1054.58/291.50	WORST_CASE(Omega(n^1), ?)
1054.58/291.51	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1054.58/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1054.58/291.51	
1054.58/291.51	(0) CpxTRS
1054.58/291.51	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1054.58/291.51	(2) CpxTRS
1054.58/291.51	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1054.58/291.51	(4) typed CpxTrs
1054.58/291.51	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1054.58/291.51	(6) typed CpxTrs
1054.58/291.51	(7) RewriteLemmaProof [LOWER BOUND(ID), 301 ms]
1054.58/291.51	(8) BEST
1054.58/291.51	    (9) proven lower bound
1054.58/291.51	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1054.58/291.51	        (11) BOUNDS(n^1, INF)
1054.58/291.51	    (12) typed CpxTrs
1054.58/291.51	        (13) RewriteLemmaProof [LOWER BOUND(ID), 59 ms]
1054.58/291.51	        (14) typed CpxTrs
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(0)
1054.58/291.51	Obligation:
1054.58/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	The TRS R consists of the following rules:
1054.58/291.51	
1054.58/291.51	   div(x, s(y)) -> d(x, s(y), 0)
1054.58/291.51	   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
1054.58/291.51	   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
1054.58/291.51	   cond(false, x, y, z) -> 0
1054.58/291.51	   ge(u, 0) -> true
1054.58/291.51	   ge(0, s(v)) -> false
1054.58/291.51	   ge(s(u), s(v)) -> ge(u, v)
1054.58/291.51	   plus(n, 0) -> n
1054.58/291.51	   plus(n, s(m)) -> s(plus(n, m))
1054.58/291.51	
1054.58/291.51	S is empty.
1054.58/291.51	Rewrite Strategy: INNERMOST
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1054.58/291.51	Renamed function symbols to avoid clashes with predefined symbol.
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(2)
1054.58/291.51	Obligation:
1054.58/291.51	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	The TRS R consists of the following rules:
1054.58/291.51	
1054.58/291.51	   div(x, s(y)) -> d(x, s(y), 0')
1054.58/291.51	   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
1054.58/291.51	   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
1054.58/291.51	   cond(false, x, y, z) -> 0'
1054.58/291.51	   ge(u, 0') -> true
1054.58/291.51	   ge(0', s(v)) -> false
1054.58/291.51	   ge(s(u), s(v)) -> ge(u, v)
1054.58/291.51	   plus(n, 0') -> n
1054.58/291.51	   plus(n, s(m)) -> s(plus(n, m))
1054.58/291.51	
1054.58/291.51	S is empty.
1054.58/291.51	Rewrite Strategy: INNERMOST
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1054.58/291.51	Infered types.
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(4)
1054.58/291.51	Obligation:
1054.58/291.51	Innermost TRS:
1054.58/291.51	Rules:
1054.58/291.51	div(x, s(y)) -> d(x, s(y), 0')
1054.58/291.51	d(x, s(y), z) -> cond(ge(x, z), x, y, z)
1054.58/291.51	cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
1054.58/291.51	cond(false, x, y, z) -> 0'
1054.58/291.51	ge(u, 0') -> true
1054.58/291.51	ge(0', s(v)) -> false
1054.58/291.51	ge(s(u), s(v)) -> ge(u, v)
1054.58/291.51	plus(n, 0') -> n
1054.58/291.51	plus(n, s(m)) -> s(plus(n, m))
1054.58/291.51	
1054.58/291.51	Types:
1054.58/291.51	div :: s:0' -> s:0' -> s:0'
1054.58/291.51	s :: s:0' -> s:0'
1054.58/291.51	d :: s:0' -> s:0' -> s:0' -> s:0'
1054.58/291.51	0' :: s:0'
1054.58/291.51	cond :: true:false -> s:0' -> s:0' -> s:0' -> s:0'
1054.58/291.51	ge :: s:0' -> s:0' -> true:false
1054.58/291.51	true :: true:false
1054.58/291.51	plus :: s:0' -> s:0' -> s:0'
1054.58/291.51	false :: true:false
1054.58/291.51	hole_s:0'1_0 :: s:0'
1054.58/291.51	hole_true:false2_0 :: true:false
1054.58/291.51	gen_s:0'3_0 :: Nat -> s:0'
1054.58/291.51	
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(5) OrderProof (LOWER BOUND(ID))
1054.58/291.51	Heuristically decided to analyse the following defined symbols:
1054.58/291.51	d, ge, plus
1054.58/291.51	
1054.58/291.51	They will be analysed ascendingly in the following order:
1054.58/291.51	ge < d
1054.58/291.51	plus < d
1054.58/291.51	
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(6)
1054.58/291.51	Obligation:
1054.58/291.51	Innermost TRS:
1054.58/291.51	Rules:
1054.58/291.51	div(x, s(y)) -> d(x, s(y), 0')
1054.58/291.51	d(x, s(y), z) -> cond(ge(x, z), x, y, z)
1054.58/291.51	cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
1054.58/291.51	cond(false, x, y, z) -> 0'
1054.58/291.51	ge(u, 0') -> true
1054.58/291.51	ge(0', s(v)) -> false
1054.58/291.51	ge(s(u), s(v)) -> ge(u, v)
1054.58/291.51	plus(n, 0') -> n
1054.58/291.51	plus(n, s(m)) -> s(plus(n, m))
1054.58/291.51	
1054.58/291.51	Types:
1054.58/291.51	div :: s:0' -> s:0' -> s:0'
1054.58/291.51	s :: s:0' -> s:0'
1054.58/291.51	d :: s:0' -> s:0' -> s:0' -> s:0'
1054.58/291.51	0' :: s:0'
1054.58/291.51	cond :: true:false -> s:0' -> s:0' -> s:0' -> s:0'
1054.58/291.51	ge :: s:0' -> s:0' -> true:false
1054.58/291.51	true :: true:false
1054.58/291.51	plus :: s:0' -> s:0' -> s:0'
1054.58/291.51	false :: true:false
1054.58/291.51	hole_s:0'1_0 :: s:0'
1054.58/291.51	hole_true:false2_0 :: true:false
1054.58/291.51	gen_s:0'3_0 :: Nat -> s:0'
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	Generator Equations:
1054.58/291.51	gen_s:0'3_0(0) <=> 0'
1054.58/291.51	gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	The following defined symbols remain to be analysed:
1054.58/291.51	ge, d, plus
1054.58/291.51	
1054.58/291.51	They will be analysed ascendingly in the following order:
1054.58/291.51	ge < d
1054.58/291.51	plus < d
1054.58/291.51	
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(7) RewriteLemmaProof (LOWER BOUND(ID))
1054.58/291.51	Proved the following rewrite lemma:
1054.58/291.51	ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
1054.58/291.51	
1054.58/291.51	Induction Base:
1054.58/291.51	ge(gen_s:0'3_0(0), gen_s:0'3_0(0)) ->_R^Omega(1)
1054.58/291.51	true
1054.58/291.51	
1054.58/291.51	Induction Step:
1054.58/291.51	ge(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1)
1054.58/291.51	ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) ->_IH
1054.58/291.51	true
1054.58/291.51	
1054.58/291.51	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(8)
1054.58/291.51	Complex Obligation (BEST)
1054.58/291.51	
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(9)
1054.58/291.51	Obligation:
1054.58/291.51	Proved the lower bound n^1 for the following obligation:
1054.58/291.51	
1054.58/291.51	Innermost TRS:
1054.58/291.51	Rules:
1054.58/291.51	div(x, s(y)) -> d(x, s(y), 0')
1054.58/291.51	d(x, s(y), z) -> cond(ge(x, z), x, y, z)
1054.58/291.51	cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
1054.58/291.51	cond(false, x, y, z) -> 0'
1054.58/291.51	ge(u, 0') -> true
1054.58/291.51	ge(0', s(v)) -> false
1054.58/291.51	ge(s(u), s(v)) -> ge(u, v)
1054.58/291.51	plus(n, 0') -> n
1054.58/291.51	plus(n, s(m)) -> s(plus(n, m))
1054.58/291.51	
1054.58/291.51	Types:
1054.58/291.51	div :: s:0' -> s:0' -> s:0'
1054.58/291.51	s :: s:0' -> s:0'
1054.58/291.51	d :: s:0' -> s:0' -> s:0' -> s:0'
1054.58/291.51	0' :: s:0'
1054.58/291.51	cond :: true:false -> s:0' -> s:0' -> s:0' -> s:0'
1054.58/291.51	ge :: s:0' -> s:0' -> true:false
1054.58/291.51	true :: true:false
1054.58/291.51	plus :: s:0' -> s:0' -> s:0'
1054.58/291.51	false :: true:false
1054.58/291.51	hole_s:0'1_0 :: s:0'
1054.58/291.51	hole_true:false2_0 :: true:false
1054.58/291.51	gen_s:0'3_0 :: Nat -> s:0'
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	Generator Equations:
1054.58/291.51	gen_s:0'3_0(0) <=> 0'
1054.58/291.51	gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	The following defined symbols remain to be analysed:
1054.58/291.51	ge, d, plus
1054.58/291.51	
1054.58/291.51	They will be analysed ascendingly in the following order:
1054.58/291.51	ge < d
1054.58/291.51	plus < d
1054.58/291.51	
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(10) LowerBoundPropagationProof (FINISHED)
1054.58/291.51	Propagated lower bound.
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(11)
1054.58/291.51	BOUNDS(n^1, INF)
1054.58/291.51	
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(12)
1054.58/291.51	Obligation:
1054.58/291.51	Innermost TRS:
1054.58/291.51	Rules:
1054.58/291.51	div(x, s(y)) -> d(x, s(y), 0')
1054.58/291.51	d(x, s(y), z) -> cond(ge(x, z), x, y, z)
1054.58/291.51	cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
1054.58/291.51	cond(false, x, y, z) -> 0'
1054.58/291.51	ge(u, 0') -> true
1054.58/291.51	ge(0', s(v)) -> false
1054.58/291.51	ge(s(u), s(v)) -> ge(u, v)
1054.58/291.51	plus(n, 0') -> n
1054.58/291.51	plus(n, s(m)) -> s(plus(n, m))
1054.58/291.51	
1054.58/291.51	Types:
1054.58/291.51	div :: s:0' -> s:0' -> s:0'
1054.58/291.51	s :: s:0' -> s:0'
1054.58/291.51	d :: s:0' -> s:0' -> s:0' -> s:0'
1054.58/291.51	0' :: s:0'
1054.58/291.51	cond :: true:false -> s:0' -> s:0' -> s:0' -> s:0'
1054.58/291.51	ge :: s:0' -> s:0' -> true:false
1054.58/291.51	true :: true:false
1054.58/291.51	plus :: s:0' -> s:0' -> s:0'
1054.58/291.51	false :: true:false
1054.58/291.51	hole_s:0'1_0 :: s:0'
1054.58/291.51	hole_true:false2_0 :: true:false
1054.58/291.51	gen_s:0'3_0 :: Nat -> s:0'
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	Lemmas:
1054.58/291.51	ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	Generator Equations:
1054.58/291.51	gen_s:0'3_0(0) <=> 0'
1054.58/291.51	gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	The following defined symbols remain to be analysed:
1054.58/291.51	plus, d
1054.58/291.51	
1054.58/291.51	They will be analysed ascendingly in the following order:
1054.58/291.51	plus < d
1054.58/291.51	
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(13) RewriteLemmaProof (LOWER BOUND(ID))
1054.58/291.51	Proved the following rewrite lemma:
1054.58/291.51	plus(gen_s:0'3_0(a), gen_s:0'3_0(n281_0)) -> gen_s:0'3_0(+(n281_0, a)), rt in Omega(1 + n281_0)
1054.58/291.51	
1054.58/291.51	Induction Base:
1054.58/291.51	plus(gen_s:0'3_0(a), gen_s:0'3_0(0)) ->_R^Omega(1)
1054.58/291.51	gen_s:0'3_0(a)
1054.58/291.51	
1054.58/291.51	Induction Step:
1054.58/291.51	plus(gen_s:0'3_0(a), gen_s:0'3_0(+(n281_0, 1))) ->_R^Omega(1)
1054.58/291.51	s(plus(gen_s:0'3_0(a), gen_s:0'3_0(n281_0))) ->_IH
1054.58/291.51	s(gen_s:0'3_0(+(a, c282_0)))
1054.58/291.51	
1054.58/291.51	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1054.58/291.51	----------------------------------------
1054.58/291.51	
1054.58/291.51	(14)
1054.58/291.51	Obligation:
1054.58/291.51	Innermost TRS:
1054.58/291.51	Rules:
1054.58/291.51	div(x, s(y)) -> d(x, s(y), 0')
1054.58/291.51	d(x, s(y), z) -> cond(ge(x, z), x, y, z)
1054.58/291.51	cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
1054.58/291.51	cond(false, x, y, z) -> 0'
1054.58/291.51	ge(u, 0') -> true
1054.58/291.51	ge(0', s(v)) -> false
1054.58/291.51	ge(s(u), s(v)) -> ge(u, v)
1054.58/291.51	plus(n, 0') -> n
1054.58/291.51	plus(n, s(m)) -> s(plus(n, m))
1054.58/291.51	
1054.58/291.51	Types:
1054.58/291.51	div :: s:0' -> s:0' -> s:0'
1054.58/291.51	s :: s:0' -> s:0'
1054.58/291.51	d :: s:0' -> s:0' -> s:0' -> s:0'
1054.58/291.51	0' :: s:0'
1054.58/291.51	cond :: true:false -> s:0' -> s:0' -> s:0' -> s:0'
1054.58/291.51	ge :: s:0' -> s:0' -> true:false
1054.58/291.51	true :: true:false
1054.58/291.51	plus :: s:0' -> s:0' -> s:0'
1054.58/291.51	false :: true:false
1054.58/291.51	hole_s:0'1_0 :: s:0'
1054.58/291.51	hole_true:false2_0 :: true:false
1054.58/291.51	gen_s:0'3_0 :: Nat -> s:0'
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	Lemmas:
1054.58/291.51	ge(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
1054.58/291.51	plus(gen_s:0'3_0(a), gen_s:0'3_0(n281_0)) -> gen_s:0'3_0(+(n281_0, a)), rt in Omega(1 + n281_0)
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	Generator Equations:
1054.58/291.51	gen_s:0'3_0(0) <=> 0'
1054.58/291.51	gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))
1054.58/291.51	
1054.58/291.51	
1054.58/291.51	The following defined symbols remain to be analysed:
1054.58/291.51	d
1054.85/291.55	EOF