981.28/291.55	WORST_CASE(Omega(n^1), ?)
981.28/291.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
981.28/291.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
981.28/291.55	
981.28/291.55	
981.28/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
981.28/291.55	
981.28/291.55	(0) CpxTRS
981.28/291.55	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
981.28/291.55	(2) CpxTRS
981.28/291.55	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
981.28/291.55	(4) typed CpxTrs
981.28/291.55	(5) OrderProof [LOWER BOUND(ID), 0 ms]
981.28/291.55	(6) typed CpxTrs
981.28/291.55	(7) RewriteLemmaProof [LOWER BOUND(ID), 931 ms]
981.28/291.55	(8) BEST
981.28/291.55	    (9) proven lower bound
981.28/291.55	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
981.28/291.55	        (11) BOUNDS(n^1, INF)
981.28/291.55	    (12) typed CpxTrs
981.28/291.55	
981.28/291.55	
981.28/291.55	----------------------------------------
981.28/291.55	
981.28/291.55	(0)
981.28/291.55	Obligation:
981.28/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
981.28/291.55	
981.28/291.55	
981.28/291.55	The TRS R consists of the following rules:
981.28/291.55	
981.28/291.55	   ack(0, y) -> s(y)
981.28/291.55	   ack(s(x), 0) -> ack(x, s(0))
981.28/291.55	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
981.28/291.55	   f(s(x), y) -> f(x, s(x))
981.28/291.55	   f(x, s(y)) -> f(y, x)
981.28/291.55	   f(x, y) -> ack(x, y)
981.28/291.55	   ack(s(x), y) -> f(x, x)
981.28/291.55	
981.28/291.55	S is empty.
981.28/291.55	Rewrite Strategy: INNERMOST
981.28/291.55	----------------------------------------
981.28/291.55	
981.28/291.55	(1) RenamingProof (BOTH BOUNDS(ID, ID))
981.28/291.55	Renamed function symbols to avoid clashes with predefined symbol.
981.28/291.55	----------------------------------------
981.28/291.55	
981.28/291.55	(2)
981.28/291.55	Obligation:
981.28/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
981.28/291.55	
981.28/291.55	
981.28/291.55	The TRS R consists of the following rules:
981.28/291.55	
981.28/291.55	   ack(0', y) -> s(y)
981.28/291.55	   ack(s(x), 0') -> ack(x, s(0'))
981.28/291.55	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
981.28/291.55	   f(s(x), y) -> f(x, s(x))
981.28/291.55	   f(x, s(y)) -> f(y, x)
981.28/291.55	   f(x, y) -> ack(x, y)
981.28/291.55	   ack(s(x), y) -> f(x, x)
981.28/291.55	
981.28/291.55	S is empty.
981.28/291.55	Rewrite Strategy: INNERMOST
981.28/291.55	----------------------------------------
981.28/291.55	
981.28/291.55	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
981.28/291.55	Infered types.
981.28/291.55	----------------------------------------
981.28/291.55	
981.28/291.55	(4)
981.28/291.55	Obligation:
981.28/291.55	Innermost TRS:
981.28/291.55	Rules:
981.28/291.55	ack(0', y) -> s(y)
981.28/291.55	ack(s(x), 0') -> ack(x, s(0'))
981.28/291.55	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
981.28/291.55	f(s(x), y) -> f(x, s(x))
981.28/291.55	f(x, s(y)) -> f(y, x)
981.28/291.55	f(x, y) -> ack(x, y)
981.28/291.55	ack(s(x), y) -> f(x, x)
981.28/291.55	
981.28/291.55	Types:
981.28/291.55	ack :: 0':s -> 0':s -> 0':s
981.28/291.55	0' :: 0':s
981.28/291.55	s :: 0':s -> 0':s
981.28/291.55	f :: 0':s -> 0':s -> 0':s
981.28/291.55	hole_0':s1_0 :: 0':s
981.28/291.55	gen_0':s2_0 :: Nat -> 0':s
981.28/291.55	
981.28/291.55	----------------------------------------
981.28/291.55	
981.28/291.55	(5) OrderProof (LOWER BOUND(ID))
981.28/291.55	Heuristically decided to analyse the following defined symbols:
981.28/291.55	ack, f
981.28/291.55	
981.28/291.55	They will be analysed ascendingly in the following order:
981.28/291.55	ack = f
981.28/291.55	
981.28/291.55	----------------------------------------
981.28/291.55	
981.28/291.55	(6)
981.28/291.55	Obligation:
981.28/291.55	Innermost TRS:
981.28/291.55	Rules:
981.28/291.55	ack(0', y) -> s(y)
981.28/291.55	ack(s(x), 0') -> ack(x, s(0'))
981.28/291.55	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
981.28/291.55	f(s(x), y) -> f(x, s(x))
981.28/291.55	f(x, s(y)) -> f(y, x)
981.28/291.55	f(x, y) -> ack(x, y)
981.28/291.55	ack(s(x), y) -> f(x, x)
981.28/291.55	
981.28/291.55	Types:
981.28/291.55	ack :: 0':s -> 0':s -> 0':s
981.28/291.55	0' :: 0':s
981.28/291.55	s :: 0':s -> 0':s
981.28/291.55	f :: 0':s -> 0':s -> 0':s
981.28/291.55	hole_0':s1_0 :: 0':s
981.28/291.55	gen_0':s2_0 :: Nat -> 0':s
981.28/291.55	
981.28/291.55	
981.28/291.55	Generator Equations:
981.28/291.55	gen_0':s2_0(0) <=> 0'
981.28/291.55	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
981.28/291.55	
981.28/291.55	
981.28/291.55	The following defined symbols remain to be analysed:
981.28/291.55	f, ack
981.28/291.55	
981.28/291.55	They will be analysed ascendingly in the following order:
981.28/291.55	ack = f
981.28/291.55	
981.28/291.55	----------------------------------------
981.28/291.55	
981.28/291.55	(7) RewriteLemmaProof (LOWER BOUND(ID))
981.28/291.56	Proved the following rewrite lemma:
981.28/291.56	ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n2381_0))) -> *3_0, rt in Omega(n2381_0)
981.28/291.56	
981.28/291.56	Induction Base:
981.28/291.56	ack(gen_0':s2_0(1), gen_0':s2_0(+(1, 0)))
981.28/291.56	
981.28/291.56	Induction Step:
981.28/291.56	ack(gen_0':s2_0(1), gen_0':s2_0(+(1, +(n2381_0, 1)))) ->_R^Omega(1)
981.28/291.56	ack(gen_0':s2_0(0), ack(s(gen_0':s2_0(0)), gen_0':s2_0(+(1, n2381_0)))) ->_IH
981.28/291.56	ack(gen_0':s2_0(0), *3_0)
981.28/291.56	
981.28/291.56	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
981.28/291.56	----------------------------------------
981.28/291.56	
981.28/291.56	(8)
981.28/291.56	Complex Obligation (BEST)
981.28/291.56	
981.28/291.56	----------------------------------------
981.28/291.56	
981.28/291.56	(9)
981.28/291.56	Obligation:
981.28/291.56	Proved the lower bound n^1 for the following obligation:
981.28/291.56	
981.28/291.56	Innermost TRS:
981.28/291.56	Rules:
981.28/291.56	ack(0', y) -> s(y)
981.28/291.56	ack(s(x), 0') -> ack(x, s(0'))
981.28/291.56	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
981.28/291.56	f(s(x), y) -> f(x, s(x))
981.28/291.56	f(x, s(y)) -> f(y, x)
981.28/291.56	f(x, y) -> ack(x, y)
981.28/291.56	ack(s(x), y) -> f(x, x)
981.28/291.56	
981.28/291.56	Types:
981.28/291.56	ack :: 0':s -> 0':s -> 0':s
981.28/291.56	0' :: 0':s
981.28/291.56	s :: 0':s -> 0':s
981.28/291.56	f :: 0':s -> 0':s -> 0':s
981.28/291.56	hole_0':s1_0 :: 0':s
981.28/291.56	gen_0':s2_0 :: Nat -> 0':s
981.28/291.56	
981.28/291.56	
981.28/291.56	Generator Equations:
981.28/291.56	gen_0':s2_0(0) <=> 0'
981.28/291.56	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
981.28/291.56	
981.28/291.56	
981.28/291.56	The following defined symbols remain to be analysed:
981.28/291.56	ack
981.28/291.56	
981.28/291.56	They will be analysed ascendingly in the following order:
981.28/291.56	ack = f
981.28/291.56	
981.28/291.56	----------------------------------------
981.28/291.56	
981.28/291.56	(10) LowerBoundPropagationProof (FINISHED)
981.28/291.56	Propagated lower bound.
981.28/291.56	----------------------------------------
981.28/291.56	
981.28/291.56	(11)
981.28/291.56	BOUNDS(n^1, INF)
981.28/291.56	
981.28/291.56	----------------------------------------
981.28/291.56	
981.28/291.56	(12)
981.28/291.56	Obligation:
981.28/291.56	Innermost TRS:
981.28/291.56	Rules:
981.28/291.56	ack(0', y) -> s(y)
981.28/291.56	ack(s(x), 0') -> ack(x, s(0'))
981.28/291.56	ack(s(x), s(y)) -> ack(x, ack(s(x), y))
981.28/291.56	f(s(x), y) -> f(x, s(x))
981.28/291.56	f(x, s(y)) -> f(y, x)
981.28/291.56	f(x, y) -> ack(x, y)
981.28/291.56	ack(s(x), y) -> f(x, x)
981.28/291.56	
981.28/291.56	Types:
981.28/291.56	ack :: 0':s -> 0':s -> 0':s
981.28/291.56	0' :: 0':s
981.28/291.56	s :: 0':s -> 0':s
981.28/291.56	f :: 0':s -> 0':s -> 0':s
981.28/291.56	hole_0':s1_0 :: 0':s
981.28/291.56	gen_0':s2_0 :: Nat -> 0':s
981.28/291.56	
981.28/291.56	
981.28/291.56	Lemmas:
981.28/291.56	ack(gen_0':s2_0(1), gen_0':s2_0(+(1, n2381_0))) -> *3_0, rt in Omega(n2381_0)
981.28/291.56	
981.28/291.56	
981.28/291.56	Generator Equations:
981.28/291.56	gen_0':s2_0(0) <=> 0'
981.28/291.56	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
981.28/291.56	
981.28/291.56	
981.28/291.56	The following defined symbols remain to be analysed:
981.28/291.56	f
981.28/291.56	
981.28/291.56	They will be analysed ascendingly in the following order:
981.28/291.56	ack = f
981.65/291.60	EOF