291.55	   le(s(x), s(y)) -> le(x, y)
1087.42/291.55	   nr -> ack(s(s(s(s(s(s(0)))))), 0)
1087.42/291.55	   ack(0, x) -> s(x)
1087.42/291.55	   ack(s(x), 0) -> ack(x, s(0))
1087.42/291.55	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1087.42/291.55	
1087.42/291.55	S is empty.
1087.42/291.55	Rewrite Strategy: FULL
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1087.42/291.55	Renamed function symbols to avoid clashes with predefined symbol.
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(2)
1087.42/291.55	Obligation:
1087.42/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	The TRS R consists of the following rules:
1087.42/291.55	
1087.42/291.55	   numbers -> d(0')
1087.42/291.55	   d(x) -> if(le(x, nr), x)
1087.42/291.55	   if(true, x) -> cons(x, d(s(x)))
1087.42/291.55	   if(false, x) -> nil
1087.42/291.55	   le(0', y) -> true
1087.42/291.55	   le(s(x), 0') -> false
1087.42/291.55	   le(s(x), s(y)) -> le(x, y)
1087.42/291.55	   nr -> ack(s(s(s(s(s(s(0')))))), 0')
1087.42/291.55	   ack(0', x) -> s(x)
1087.42/291.55	   ack(s(x), 0') -> ack(x, s(0'))
1087.42/291.55	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1087.42/291.55	
1087.42/291.55	S is empty.
1087.42/291.55	Rewrite Strategy: FULL
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(3) SlicingProof (LOWER BOUND(ID))
1087.42/291.55	Sliced the following arguments:
1087.42/291.55	cons/0
1087.42/291.55	
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(4)
1087.42/291.55	Obligation:
1087.42/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	The TRS R consists of the following rules:
1087.42/291.55	
1087.42/291.55	   numbers -> d(0')
1087.42/291.55	   d(x) -> if(le(x, nr), x)
1087.42/291.55	   if(true, x) -> cons(d(s(x)))
1087.42/291.55	   if(false, x) -> nil
1087.42/291.55	   le(0', y) -> true
1087.42/291.55	   le(s(x), 0') -> false
1087.42/291.55	   le(s(x), s(y)) -> le(x, y)
1087.42/291.55	   nr -> ack(s(s(s(s(s(s(0')))))), 0')
1087.42/291.55	   ack(0', x) -> s(x)
1087.42/291.55	   ack(s(x), 0') -> ack(x, s(0'))
1087.42/291.55	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1087.42/291.55	
1087.42/291.55	S is empty.
1087.42/291.55	Rewrite Strategy: FULL
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1087.42/291.55	Infered types.
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(6)
1087.42/291.55	Obligation:
1087.42/291.55	TRS:
1087.42/291.55	Rules:
1087.42/291.55	numbers -> d(0')
1087.42/291.55	d(x) -> if(le(x, nr), x)
1087.42/291.55	if(true, x) -> cons(d(s(x)))
1087.42/291.55	if(false, x) -> nil
1087.42/291.55	le(0', y) -> true
1087.42/291.55	le(s(x), 0') -> false
1087.42/291.55	le(s(x), s(y)) -> le(x, y)
1087.42/291.55	nr -> ack(s(s(s(s(s(s(0')))))), 0')
1087.42/291.55	ack(0', x) -> s(x)
1087.42/291.55	ack(s(x), 0') -> ack(x, s(0'))
1087.42/291.55	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1087.42/291.55	
1087.42/291.55	Types:
1087.42/291.55	numbers :: cons:nil
1087.42/291.55	d :: 0':s -> cons:nil
1087.42/291.55	0' :: 0':s
1087.42/291.55	if :: true:false -> 0':s -> cons:nil
1087.42/291.55	le :: 0':s -> 0':s -> true:false
1087.42/291.55	nr :: 0':s
1087.42/291.55	true :: true:false
1087.42/291.55	cons :: cons:nil -> cons:nil
1087.42/291.55	s :: 0':s -> 0':s
1087.42/291.55	false :: true:false
1087.42/291.55	nil :: cons:nil
1087.42/291.55	ack :: 0':s -> 0':s -> 0':s
1087.42/291.55	hole_cons:nil1_0 :: cons:nil
1087.42/291.55	hole_0':s2_0 :: 0':s
1087.42/291.55	hole_true:false3_0 :: true:false
1087.42/291.55	gen_cons:nil4_0 :: Nat -> cons:nil
1087.42/291.55	gen_0':s5_0 :: Nat -> 0':s
1087.42/291.55	
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(7) OrderProof (LOWER BOUND(ID))
1087.42/291.55	Heuristically decided to analyse the following defined symbols:
1087.42/291.55	d, le, ack
1087.42/291.55	
1087.42/291.55	They will be analysed ascendingly in the following order:
1087.42/291.55	le < d
1087.42/291.55	
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(8)
1087.42/291.55	Obligation:
1087.42/291.55	TRS:
1087.42/291.55	Rules:
1087.42/291.55	numbers -> d(0')
1087.42/291.55	d(x) -> if(le(x, nr), x)
1087.42/291.55	if(true, x) -> cons(d(s(x)))
1087.42/291.55	if(false, x) -> nil
1087.42/291.55	le(0', y) -> true
1087.42/291.55	le(s(x), 0') -> false
1087.42/291.55	le(s(x), s(y)) -> le(x, y)
1087.42/291.55	nr -> ack(s(s(s(s(s(s(0')))))), 0')
1087.42/291.55	ack(0', x) -> s(x)
1087.42/291.55	ack(s(x), 0') -> ack(x, s(0'))
1087.42/291.55	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1087.42/291.55	
1087.42/291.55	Types:
1087.42/291.55	numbers :: cons:nil
1087.42/291.55	d :: 0':s -> cons:nil
1087.42/291.55	0' :: 0':s
1087.42/291.55	if :: true:false -> 0':s -> cons:nil
1087.42/291.55	le :: 0':s -> 0':s -> true:false
1087.42/291.55	nr :: 0':s
1087.42/291.55	true :: true:false
1087.42/291.55	cons :: cons:nil -> cons:nil
1087.42/291.55	s :: 0':s -> 0':s
1087.42/291.55	false :: true:false
1087.42/291.55	nil :: cons:nil
1087.42/291.55	ack :: 0':s -> 0':s -> 0':s
1087.42/291.55	hole_cons:nil1_0 :: cons:nil
1087.42/291.55	hole_0':s2_0 :: 0':s
1087.42/291.55	hole_true:false3_0 :: true:false
1087.42/291.55	gen_cons:nil4_0 :: Nat -> cons:nil
1087.42/291.55	gen_0':s5_0 :: Nat -> 0':s
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	Generator Equations:
1087.42/291.55	gen_cons:nil4_0(0) <=> nil
1087.42/291.55	gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
1087.42/291.55	gen_0':s5_0(0) <=> 0'
1087.42/291.55	gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	The following defined symbols remain to be analysed:
1087.42/291.55	le, d, ack
1087.42/291.55	
1087.42/291.55	They will be analysed ascendingly in the following order:
1087.42/291.55	le < d
1087.42/291.55	
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(9) RewriteLemmaProof (LOWER BOUND(ID))
1087.42/291.55	Proved the following rewrite lemma:
1087.42/291.55	le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_0)) -> true, rt in Omega(1 + n7_0)
1087.42/291.55	
1087.42/291.55	Induction Base:
1087.42/291.55	le(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1)
1087.42/291.55	true
1087.42/291.55	
1087.42/291.55	Induction Step:
1087.42/291.55	le(gen_0':s5_0(+(n7_0, 1)), gen_0':s5_0(+(n7_0, 1))) ->_R^Omega(1)
1087.42/291.55	le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_0)) ->_IH
1087.42/291.55	true
1087.42/291.55	
1087.42/291.55	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(10)
1087.42/291.55	Complex Obligation (BEST)
1087.42/291.55	
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(11)
1087.42/291.55	Obligation:
1087.42/291.55	Proved the lower bound n^1 for the following obligation:
1087.42/291.55	
1087.42/291.55	TRS:
1087.42/291.55	Rules:
1087.42/291.55	numbers -> d(0')
1087.42/291.55	d(x) -> if(le(x, nr), x)
1087.42/291.55	if(true, x) -> cons(d(s(x)))
1087.42/291.55	if(false, x) -> nil
1087.42/291.55	le(0', y) -> true
1087.42/291.55	le(s(x), 0') -> false
1087.42/291.55	le(s(x), s(y)) -> le(x, y)
1087.42/291.55	nr -> ack(s(s(s(s(s(s(0')))))), 0')
1087.42/291.55	ack(0', x) -> s(x)
1087.42/291.55	ack(s(x), 0') -> ack(x, s(0'))
1087.42/291.55	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1087.42/291.55	
1087.42/291.55	Types:
1087.42/291.55	numbers :: cons:nil
1087.42/291.55	d :: 0':s -> cons:nil
1087.42/291.55	0' :: 0':s
1087.42/291.55	if :: true:false -> 0':s -> cons:nil
1087.42/291.55	le :: 0':s -> 0':s -> true:false
1087.42/291.55	nr :: 0':s
1087.42/291.55	true :: true:false
1087.42/291.55	cons :: cons:nil -> cons:nil
1087.42/291.55	s :: 0':s -> 0':s
1087.42/291.55	false :: true:false
1087.42/291.55	nil :: cons:nil
1087.42/291.55	ack :: 0':s -> 0':s -> 0':s
1087.42/291.55	hole_cons:nil1_0 :: cons:nil
1087.42/291.55	hole_0':s2_0 :: 0':s
1087.42/291.55	hole_true:false3_0 :: true:false
1087.42/291.55	gen_cons:nil4_0 :: Nat -> cons:nil
1087.42/291.55	gen_0':s5_0 :: Nat -> 0':s
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	Generator Equations:
1087.42/291.55	gen_cons:nil4_0(0) <=> nil
1087.42/291.55	gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
1087.42/291.55	gen_0':s5_0(0) <=> 0'
1087.42/291.55	gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	The following defined symbols remain to be analysed:
1087.42/291.55	le, d, ack
1087.42/291.55	
1087.42/291.55	They will be analysed ascendingly in the following order:
1087.42/291.55	le < d
1087.42/291.55	
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(12) LowerBoundPropagationProof (FINISHED)
1087.42/291.55	Propagated lower bound.
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(13)
1087.42/291.55	BOUNDS(n^1, INF)
1087.42/291.55	
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(14)
1087.42/291.55	Obligation:
1087.42/291.55	TRS:
1087.42/291.55	Rules:
1087.42/291.55	numbers -> d(0')
1087.42/291.55	d(x) -> if(le(x, nr), x)
1087.42/291.55	if(true, x) -> cons(d(s(x)))
1087.42/291.55	if(false, x) -> nil
1087.42/291.55	le(0', y) -> true
1087.42/291.55	le(s(x), 0') -> false
1087.42/291.55	le(s(x), s(y)) -> le(x, y)
1087.42/291.55	nr -> ack(s(s(s(s(s(s(0')))))), 0')
1087.42/291.55	ack(0', x) -> s(x)
1087.42/291.55	ack(s(x), 0') -> ack(x, s(0'))
1087.42/291.55	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
1087.42/291.55	
1087.42/291.55	Types:
1087.42/291.55	numbers :: cons:nil
1087.42/291.55	d :: 0':s -> cons:nil
1087.42/291.55	0' :: 0':s
1087.42/291.55	if :: true:false -> 0':s -> cons:nil
1087.42/291.55	le :: 0':s -> 0':s -> true:false
1087.42/291.55	nr :: 0':s
1087.42/291.55	true :: true:false
1087.42/291.55	cons :: cons:nil -> cons:nil
1087.42/291.55	s :: 0':s -> 0':s
1087.42/291.55	false :: true:false
1087.42/291.55	nil :: cons:nil
1087.42/291.55	ack :: 0':s -> 0':s -> 0':s
1087.42/291.55	hole_cons:nil1_0 :: cons:nil
1087.42/291.55	hole_0':s2_0 :: 0':s
1087.42/291.55	hole_true:false3_0 :: true:false
1087.42/291.55	gen_cons:nil4_0 :: Nat -> cons:nil
1087.42/291.55	gen_0':s5_0 :: Nat -> 0':s
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	Lemmas:
1087.42/291.55	le(gen_0':s5_0(n7_0), gen_0':s5_0(n7_0)) -> true, rt in Omega(1 + n7_0)
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	Generator Equations:
1087.42/291.55	gen_cons:nil4_0(0) <=> nil
1087.42/291.55	gen_cons:nil4_0(+(x, 1)) <=> cons(gen_cons:nil4_0(x))
1087.42/291.55	gen_0':s5_0(0) <=> 0'
1087.42/291.55	gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
1087.42/291.55	
1087.42/291.55	
1087.42/291.55	The following defined symbols remain to be analysed:
1087.42/291.55	d, ack
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(15) RewriteLemmaProof (LOWER BOUND(ID))
1087.42/291.55	Proved the following rewrite lemma:
1087.42/291.55	ack(gen_0':s5_0(1), gen_0':s5_0(+(1, n910_0))) -> *6_0, rt in Omega(n910_0)
1087.42/291.55	
1087.42/291.55	Induction Base:
1087.42/291.55	ack(gen_0':s5_0(1), gen_0':s5_0(+(1, 0)))
1087.42/291.55	
1087.42/291.55	Induction Step:
1087.42/291.55	ack(gen_0':s5_0(1), gen_0':s5_0(+(1, +(n910_0, 1)))) ->_R^Omega(1)
1087.42/291.55	ack(gen_0':s5_0(0), ack(s(gen_0':s5_0(0)), gen_0':s5_0(+(1, n910_0)))) ->_IH
1087.42/291.55	ack(gen_0':s5_0(0), *6_0)
1087.42/291.55	
1087.42/291.55	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1087.42/291.55	----------------------------------------
1087.42/291.55	
1087.42/291.55	(16)
1087.42/291.55	BOUNDS(1, INF)
1087.60/291.65	EOF