308.07/291.54	WORST_CASE(Omega(n^1), ?)
308.07/291.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
308.07/291.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
308.07/291.55	
308.07/291.55	
308.07/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.07/291.55	
308.07/291.55	(0) CpxTRS
308.07/291.55	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
308.07/291.55	(2) CpxTRS
308.07/291.55	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
308.07/291.55	(4) typed CpxTrs
308.07/291.55	(5) OrderProof [LOWER BOUND(ID), 0 ms]
308.07/291.55	(6) typed CpxTrs
308.07/291.55	(7) RewriteLemmaProof [LOWER BOUND(ID), 301 ms]
308.07/291.55	(8) BEST
308.07/291.55	    (9) proven lower bound
308.07/291.55	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
308.07/291.55	        (11) BOUNDS(n^1, INF)
308.07/291.55	    (12) typed CpxTrs
308.07/291.55	        (13) RewriteLemmaProof [LOWER BOUND(ID), 72 ms]
308.07/291.55	        (14) typed CpxTrs
308.07/291.55	
308.07/291.55	
308.07/291.55	----------------------------------------
308.07/291.55	
308.07/291.55	(0)
308.07/291.55	Obligation:
308.07/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.07/291.55	
308.07/291.55	
308.07/291.55	The TRS R consists of the following rules:
308.07/291.55	
308.07/291.55	   ge(x, 0) -> true
308.07/291.55	   ge(0, s(x)) -> false
308.07/291.55	   ge(s(x), s(y)) -> ge(x, y)
308.07/291.55	   minus(x, 0) -> x
308.07/291.55	   minus(s(x), s(y)) -> minus(x, y)
308.07/291.55	   div(x, y) -> ify(ge(y, s(0)), x, y)
308.07/291.55	   ify(false, x, y) -> divByZeroError
308.07/291.55	   ify(true, x, y) -> if(ge(x, y), x, y)
308.07/291.55	   if(false, x, y) -> 0
308.07/291.55	   if(true, x, y) -> s(div(minus(x, y), y))
308.07/291.55	
308.07/291.55	S is empty.
308.07/291.55	Rewrite Strategy: FULL
308.07/291.55	----------------------------------------
308.07/291.55	
308.07/291.55	(1) RenamingProof (BOTH BOUNDS(ID, ID))
308.07/291.55	Renamed function symbols to avoid clashes with predefined symbol.
308.07/291.55	----------------------------------------
308.07/291.55	
308.07/291.55	(2)
308.07/291.55	Obligation:
308.07/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.07/291.55	
308.07/291.55	
308.07/291.55	The TRS R consists of the following rules:
308.07/291.55	
308.07/291.55	   ge(x, 0') -> true
308.07/291.55	   ge(0', s(x)) -> false
308.07/291.55	   ge(s(x), s(y)) -> ge(x, y)
308.07/291.55	   minus(x, 0') -> x
308.07/291.55	   minus(s(x), s(y)) -> minus(x, y)
308.07/291.55	   div(x, y) -> ify(ge(y, s(0')), x, y)
308.07/291.55	   ify(false, x, y) -> divByZeroError
308.07/291.55	   ify(true, x, y) -> if(ge(x, y), x, y)
308.07/291.55	   if(false, x, y) -> 0'
308.07/291.55	   if(true, x, y) -> s(div(minus(x, y), y))
308.07/291.55	
308.07/291.55	S is empty.
308.07/291.55	Rewrite Strategy: FULL
308.07/291.55	----------------------------------------
308.07/291.55	
308.07/291.55	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
308.07/291.55	Infered types.
308.07/291.55	----------------------------------------
308.07/291.55	
308.07/291.55	(4)
308.07/291.55	Obligation:
308.07/291.55	TRS:
308.07/291.55	Rules:
308.07/291.55	ge(x, 0') -> true
308.07/291.55	ge(0', s(x)) -> false
308.07/291.55	ge(s(x), s(y)) -> ge(x, y)
308.07/291.55	minus(x, 0') -> x
308.07/291.55	minus(s(x), s(y)) -> minus(x, y)
308.07/291.55	div(x, y) -> ify(ge(y, s(0')), x, y)
308.07/291.55	ify(false, x, y) -> divByZeroError
308.07/291.55	ify(true, x, y) -> if(ge(x, y), x, y)
308.07/291.55	if(false, x, y) -> 0'
308.07/291.55	if(true, x, y) -> s(div(minus(x, y), y))
308.07/291.55	
308.07/291.55	Types:
308.07/291.55	ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
308.07/291.55	0' :: 0':s:divByZeroError
308.07/291.55	true :: true:false
308.07/291.55	s :: 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	false :: true:false
308.07/291.55	minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	divByZeroError :: 0':s:divByZeroError
308.07/291.55	if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	hole_true:false1_0 :: true:false
308.07/291.55	hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
308.07/291.55	gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError
308.07/291.55	
308.07/291.55	----------------------------------------
308.07/291.55	
308.07/291.55	(5) OrderProof (LOWER BOUND(ID))
308.07/291.55	Heuristically decided to analyse the following defined symbols:
308.07/291.55	ge, minus, div
308.07/291.55	
308.07/291.55	They will be analysed ascendingly in the following order:
308.07/291.55	ge < div
308.07/291.55	minus < div
308.07/291.55	
308.07/291.55	----------------------------------------
308.07/291.55	
308.07/291.55	(6)
308.07/291.55	Obligation:
308.07/291.55	TRS:
308.07/291.55	Rules:
308.07/291.55	ge(x, 0') -> true
308.07/291.55	ge(0', s(x)) -> false
308.07/291.55	ge(s(x), s(y)) -> ge(x, y)
308.07/291.55	minus(x, 0') -> x
308.07/291.55	minus(s(x), s(y)) -> minus(x, y)
308.07/291.55	div(x, y) -> ify(ge(y, s(0')), x, y)
308.07/291.55	ify(false, x, y) -> divByZeroError
308.07/291.55	ify(true, x, y) -> if(ge(x, y), x, y)
308.07/291.55	if(false, x, y) -> 0'
308.07/291.55	if(true, x, y) -> s(div(minus(x, y), y))
308.07/291.55	
308.07/291.55	Types:
308.07/291.55	ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
308.07/291.55	0' :: 0':s:divByZeroError
308.07/291.55	true :: true:false
308.07/291.55	s :: 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	false :: true:false
308.07/291.55	minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	divByZeroError :: 0':s:divByZeroError
308.07/291.55	if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	hole_true:false1_0 :: true:false
308.07/291.55	hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
308.07/291.55	gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError
308.07/291.55	
308.07/291.55	
308.07/291.55	Generator Equations:
308.07/291.55	gen_0':s:divByZeroError3_0(0) <=> 0'
308.07/291.55	gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))
308.07/291.55	
308.07/291.55	
308.07/291.55	The following defined symbols remain to be analysed:
308.07/291.55	ge, minus, div
308.07/291.55	
308.07/291.55	They will be analysed ascendingly in the following order:
308.07/291.55	ge < div
308.07/291.55	minus < div
308.07/291.55	
308.07/291.55	----------------------------------------
308.07/291.55	
308.07/291.55	(7) RewriteLemmaProof (LOWER BOUND(ID))
308.07/291.55	Proved the following rewrite lemma:
308.07/291.55	ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
308.07/291.55	
308.07/291.55	Induction Base:
308.07/291.55	ge(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(0)) ->_R^Omega(1)
308.07/291.55	true
308.07/291.55	
308.07/291.55	Induction Step:
308.07/291.55	ge(gen_0':s:divByZeroError3_0(+(n5_0, 1)), gen_0':s:divByZeroError3_0(+(n5_0, 1))) ->_R^Omega(1)
308.07/291.55	ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) ->_IH
308.07/291.55	true
308.07/291.55	
308.07/291.55	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
308.07/291.55	----------------------------------------
308.07/291.55	
308.07/291.55	(8)
308.07/291.55	Complex Obligation (BEST)
308.07/291.55	
308.07/291.55	----------------------------------------
308.07/291.55	
308.07/291.55	(9)
308.07/291.55	Obligation:
308.07/291.55	Proved the lower bound n^1 for the following obligation:
308.07/291.55	
308.07/291.55	TRS:
308.07/291.55	Rules:
308.07/291.55	ge(x, 0') -> true
308.07/291.55	ge(0', s(x)) -> false
308.07/291.55	ge(s(x), s(y)) -> ge(x, y)
308.07/291.55	minus(x, 0') -> x
308.07/291.55	minus(s(x), s(y)) -> minus(x, y)
308.07/291.55	div(x, y) -> ify(ge(y, s(0')), x, y)
308.07/291.55	ify(false, x, y) -> divByZeroError
308.07/291.55	ify(true, x, y) -> if(ge(x, y), x, y)
308.07/291.55	if(false, x, y) -> 0'
308.07/291.55	if(true, x, y) -> s(div(minus(x, y), y))
308.07/291.55	
308.07/291.55	Types:
308.07/291.55	ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
308.07/291.55	0' :: 0':s:divByZeroError
308.07/291.55	true :: true:false
308.07/291.55	s :: 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	false :: true:false
308.07/291.55	minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.55	ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	divByZeroError :: 0':s:divByZeroError
308.07/291.56	if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	hole_true:false1_0 :: true:false
308.07/291.56	hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
308.07/291.56	gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError
308.07/291.56	
308.07/291.56	
308.07/291.56	Generator Equations:
308.07/291.56	gen_0':s:divByZeroError3_0(0) <=> 0'
308.07/291.56	gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))
308.07/291.56	
308.07/291.56	
308.07/291.56	The following defined symbols remain to be analysed:
308.07/291.56	ge, minus, div
308.07/291.56	
308.07/291.56	They will be analysed ascendingly in the following order:
308.07/291.56	ge < div
308.07/291.56	minus < div
308.07/291.56	
308.07/291.56	----------------------------------------
308.07/291.56	
308.07/291.56	(10) LowerBoundPropagationProof (FINISHED)
308.07/291.56	Propagated lower bound.
308.07/291.56	----------------------------------------
308.07/291.56	
308.07/291.56	(11)
308.07/291.56	BOUNDS(n^1, INF)
308.07/291.56	
308.07/291.56	----------------------------------------
308.07/291.56	
308.07/291.56	(12)
308.07/291.56	Obligation:
308.07/291.56	TRS:
308.07/291.56	Rules:
308.07/291.56	ge(x, 0') -> true
308.07/291.56	ge(0', s(x)) -> false
308.07/291.56	ge(s(x), s(y)) -> ge(x, y)
308.07/291.56	minus(x, 0') -> x
308.07/291.56	minus(s(x), s(y)) -> minus(x, y)
308.07/291.56	div(x, y) -> ify(ge(y, s(0')), x, y)
308.07/291.56	ify(false, x, y) -> divByZeroError
308.07/291.56	ify(true, x, y) -> if(ge(x, y), x, y)
308.07/291.56	if(false, x, y) -> 0'
308.07/291.56	if(true, x, y) -> s(div(minus(x, y), y))
308.07/291.56	
308.07/291.56	Types:
308.07/291.56	ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
308.07/291.56	0' :: 0':s:divByZeroError
308.07/291.56	true :: true:false
308.07/291.56	s :: 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	false :: true:false
308.07/291.56	minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	divByZeroError :: 0':s:divByZeroError
308.07/291.56	if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	hole_true:false1_0 :: true:false
308.07/291.56	hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
308.07/291.56	gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError
308.07/291.56	
308.07/291.56	
308.07/291.56	Lemmas:
308.07/291.56	ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
308.07/291.56	
308.07/291.56	
308.07/291.56	Generator Equations:
308.07/291.56	gen_0':s:divByZeroError3_0(0) <=> 0'
308.07/291.56	gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))
308.07/291.56	
308.07/291.56	
308.07/291.56	The following defined symbols remain to be analysed:
308.07/291.56	minus, div
308.07/291.56	
308.07/291.56	They will be analysed ascendingly in the following order:
308.07/291.56	minus < div
308.07/291.56	
308.07/291.56	----------------------------------------
308.07/291.56	
308.07/291.56	(13) RewriteLemmaProof (LOWER BOUND(ID))
308.07/291.56	Proved the following rewrite lemma:
308.07/291.56	minus(gen_0':s:divByZeroError3_0(n251_0), gen_0':s:divByZeroError3_0(n251_0)) -> gen_0':s:divByZeroError3_0(0), rt in Omega(1 + n251_0)
308.07/291.56	
308.07/291.56	Induction Base:
308.07/291.56	minus(gen_0':s:divByZeroError3_0(0), gen_0':s:divByZeroError3_0(0)) ->_R^Omega(1)
308.07/291.56	gen_0':s:divByZeroError3_0(0)
308.07/291.56	
308.07/291.56	Induction Step:
308.07/291.56	minus(gen_0':s:divByZeroError3_0(+(n251_0, 1)), gen_0':s:divByZeroError3_0(+(n251_0, 1))) ->_R^Omega(1)
308.07/291.56	minus(gen_0':s:divByZeroError3_0(n251_0), gen_0':s:divByZeroError3_0(n251_0)) ->_IH
308.07/291.56	gen_0':s:divByZeroError3_0(0)
308.07/291.56	
308.07/291.56	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
308.07/291.56	----------------------------------------
308.07/291.56	
308.07/291.56	(14)
308.07/291.56	Obligation:
308.07/291.56	TRS:
308.07/291.56	Rules:
308.07/291.56	ge(x, 0') -> true
308.07/291.56	ge(0', s(x)) -> false
308.07/291.56	ge(s(x), s(y)) -> ge(x, y)
308.07/291.56	minus(x, 0') -> x
308.07/291.56	minus(s(x), s(y)) -> minus(x, y)
308.07/291.56	div(x, y) -> ify(ge(y, s(0')), x, y)
308.07/291.56	ify(false, x, y) -> divByZeroError
308.07/291.56	ify(true, x, y) -> if(ge(x, y), x, y)
308.07/291.56	if(false, x, y) -> 0'
308.07/291.56	if(true, x, y) -> s(div(minus(x, y), y))
308.07/291.56	
308.07/291.56	Types:
308.07/291.56	ge :: 0':s:divByZeroError -> 0':s:divByZeroError -> true:false
308.07/291.56	0' :: 0':s:divByZeroError
308.07/291.56	true :: true:false
308.07/291.56	s :: 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	false :: true:false
308.07/291.56	minus :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	div :: 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	ify :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	divByZeroError :: 0':s:divByZeroError
308.07/291.56	if :: true:false -> 0':s:divByZeroError -> 0':s:divByZeroError -> 0':s:divByZeroError
308.07/291.56	hole_true:false1_0 :: true:false
308.07/291.56	hole_0':s:divByZeroError2_0 :: 0':s:divByZeroError
308.07/291.56	gen_0':s:divByZeroError3_0 :: Nat -> 0':s:divByZeroError
308.07/291.56	
308.07/291.56	
308.07/291.56	Lemmas:
308.07/291.56	ge(gen_0':s:divByZeroError3_0(n5_0), gen_0':s:divByZeroError3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
308.07/291.56	minus(gen_0':s:divByZeroError3_0(n251_0), gen_0':s:divByZeroError3_0(n251_0)) -> gen_0':s:divByZeroError3_0(0), rt in Omega(1 + n251_0)
308.07/291.56	
308.07/291.56	
308.07/291.56	Generator Equations:
308.07/291.56	gen_0':s:divByZeroError3_0(0) <=> 0'
308.07/291.56	gen_0':s:divByZeroError3_0(+(x, 1)) <=> s(gen_0':s:divByZeroError3_0(x))
308.07/291.56	
308.07/291.56	
308.07/291.56	The following defined symbols remain to be analysed:
308.07/291.56	div
308.16/291.59	EOF