301.41/291.50	WORST_CASE(Omega(n^1), ?)
301.54/291.55	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
301.54/291.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
301.54/291.55	
301.54/291.55	
301.54/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.54/291.55	
301.54/291.55	(0) CpxTRS
301.54/291.55	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
301.54/291.55	(2) CpxTRS
301.54/291.55	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
301.54/291.55	(4) typed CpxTrs
301.54/291.55	(5) OrderProof [LOWER BOUND(ID), 0 ms]
301.54/291.55	(6) typed CpxTrs
301.54/291.55	(7) RewriteLemmaProof [LOWER BOUND(ID), 234 ms]
301.54/291.55	(8) BEST
301.54/291.55	    (9) proven lower bound
301.54/291.55	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
301.54/291.55	        (11) BOUNDS(n^1, INF)
301.54/291.55	    (12) typed CpxTrs
301.54/291.55	        (13) RewriteLemmaProof [LOWER BOUND(ID), 85 ms]
301.54/291.55	        (14) typed CpxTrs
301.54/291.55	
301.54/291.55	
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(0)
301.54/291.55	Obligation:
301.54/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.54/291.55	
301.54/291.55	
301.54/291.55	The TRS R consists of the following rules:
301.54/291.55	
301.54/291.55	   cond1(true, x, y) -> cond2(gr(x, y), x, y)
301.54/291.55	   cond2(true, x, y) -> cond1(neq(x, 0), y, y)
301.54/291.55	   cond2(false, x, y) -> cond1(neq(x, 0), p(x), y)
301.54/291.55	   gr(0, x) -> false
301.54/291.55	   gr(s(x), 0) -> true
301.54/291.55	   gr(s(x), s(y)) -> gr(x, y)
301.54/291.55	   neq(0, 0) -> false
301.54/291.55	   neq(0, s(x)) -> true
301.54/291.55	   neq(s(x), 0) -> true
301.54/291.55	   neq(s(x), s(y)) -> neq(x, y)
301.54/291.55	   p(0) -> 0
301.54/291.55	   p(s(x)) -> x
301.54/291.55	
301.54/291.55	S is empty.
301.54/291.55	Rewrite Strategy: FULL
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(1) RenamingProof (BOTH BOUNDS(ID, ID))
301.54/291.55	Renamed function symbols to avoid clashes with predefined symbol.
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(2)
301.54/291.55	Obligation:
301.54/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.54/291.55	
301.54/291.55	
301.54/291.55	The TRS R consists of the following rules:
301.54/291.55	
301.54/291.55	   cond1(true, x, y) -> cond2(gr(x, y), x, y)
301.54/291.55	   cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
301.54/291.55	   cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
301.54/291.55	   gr(0', x) -> false
301.54/291.55	   gr(s(x), 0') -> true
301.54/291.55	   gr(s(x), s(y)) -> gr(x, y)
301.54/291.55	   neq(0', 0') -> false
301.54/291.55	   neq(0', s(x)) -> true
301.54/291.55	   neq(s(x), 0') -> true
301.54/291.55	   neq(s(x), s(y)) -> neq(x, y)
301.54/291.55	   p(0') -> 0'
301.54/291.55	   p(s(x)) -> x
301.54/291.55	
301.54/291.55	S is empty.
301.54/291.55	Rewrite Strategy: FULL
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
301.54/291.55	Infered types.
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(4)
301.54/291.55	Obligation:
301.54/291.55	TRS:
301.54/291.55	Rules:
301.54/291.55	cond1(true, x, y) -> cond2(gr(x, y), x, y)
301.54/291.55	cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
301.54/291.55	cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
301.54/291.55	gr(0', x) -> false
301.54/291.55	gr(s(x), 0') -> true
301.54/291.55	gr(s(x), s(y)) -> gr(x, y)
301.54/291.55	neq(0', 0') -> false
301.54/291.55	neq(0', s(x)) -> true
301.54/291.55	neq(s(x), 0') -> true
301.54/291.55	neq(s(x), s(y)) -> neq(x, y)
301.54/291.55	p(0') -> 0'
301.54/291.55	p(s(x)) -> x
301.54/291.55	
301.54/291.55	Types:
301.54/291.55	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
301.54/291.55	true :: true:false
301.54/291.55	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
301.54/291.55	gr :: 0':s -> 0':s -> true:false
301.54/291.55	neq :: 0':s -> 0':s -> true:false
301.54/291.55	0' :: 0':s
301.54/291.55	false :: true:false
301.54/291.55	p :: 0':s -> 0':s
301.54/291.55	s :: 0':s -> 0':s
301.54/291.55	hole_cond1:cond21_0 :: cond1:cond2
301.54/291.55	hole_true:false2_0 :: true:false
301.54/291.55	hole_0':s3_0 :: 0':s
301.54/291.55	gen_0':s4_0 :: Nat -> 0':s
301.54/291.55	
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(5) OrderProof (LOWER BOUND(ID))
301.54/291.55	Heuristically decided to analyse the following defined symbols:
301.54/291.55	cond1, cond2, gr, neq
301.54/291.55	
301.54/291.55	They will be analysed ascendingly in the following order:
301.54/291.55	cond1 = cond2
301.54/291.55	gr < cond1
301.54/291.55	neq < cond2
301.54/291.55	
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(6)
301.54/291.55	Obligation:
301.54/291.55	TRS:
301.54/291.55	Rules:
301.54/291.55	cond1(true, x, y) -> cond2(gr(x, y), x, y)
301.54/291.55	cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
301.54/291.55	cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
301.54/291.55	gr(0', x) -> false
301.54/291.55	gr(s(x), 0') -> true
301.54/291.55	gr(s(x), s(y)) -> gr(x, y)
301.54/291.55	neq(0', 0') -> false
301.54/291.55	neq(0', s(x)) -> true
301.54/291.55	neq(s(x), 0') -> true
301.54/291.55	neq(s(x), s(y)) -> neq(x, y)
301.54/291.55	p(0') -> 0'
301.54/291.55	p(s(x)) -> x
301.54/291.55	
301.54/291.55	Types:
301.54/291.55	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
301.54/291.55	true :: true:false
301.54/291.55	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
301.54/291.55	gr :: 0':s -> 0':s -> true:false
301.54/291.55	neq :: 0':s -> 0':s -> true:false
301.54/291.55	0' :: 0':s
301.54/291.55	false :: true:false
301.54/291.55	p :: 0':s -> 0':s
301.54/291.55	s :: 0':s -> 0':s
301.54/291.55	hole_cond1:cond21_0 :: cond1:cond2
301.54/291.55	hole_true:false2_0 :: true:false
301.54/291.55	hole_0':s3_0 :: 0':s
301.54/291.55	gen_0':s4_0 :: Nat -> 0':s
301.54/291.55	
301.54/291.55	
301.54/291.55	Generator Equations:
301.54/291.55	gen_0':s4_0(0) <=> 0'
301.54/291.55	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
301.54/291.55	
301.54/291.55	
301.54/291.55	The following defined symbols remain to be analysed:
301.54/291.55	gr, cond1, cond2, neq
301.54/291.55	
301.54/291.55	They will be analysed ascendingly in the following order:
301.54/291.55	cond1 = cond2
301.54/291.55	gr < cond1
301.54/291.55	neq < cond2
301.54/291.55	
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(7) RewriteLemmaProof (LOWER BOUND(ID))
301.54/291.55	Proved the following rewrite lemma:
301.54/291.55	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
301.54/291.55	
301.54/291.55	Induction Base:
301.54/291.55	gr(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
301.54/291.55	false
301.54/291.55	
301.54/291.55	Induction Step:
301.54/291.55	gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
301.54/291.55	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH
301.54/291.55	false
301.54/291.55	
301.54/291.55	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(8)
301.54/291.55	Complex Obligation (BEST)
301.54/291.55	
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(9)
301.54/291.55	Obligation:
301.54/291.55	Proved the lower bound n^1 for the following obligation:
301.54/291.55	
301.54/291.55	TRS:
301.54/291.55	Rules:
301.54/291.55	cond1(true, x, y) -> cond2(gr(x, y), x, y)
301.54/291.55	cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
301.54/291.55	cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
301.54/291.55	gr(0', x) -> false
301.54/291.55	gr(s(x), 0') -> true
301.54/291.55	gr(s(x), s(y)) -> gr(x, y)
301.54/291.55	neq(0', 0') -> false
301.54/291.55	neq(0', s(x)) -> true
301.54/291.55	neq(s(x), 0') -> true
301.54/291.55	neq(s(x), s(y)) -> neq(x, y)
301.54/291.55	p(0') -> 0'
301.54/291.55	p(s(x)) -> x
301.54/291.55	
301.54/291.55	Types:
301.54/291.55	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
301.54/291.55	true :: true:false
301.54/291.55	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
301.54/291.55	gr :: 0':s -> 0':s -> true:false
301.54/291.55	neq :: 0':s -> 0':s -> true:false
301.54/291.55	0' :: 0':s
301.54/291.55	false :: true:false
301.54/291.55	p :: 0':s -> 0':s
301.54/291.55	s :: 0':s -> 0':s
301.54/291.55	hole_cond1:cond21_0 :: cond1:cond2
301.54/291.55	hole_true:false2_0 :: true:false
301.54/291.55	hole_0':s3_0 :: 0':s
301.54/291.55	gen_0':s4_0 :: Nat -> 0':s
301.54/291.55	
301.54/291.55	
301.54/291.55	Generator Equations:
301.54/291.55	gen_0':s4_0(0) <=> 0'
301.54/291.55	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
301.54/291.55	
301.54/291.55	
301.54/291.55	The following defined symbols remain to be analysed:
301.54/291.55	gr, cond1, cond2, neq
301.54/291.55	
301.54/291.55	They will be analysed ascendingly in the following order:
301.54/291.55	cond1 = cond2
301.54/291.55	gr < cond1
301.54/291.55	neq < cond2
301.54/291.55	
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(10) LowerBoundPropagationProof (FINISHED)
301.54/291.55	Propagated lower bound.
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(11)
301.54/291.55	BOUNDS(n^1, INF)
301.54/291.55	
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(12)
301.54/291.55	Obligation:
301.54/291.55	TRS:
301.54/291.55	Rules:
301.54/291.55	cond1(true, x, y) -> cond2(gr(x, y), x, y)
301.54/291.55	cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
301.54/291.55	cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
301.54/291.55	gr(0', x) -> false
301.54/291.55	gr(s(x), 0') -> true
301.54/291.55	gr(s(x), s(y)) -> gr(x, y)
301.54/291.55	neq(0', 0') -> false
301.54/291.55	neq(0', s(x)) -> true
301.54/291.55	neq(s(x), 0') -> true
301.54/291.55	neq(s(x), s(y)) -> neq(x, y)
301.54/291.55	p(0') -> 0'
301.54/291.55	p(s(x)) -> x
301.54/291.55	
301.54/291.55	Types:
301.54/291.55	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
301.54/291.55	true :: true:false
301.54/291.55	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
301.54/291.55	gr :: 0':s -> 0':s -> true:false
301.54/291.55	neq :: 0':s -> 0':s -> true:false
301.54/291.55	0' :: 0':s
301.54/291.55	false :: true:false
301.54/291.55	p :: 0':s -> 0':s
301.54/291.55	s :: 0':s -> 0':s
301.54/291.55	hole_cond1:cond21_0 :: cond1:cond2
301.54/291.55	hole_true:false2_0 :: true:false
301.54/291.55	hole_0':s3_0 :: 0':s
301.54/291.55	gen_0':s4_0 :: Nat -> 0':s
301.54/291.55	
301.54/291.55	
301.54/291.55	Lemmas:
301.54/291.55	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
301.54/291.55	
301.54/291.55	
301.54/291.55	Generator Equations:
301.54/291.55	gen_0':s4_0(0) <=> 0'
301.54/291.55	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
301.54/291.55	
301.54/291.55	
301.54/291.55	The following defined symbols remain to be analysed:
301.54/291.55	neq, cond1, cond2
301.54/291.55	
301.54/291.55	They will be analysed ascendingly in the following order:
301.54/291.55	cond1 = cond2
301.54/291.55	neq < cond2
301.54/291.55	
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(13) RewriteLemmaProof (LOWER BOUND(ID))
301.54/291.55	Proved the following rewrite lemma:
301.54/291.55	neq(gen_0':s4_0(n265_0), gen_0':s4_0(n265_0)) -> false, rt in Omega(1 + n265_0)
301.54/291.55	
301.54/291.55	Induction Base:
301.54/291.55	neq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
301.54/291.55	false
301.54/291.55	
301.54/291.55	Induction Step:
301.54/291.55	neq(gen_0':s4_0(+(n265_0, 1)), gen_0':s4_0(+(n265_0, 1))) ->_R^Omega(1)
301.54/291.55	neq(gen_0':s4_0(n265_0), gen_0':s4_0(n265_0)) ->_IH
301.54/291.55	false
301.54/291.55	
301.54/291.55	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
301.54/291.55	----------------------------------------
301.54/291.55	
301.54/291.55	(14)
301.54/291.55	Obligation:
301.54/291.55	TRS:
301.54/291.55	Rules:
301.54/291.55	cond1(true, x, y) -> cond2(gr(x, y), x, y)
301.54/291.55	cond2(true, x, y) -> cond1(neq(x, 0'), y, y)
301.54/291.55	cond2(false, x, y) -> cond1(neq(x, 0'), p(x), y)
301.54/291.55	gr(0', x) -> false
301.54/291.55	gr(s(x), 0') -> true
301.54/291.55	gr(s(x), s(y)) -> gr(x, y)
301.54/291.55	neq(0', 0') -> false
301.54/291.55	neq(0', s(x)) -> true
301.54/291.55	neq(s(x), 0') -> true
301.54/291.55	neq(s(x), s(y)) -> neq(x, y)
301.54/291.55	p(0') -> 0'
301.54/291.55	p(s(x)) -> x
301.54/291.55	
301.54/291.55	Types:
301.54/291.55	cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
301.54/291.55	true :: true:false
301.54/291.55	cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
301.54/291.55	gr :: 0':s -> 0':s -> true:false
301.54/291.55	neq :: 0':s -> 0':s -> true:false
301.54/291.55	0' :: 0':s
301.54/291.55	false :: true:false
301.54/291.55	p :: 0':s -> 0':s
301.54/291.55	s :: 0':s -> 0':s
301.54/291.55	hole_cond1:cond21_0 :: cond1:cond2
301.54/291.55	hole_true:false2_0 :: true:false
301.54/291.55	hole_0':s3_0 :: 0':s
301.54/291.55	gen_0':s4_0 :: Nat -> 0':s
301.54/291.55	
301.54/291.55	
301.54/291.55	Lemmas:
301.54/291.55	gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
301.54/291.55	neq(gen_0':s4_0(n265_0), gen_0':s4_0(n265_0)) -> false, rt in Omega(1 + n265_0)
301.54/291.55	
301.54/291.55	
301.54/291.55	Generator Equations:
301.54/291.55	gen_0':s4_0(0) <=> 0'
301.54/291.55	gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
301.54/291.55	
301.54/291.55	
301.54/291.55	The following defined symbols remain to be analysed:
301.54/291.55	cond2, cond1
301.54/291.55	
301.54/291.55	They will be analysed ascendingly in the following order:
301.54/291.55	cond1 = cond2
301.54/291.58	EOF