314.71/291.45	WORST_CASE(Omega(n^1), ?)
314.71/291.46	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
314.71/291.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
314.71/291.46	
314.71/291.46	
314.71/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.71/291.46	
314.71/291.46	(0) CpxTRS
314.71/291.46	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
314.71/291.46	(2) CpxTRS
314.71/291.46	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
314.71/291.46	(4) typed CpxTrs
314.71/291.46	(5) OrderProof [LOWER BOUND(ID), 0 ms]
314.71/291.46	(6) typed CpxTrs
314.71/291.46	(7) RewriteLemmaProof [LOWER BOUND(ID), 267 ms]
314.71/291.46	(8) BEST
314.71/291.46	    (9) proven lower bound
314.71/291.46	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
314.71/291.46	        (11) BOUNDS(n^1, INF)
314.71/291.46	    (12) typed CpxTrs
314.71/291.46	        (13) RewriteLemmaProof [LOWER BOUND(ID), 1336 ms]
314.71/291.46	        (14) typed CpxTrs
314.71/291.46	        (15) RewriteLemmaProof [LOWER BOUND(ID), 2075 ms]
314.71/291.46	        (16) typed CpxTrs
314.71/291.46	        (17) RewriteLemmaProof [LOWER BOUND(ID), 2519 ms]
314.71/291.46	        (18) BOUNDS(1, INF)
314.71/291.46	
314.71/291.46	
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(0)
314.71/291.46	Obligation:
314.71/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.71/291.46	
314.71/291.46	
314.71/291.46	The TRS R consists of the following rules:
314.71/291.46	
314.71/291.46	   minus(x, 0) -> x
314.71/291.46	   minus(s(x), s(y)) -> minus(x, y)
314.71/291.46	   f(0) -> s(0)
314.71/291.46	   f(s(x)) -> minus(s(x), g(f(x)))
314.71/291.46	   g(0) -> 0
314.71/291.46	   g(s(x)) -> minus(s(x), f(g(x)))
314.71/291.46	
314.71/291.46	S is empty.
314.71/291.46	Rewrite Strategy: FULL
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(1) RenamingProof (BOTH BOUNDS(ID, ID))
314.71/291.46	Renamed function symbols to avoid clashes with predefined symbol.
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(2)
314.71/291.46	Obligation:
314.71/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.71/291.46	
314.71/291.46	
314.71/291.46	The TRS R consists of the following rules:
314.71/291.46	
314.71/291.46	   minus(x, 0') -> x
314.71/291.46	   minus(s(x), s(y)) -> minus(x, y)
314.71/291.46	   f(0') -> s(0')
314.71/291.46	   f(s(x)) -> minus(s(x), g(f(x)))
314.71/291.46	   g(0') -> 0'
314.71/291.46	   g(s(x)) -> minus(s(x), f(g(x)))
314.71/291.46	
314.71/291.46	S is empty.
314.71/291.46	Rewrite Strategy: FULL
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
314.71/291.46	Infered types.
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(4)
314.71/291.46	Obligation:
314.71/291.46	TRS:
314.71/291.46	Rules:
314.71/291.46	minus(x, 0') -> x
314.71/291.46	minus(s(x), s(y)) -> minus(x, y)
314.71/291.46	f(0') -> s(0')
314.71/291.46	f(s(x)) -> minus(s(x), g(f(x)))
314.71/291.46	g(0') -> 0'
314.71/291.46	g(s(x)) -> minus(s(x), f(g(x)))
314.71/291.46	
314.71/291.46	Types:
314.71/291.46	minus :: 0':s -> 0':s -> 0':s
314.71/291.46	0' :: 0':s
314.71/291.46	s :: 0':s -> 0':s
314.71/291.46	f :: 0':s -> 0':s
314.71/291.46	g :: 0':s -> 0':s
314.71/291.46	hole_0':s1_0 :: 0':s
314.71/291.46	gen_0':s2_0 :: Nat -> 0':s
314.71/291.46	
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(5) OrderProof (LOWER BOUND(ID))
314.71/291.46	Heuristically decided to analyse the following defined symbols:
314.71/291.46	minus, f, g
314.71/291.46	
314.71/291.46	They will be analysed ascendingly in the following order:
314.71/291.46	minus < f
314.71/291.46	minus < g
314.71/291.46	f = g
314.71/291.46	
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(6)
314.71/291.46	Obligation:
314.71/291.46	TRS:
314.71/291.46	Rules:
314.71/291.46	minus(x, 0') -> x
314.71/291.46	minus(s(x), s(y)) -> minus(x, y)
314.71/291.46	f(0') -> s(0')
314.71/291.46	f(s(x)) -> minus(s(x), g(f(x)))
314.71/291.46	g(0') -> 0'
314.71/291.46	g(s(x)) -> minus(s(x), f(g(x)))
314.71/291.46	
314.71/291.46	Types:
314.71/291.46	minus :: 0':s -> 0':s -> 0':s
314.71/291.46	0' :: 0':s
314.71/291.46	s :: 0':s -> 0':s
314.71/291.46	f :: 0':s -> 0':s
314.71/291.46	g :: 0':s -> 0':s
314.71/291.46	hole_0':s1_0 :: 0':s
314.71/291.46	gen_0':s2_0 :: Nat -> 0':s
314.71/291.46	
314.71/291.46	
314.71/291.46	Generator Equations:
314.71/291.46	gen_0':s2_0(0) <=> 0'
314.71/291.46	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
314.71/291.46	
314.71/291.46	
314.71/291.46	The following defined symbols remain to be analysed:
314.71/291.46	minus, f, g
314.71/291.46	
314.71/291.46	They will be analysed ascendingly in the following order:
314.71/291.46	minus < f
314.71/291.46	minus < g
314.71/291.46	f = g
314.71/291.46	
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(7) RewriteLemmaProof (LOWER BOUND(ID))
314.71/291.46	Proved the following rewrite lemma:
314.71/291.46	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
314.71/291.46	
314.71/291.46	Induction Base:
314.71/291.46	minus(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
314.71/291.46	gen_0':s2_0(0)
314.71/291.46	
314.71/291.46	Induction Step:
314.71/291.46	minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
314.71/291.46	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH
314.71/291.46	gen_0':s2_0(0)
314.71/291.46	
314.71/291.46	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(8)
314.71/291.46	Complex Obligation (BEST)
314.71/291.46	
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(9)
314.71/291.46	Obligation:
314.71/291.46	Proved the lower bound n^1 for the following obligation:
314.71/291.46	
314.71/291.46	TRS:
314.71/291.46	Rules:
314.71/291.46	minus(x, 0') -> x
314.71/291.46	minus(s(x), s(y)) -> minus(x, y)
314.71/291.46	f(0') -> s(0')
314.71/291.46	f(s(x)) -> minus(s(x), g(f(x)))
314.71/291.46	g(0') -> 0'
314.71/291.46	g(s(x)) -> minus(s(x), f(g(x)))
314.71/291.46	
314.71/291.46	Types:
314.71/291.46	minus :: 0':s -> 0':s -> 0':s
314.71/291.46	0' :: 0':s
314.71/291.46	s :: 0':s -> 0':s
314.71/291.46	f :: 0':s -> 0':s
314.71/291.46	g :: 0':s -> 0':s
314.71/291.46	hole_0':s1_0 :: 0':s
314.71/291.46	gen_0':s2_0 :: Nat -> 0':s
314.71/291.46	
314.71/291.46	
314.71/291.46	Generator Equations:
314.71/291.46	gen_0':s2_0(0) <=> 0'
314.71/291.46	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
314.71/291.46	
314.71/291.46	
314.71/291.46	The following defined symbols remain to be analysed:
314.71/291.46	minus, f, g
314.71/291.46	
314.71/291.46	They will be analysed ascendingly in the following order:
314.71/291.46	minus < f
314.71/291.46	minus < g
314.71/291.46	f = g
314.71/291.46	
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(10) LowerBoundPropagationProof (FINISHED)
314.71/291.46	Propagated lower bound.
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(11)
314.71/291.46	BOUNDS(n^1, INF)
314.71/291.46	
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(12)
314.71/291.46	Obligation:
314.71/291.46	TRS:
314.71/291.46	Rules:
314.71/291.46	minus(x, 0') -> x
314.71/291.46	minus(s(x), s(y)) -> minus(x, y)
314.71/291.46	f(0') -> s(0')
314.71/291.46	f(s(x)) -> minus(s(x), g(f(x)))
314.71/291.46	g(0') -> 0'
314.71/291.46	g(s(x)) -> minus(s(x), f(g(x)))
314.71/291.46	
314.71/291.46	Types:
314.71/291.46	minus :: 0':s -> 0':s -> 0':s
314.71/291.46	0' :: 0':s
314.71/291.46	s :: 0':s -> 0':s
314.71/291.46	f :: 0':s -> 0':s
314.71/291.46	g :: 0':s -> 0':s
314.71/291.46	hole_0':s1_0 :: 0':s
314.71/291.46	gen_0':s2_0 :: Nat -> 0':s
314.71/291.46	
314.71/291.46	
314.71/291.46	Lemmas:
314.71/291.46	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
314.71/291.46	
314.71/291.46	
314.71/291.46	Generator Equations:
314.71/291.46	gen_0':s2_0(0) <=> 0'
314.71/291.46	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
314.71/291.46	
314.71/291.46	
314.71/291.46	The following defined symbols remain to be analysed:
314.71/291.46	g, f
314.71/291.46	
314.71/291.46	They will be analysed ascendingly in the following order:
314.71/291.46	f = g
314.71/291.46	
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(13) RewriteLemmaProof (LOWER BOUND(ID))
314.71/291.46	Proved the following rewrite lemma:
314.71/291.46	g(gen_0':s2_0(+(1, n216_0))) -> *3_0, rt in Omega(n216_0)
314.71/291.46	
314.71/291.46	Induction Base:
314.71/291.46	g(gen_0':s2_0(+(1, 0)))
314.71/291.46	
314.71/291.46	Induction Step:
314.71/291.46	g(gen_0':s2_0(+(1, +(n216_0, 1)))) ->_R^Omega(1)
314.71/291.46	minus(s(gen_0':s2_0(+(1, n216_0))), f(g(gen_0':s2_0(+(1, n216_0))))) ->_IH
314.71/291.46	minus(s(gen_0':s2_0(+(1, n216_0))), f(*3_0))
314.71/291.46	
314.71/291.46	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(14)
314.71/291.46	Obligation:
314.71/291.46	TRS:
314.71/291.46	Rules:
314.71/291.46	minus(x, 0') -> x
314.71/291.46	minus(s(x), s(y)) -> minus(x, y)
314.71/291.46	f(0') -> s(0')
314.71/291.46	f(s(x)) -> minus(s(x), g(f(x)))
314.71/291.46	g(0') -> 0'
314.71/291.46	g(s(x)) -> minus(s(x), f(g(x)))
314.71/291.46	
314.71/291.46	Types:
314.71/291.46	minus :: 0':s -> 0':s -> 0':s
314.71/291.46	0' :: 0':s
314.71/291.46	s :: 0':s -> 0':s
314.71/291.46	f :: 0':s -> 0':s
314.71/291.46	g :: 0':s -> 0':s
314.71/291.46	hole_0':s1_0 :: 0':s
314.71/291.46	gen_0':s2_0 :: Nat -> 0':s
314.71/291.46	
314.71/291.46	
314.71/291.46	Lemmas:
314.71/291.46	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
314.71/291.46	g(gen_0':s2_0(+(1, n216_0))) -> *3_0, rt in Omega(n216_0)
314.71/291.46	
314.71/291.46	
314.71/291.46	Generator Equations:
314.71/291.46	gen_0':s2_0(0) <=> 0'
314.71/291.46	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
314.71/291.46	
314.71/291.46	
314.71/291.46	The following defined symbols remain to be analysed:
314.71/291.46	f
314.71/291.46	
314.71/291.46	They will be analysed ascendingly in the following order:
314.71/291.46	f = g
314.71/291.46	
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(15) RewriteLemmaProof (LOWER BOUND(ID))
314.71/291.46	Proved the following rewrite lemma:
314.71/291.46	f(gen_0':s2_0(+(1, n3745_0))) -> *3_0, rt in Omega(n3745_0)
314.71/291.46	
314.71/291.46	Induction Base:
314.71/291.46	f(gen_0':s2_0(+(1, 0)))
314.71/291.46	
314.71/291.46	Induction Step:
314.71/291.46	f(gen_0':s2_0(+(1, +(n3745_0, 1)))) ->_R^Omega(1)
314.71/291.46	minus(s(gen_0':s2_0(+(1, n3745_0))), g(f(gen_0':s2_0(+(1, n3745_0))))) ->_IH
314.71/291.46	minus(s(gen_0':s2_0(+(1, n3745_0))), g(*3_0))
314.71/291.46	
314.71/291.46	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(16)
314.71/291.46	Obligation:
314.71/291.46	TRS:
314.71/291.46	Rules:
314.71/291.46	minus(x, 0') -> x
314.71/291.46	minus(s(x), s(y)) -> minus(x, y)
314.71/291.46	f(0') -> s(0')
314.71/291.46	f(s(x)) -> minus(s(x), g(f(x)))
314.71/291.46	g(0') -> 0'
314.71/291.46	g(s(x)) -> minus(s(x), f(g(x)))
314.71/291.46	
314.71/291.46	Types:
314.71/291.46	minus :: 0':s -> 0':s -> 0':s
314.71/291.46	0' :: 0':s
314.71/291.46	s :: 0':s -> 0':s
314.71/291.46	f :: 0':s -> 0':s
314.71/291.46	g :: 0':s -> 0':s
314.71/291.46	hole_0':s1_0 :: 0':s
314.71/291.46	gen_0':s2_0 :: Nat -> 0':s
314.71/291.46	
314.71/291.46	
314.71/291.46	Lemmas:
314.71/291.46	minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
314.71/291.46	g(gen_0':s2_0(+(1, n216_0))) -> *3_0, rt in Omega(n216_0)
314.71/291.46	f(gen_0':s2_0(+(1, n3745_0))) -> *3_0, rt in Omega(n3745_0)
314.71/291.46	
314.71/291.46	
314.71/291.46	Generator Equations:
314.71/291.46	gen_0':s2_0(0) <=> 0'
314.71/291.46	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
314.71/291.46	
314.71/291.46	
314.71/291.46	The following defined symbols remain to be analysed:
314.71/291.46	g
314.71/291.46	
314.71/291.46	They will be analysed ascendingly in the following order:
314.71/291.46	f = g
314.71/291.46	
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(17) RewriteLemmaProof (LOWER BOUND(ID))
314.71/291.46	Proved the following rewrite lemma:
314.71/291.46	g(gen_0':s2_0(+(1, n63581_0))) -> *3_0, rt in Omega(n63581_0)
314.71/291.46	
314.71/291.46	Induction Base:
314.71/291.46	g(gen_0':s2_0(+(1, 0)))
314.71/291.46	
314.71/291.46	Induction Step:
314.71/291.46	g(gen_0':s2_0(+(1, +(n63581_0, 1)))) ->_R^Omega(1)
314.71/291.46	minus(s(gen_0':s2_0(+(1, n63581_0))), f(g(gen_0':s2_0(+(1, n63581_0))))) ->_IH
314.71/291.46	minus(s(gen_0':s2_0(+(1, n63581_0))), f(*3_0))
314.71/291.46	
314.71/291.46	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
314.71/291.46	----------------------------------------
314.71/291.46	
314.71/291.46	(18)
314.71/291.46	BOUNDS(1, INF)
314.71/291.50	EOF