303.88/291.56	WORST_CASE(Omega(n^1), ?)
303.88/291.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
303.88/291.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
303.88/291.57	
303.88/291.57	
303.88/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.88/291.57	
303.88/291.57	(0) CpxTRS
303.88/291.57	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
303.88/291.57	(2) CpxTRS
303.88/291.57	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
303.88/291.57	(4) typed CpxTrs
303.88/291.57	(5) OrderProof [LOWER BOUND(ID), 0 ms]
303.88/291.57	(6) typed CpxTrs
303.88/291.57	(7) RewriteLemmaProof [LOWER BOUND(ID), 290 ms]
303.88/291.57	(8) BEST
303.88/291.57	    (9) proven lower bound
303.88/291.57	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
303.88/291.57	        (11) BOUNDS(n^1, INF)
303.88/291.57	    (12) typed CpxTrs
303.88/291.57	        (13) RewriteLemmaProof [LOWER BOUND(ID), 68 ms]
303.88/291.57	        (14) typed CpxTrs
303.88/291.57	
303.88/291.57	
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(0)
303.88/291.57	Obligation:
303.88/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.88/291.57	
303.88/291.57	
303.88/291.57	The TRS R consists of the following rules:
303.88/291.57	
303.88/291.57	   ge(x, 0) -> true
303.88/291.57	   ge(0, s(y)) -> false
303.88/291.57	   ge(s(x), s(y)) -> ge(x, y)
303.88/291.57	   minus(x, 0) -> x
303.88/291.57	   minus(0, y) -> 0
303.88/291.57	   minus(s(x), s(y)) -> minus(x, y)
303.88/291.57	   id_inc(x) -> x
303.88/291.57	   id_inc(x) -> s(x)
303.88/291.57	   div(x, y) -> if(ge(y, s(0)), ge(x, y), x, y)
303.88/291.57	   if(false, b, x, y) -> div_by_zero
303.88/291.57	   if(true, false, x, y) -> 0
303.88/291.57	   if(true, true, x, y) -> id_inc(div(minus(x, y), y))
303.88/291.57	
303.88/291.57	S is empty.
303.88/291.57	Rewrite Strategy: FULL
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(1) RenamingProof (BOTH BOUNDS(ID, ID))
303.88/291.57	Renamed function symbols to avoid clashes with predefined symbol.
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(2)
303.88/291.57	Obligation:
303.88/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.88/291.57	
303.88/291.57	
303.88/291.57	The TRS R consists of the following rules:
303.88/291.57	
303.88/291.57	   ge(x, 0') -> true
303.88/291.57	   ge(0', s(y)) -> false
303.88/291.57	   ge(s(x), s(y)) -> ge(x, y)
303.88/291.57	   minus(x, 0') -> x
303.88/291.57	   minus(0', y) -> 0'
303.88/291.57	   minus(s(x), s(y)) -> minus(x, y)
303.88/291.57	   id_inc(x) -> x
303.88/291.57	   id_inc(x) -> s(x)
303.88/291.57	   div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
303.88/291.57	   if(false, b, x, y) -> div_by_zero
303.88/291.57	   if(true, false, x, y) -> 0'
303.88/291.57	   if(true, true, x, y) -> id_inc(div(minus(x, y), y))
303.88/291.57	
303.88/291.57	S is empty.
303.88/291.57	Rewrite Strategy: FULL
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
303.88/291.57	Infered types.
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(4)
303.88/291.57	Obligation:
303.88/291.57	TRS:
303.88/291.57	Rules:
303.88/291.57	ge(x, 0') -> true
303.88/291.57	ge(0', s(y)) -> false
303.88/291.57	ge(s(x), s(y)) -> ge(x, y)
303.88/291.57	minus(x, 0') -> x
303.88/291.57	minus(0', y) -> 0'
303.88/291.57	minus(s(x), s(y)) -> minus(x, y)
303.88/291.57	id_inc(x) -> x
303.88/291.57	id_inc(x) -> s(x)
303.88/291.57	div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
303.88/291.57	if(false, b, x, y) -> div_by_zero
303.88/291.57	if(true, false, x, y) -> 0'
303.88/291.57	if(true, true, x, y) -> id_inc(div(minus(x, y), y))
303.88/291.57	
303.88/291.57	Types:
303.88/291.57	ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
303.88/291.57	0' :: 0':s:div_by_zero
303.88/291.57	true :: true:false
303.88/291.57	s :: 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	false :: true:false
303.88/291.57	minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	div_by_zero :: 0':s:div_by_zero
303.88/291.57	hole_true:false1_0 :: true:false
303.88/291.57	hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
303.88/291.57	gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero
303.88/291.57	
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(5) OrderProof (LOWER BOUND(ID))
303.88/291.57	Heuristically decided to analyse the following defined symbols:
303.88/291.57	ge, minus, div
303.88/291.57	
303.88/291.57	They will be analysed ascendingly in the following order:
303.88/291.57	ge < div
303.88/291.57	minus < div
303.88/291.57	
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(6)
303.88/291.57	Obligation:
303.88/291.57	TRS:
303.88/291.57	Rules:
303.88/291.57	ge(x, 0') -> true
303.88/291.57	ge(0', s(y)) -> false
303.88/291.57	ge(s(x), s(y)) -> ge(x, y)
303.88/291.57	minus(x, 0') -> x
303.88/291.57	minus(0', y) -> 0'
303.88/291.57	minus(s(x), s(y)) -> minus(x, y)
303.88/291.57	id_inc(x) -> x
303.88/291.57	id_inc(x) -> s(x)
303.88/291.57	div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
303.88/291.57	if(false, b, x, y) -> div_by_zero
303.88/291.57	if(true, false, x, y) -> 0'
303.88/291.57	if(true, true, x, y) -> id_inc(div(minus(x, y), y))
303.88/291.57	
303.88/291.57	Types:
303.88/291.57	ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
303.88/291.57	0' :: 0':s:div_by_zero
303.88/291.57	true :: true:false
303.88/291.57	s :: 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	false :: true:false
303.88/291.57	minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	div_by_zero :: 0':s:div_by_zero
303.88/291.57	hole_true:false1_0 :: true:false
303.88/291.57	hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
303.88/291.57	gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero
303.88/291.57	
303.88/291.57	
303.88/291.57	Generator Equations:
303.88/291.57	gen_0':s:div_by_zero3_0(0) <=> 0'
303.88/291.57	gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))
303.88/291.57	
303.88/291.57	
303.88/291.57	The following defined symbols remain to be analysed:
303.88/291.57	ge, minus, div
303.88/291.57	
303.88/291.57	They will be analysed ascendingly in the following order:
303.88/291.57	ge < div
303.88/291.57	minus < div
303.88/291.57	
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(7) RewriteLemmaProof (LOWER BOUND(ID))
303.88/291.57	Proved the following rewrite lemma:
303.88/291.57	ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
303.88/291.57	
303.88/291.57	Induction Base:
303.88/291.57	ge(gen_0':s:div_by_zero3_0(0), gen_0':s:div_by_zero3_0(0)) ->_R^Omega(1)
303.88/291.57	true
303.88/291.57	
303.88/291.57	Induction Step:
303.88/291.57	ge(gen_0':s:div_by_zero3_0(+(n5_0, 1)), gen_0':s:div_by_zero3_0(+(n5_0, 1))) ->_R^Omega(1)
303.88/291.57	ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) ->_IH
303.88/291.57	true
303.88/291.57	
303.88/291.57	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(8)
303.88/291.57	Complex Obligation (BEST)
303.88/291.57	
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(9)
303.88/291.57	Obligation:
303.88/291.57	Proved the lower bound n^1 for the following obligation:
303.88/291.57	
303.88/291.57	TRS:
303.88/291.57	Rules:
303.88/291.57	ge(x, 0') -> true
303.88/291.57	ge(0', s(y)) -> false
303.88/291.57	ge(s(x), s(y)) -> ge(x, y)
303.88/291.57	minus(x, 0') -> x
303.88/291.57	minus(0', y) -> 0'
303.88/291.57	minus(s(x), s(y)) -> minus(x, y)
303.88/291.57	id_inc(x) -> x
303.88/291.57	id_inc(x) -> s(x)
303.88/291.57	div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
303.88/291.57	if(false, b, x, y) -> div_by_zero
303.88/291.57	if(true, false, x, y) -> 0'
303.88/291.57	if(true, true, x, y) -> id_inc(div(minus(x, y), y))
303.88/291.57	
303.88/291.57	Types:
303.88/291.57	ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
303.88/291.57	0' :: 0':s:div_by_zero
303.88/291.57	true :: true:false
303.88/291.57	s :: 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	false :: true:false
303.88/291.57	minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	div_by_zero :: 0':s:div_by_zero
303.88/291.57	hole_true:false1_0 :: true:false
303.88/291.57	hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
303.88/291.57	gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero
303.88/291.57	
303.88/291.57	
303.88/291.57	Generator Equations:
303.88/291.57	gen_0':s:div_by_zero3_0(0) <=> 0'
303.88/291.57	gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))
303.88/291.57	
303.88/291.57	
303.88/291.57	The following defined symbols remain to be analysed:
303.88/291.57	ge, minus, div
303.88/291.57	
303.88/291.57	They will be analysed ascendingly in the following order:
303.88/291.57	ge < div
303.88/291.57	minus < div
303.88/291.57	
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(10) LowerBoundPropagationProof (FINISHED)
303.88/291.57	Propagated lower bound.
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(11)
303.88/291.57	BOUNDS(n^1, INF)
303.88/291.57	
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(12)
303.88/291.57	Obligation:
303.88/291.57	TRS:
303.88/291.57	Rules:
303.88/291.57	ge(x, 0') -> true
303.88/291.57	ge(0', s(y)) -> false
303.88/291.57	ge(s(x), s(y)) -> ge(x, y)
303.88/291.57	minus(x, 0') -> x
303.88/291.57	minus(0', y) -> 0'
303.88/291.57	minus(s(x), s(y)) -> minus(x, y)
303.88/291.57	id_inc(x) -> x
303.88/291.57	id_inc(x) -> s(x)
303.88/291.57	div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
303.88/291.57	if(false, b, x, y) -> div_by_zero
303.88/291.57	if(true, false, x, y) -> 0'
303.88/291.57	if(true, true, x, y) -> id_inc(div(minus(x, y), y))
303.88/291.57	
303.88/291.57	Types:
303.88/291.57	ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
303.88/291.57	0' :: 0':s:div_by_zero
303.88/291.57	true :: true:false
303.88/291.57	s :: 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	false :: true:false
303.88/291.57	minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	div_by_zero :: 0':s:div_by_zero
303.88/291.57	hole_true:false1_0 :: true:false
303.88/291.57	hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
303.88/291.57	gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero
303.88/291.57	
303.88/291.57	
303.88/291.57	Lemmas:
303.88/291.57	ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
303.88/291.57	
303.88/291.57	
303.88/291.57	Generator Equations:
303.88/291.57	gen_0':s:div_by_zero3_0(0) <=> 0'
303.88/291.57	gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))
303.88/291.57	
303.88/291.57	
303.88/291.57	The following defined symbols remain to be analysed:
303.88/291.57	minus, div
303.88/291.57	
303.88/291.57	They will be analysed ascendingly in the following order:
303.88/291.57	minus < div
303.88/291.57	
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(13) RewriteLemmaProof (LOWER BOUND(ID))
303.88/291.57	Proved the following rewrite lemma:
303.88/291.57	minus(gen_0':s:div_by_zero3_0(n263_0), gen_0':s:div_by_zero3_0(n263_0)) -> gen_0':s:div_by_zero3_0(0), rt in Omega(1 + n263_0)
303.88/291.57	
303.88/291.57	Induction Base:
303.88/291.57	minus(gen_0':s:div_by_zero3_0(0), gen_0':s:div_by_zero3_0(0)) ->_R^Omega(1)
303.88/291.57	gen_0':s:div_by_zero3_0(0)
303.88/291.57	
303.88/291.57	Induction Step:
303.88/291.57	minus(gen_0':s:div_by_zero3_0(+(n263_0, 1)), gen_0':s:div_by_zero3_0(+(n263_0, 1))) ->_R^Omega(1)
303.88/291.57	minus(gen_0':s:div_by_zero3_0(n263_0), gen_0':s:div_by_zero3_0(n263_0)) ->_IH
303.88/291.57	gen_0':s:div_by_zero3_0(0)
303.88/291.57	
303.88/291.57	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
303.88/291.57	----------------------------------------
303.88/291.57	
303.88/291.57	(14)
303.88/291.57	Obligation:
303.88/291.57	TRS:
303.88/291.57	Rules:
303.88/291.57	ge(x, 0') -> true
303.88/291.57	ge(0', s(y)) -> false
303.88/291.57	ge(s(x), s(y)) -> ge(x, y)
303.88/291.57	minus(x, 0') -> x
303.88/291.57	minus(0', y) -> 0'
303.88/291.57	minus(s(x), s(y)) -> minus(x, y)
303.88/291.57	id_inc(x) -> x
303.88/291.57	id_inc(x) -> s(x)
303.88/291.57	div(x, y) -> if(ge(y, s(0')), ge(x, y), x, y)
303.88/291.57	if(false, b, x, y) -> div_by_zero
303.88/291.57	if(true, false, x, y) -> 0'
303.88/291.57	if(true, true, x, y) -> id_inc(div(minus(x, y), y))
303.88/291.57	
303.88/291.57	Types:
303.88/291.57	ge :: 0':s:div_by_zero -> 0':s:div_by_zero -> true:false
303.88/291.57	0' :: 0':s:div_by_zero
303.88/291.57	true :: true:false
303.88/291.57	s :: 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	false :: true:false
303.88/291.57	minus :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	id_inc :: 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	div :: 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	if :: true:false -> true:false -> 0':s:div_by_zero -> 0':s:div_by_zero -> 0':s:div_by_zero
303.88/291.57	div_by_zero :: 0':s:div_by_zero
303.88/291.57	hole_true:false1_0 :: true:false
303.88/291.57	hole_0':s:div_by_zero2_0 :: 0':s:div_by_zero
303.88/291.57	gen_0':s:div_by_zero3_0 :: Nat -> 0':s:div_by_zero
303.88/291.57	
303.88/291.57	
303.88/291.57	Lemmas:
303.88/291.57	ge(gen_0':s:div_by_zero3_0(n5_0), gen_0':s:div_by_zero3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
303.88/291.57	minus(gen_0':s:div_by_zero3_0(n263_0), gen_0':s:div_by_zero3_0(n263_0)) -> gen_0':s:div_by_zero3_0(0), rt in Omega(1 + n263_0)
303.88/291.57	
303.88/291.57	
303.88/291.57	Generator Equations:
303.88/291.57	gen_0':s:div_by_zero3_0(0) <=> 0'
303.88/291.57	gen_0':s:div_by_zero3_0(+(x, 1)) <=> s(gen_0':s:div_by_zero3_0(x))
303.88/291.57	
303.88/291.57	
303.88/291.57	The following defined symbols remain to be analysed:
303.88/291.57	div
303.97/291.59	EOF