1040.81/291.47	WORST_CASE(Omega(n^1), ?)
1040.81/291.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1040.81/291.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1040.81/291.48	
1040.81/291.48	(0) CpxTRS
1040.81/291.48	(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1040.81/291.48	(2) CpxTRS
1040.81/291.48	(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1040.81/291.48	(4) typed CpxTrs
1040.81/291.48	(5) OrderProof [LOWER BOUND(ID), 0 ms]
1040.81/291.48	(6) typed CpxTrs
1040.81/291.48	(7) RewriteLemmaProof [LOWER BOUND(ID), 313 ms]
1040.81/291.48	(8) BEST
1040.81/291.48	    (9) proven lower bound
1040.81/291.48	        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
1040.81/291.48	        (11) BOUNDS(n^1, INF)
1040.81/291.48	    (12) typed CpxTrs
1040.81/291.48	        (13) RewriteLemmaProof [LOWER BOUND(ID), 1199 ms]
1040.81/291.48	        (14) typed CpxTrs
1040.81/291.48	        (15) RewriteLemmaProof [LOWER BOUND(ID), 1959 ms]
1040.81/291.48	        (16) typed CpxTrs
1040.81/291.48	        (17) RewriteLemmaProof [LOWER BOUND(ID), 2462 ms]
1040.81/291.48	        (18) BOUNDS(1, INF)
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(0)
1040.81/291.48	Obligation:
1040.81/291.48	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	The TRS R consists of the following rules:
1040.81/291.48	
1040.81/291.48	   -(x, 0) -> x
1040.81/291.48	   -(0, s(y)) -> 0
1040.81/291.48	   -(s(x), s(y)) -> -(x, y)
1040.81/291.48	   f(0) -> 0
1040.81/291.48	   f(s(x)) -> -(s(x), g(f(x)))
1040.81/291.48	   g(0) -> s(0)
1040.81/291.48	   g(s(x)) -> -(s(x), f(g(x)))
1040.81/291.48	
1040.81/291.48	S is empty.
1040.81/291.48	Rewrite Strategy: INNERMOST
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(1) RenamingProof (BOTH BOUNDS(ID, ID))
1040.81/291.48	Renamed function symbols to avoid clashes with predefined symbol.
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(2)
1040.81/291.48	Obligation:
1040.81/291.48	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	The TRS R consists of the following rules:
1040.81/291.48	
1040.81/291.48	   -(x, 0') -> x
1040.81/291.48	   -(0', s(y)) -> 0'
1040.81/291.48	   -(s(x), s(y)) -> -(x, y)
1040.81/291.48	   f(0') -> 0'
1040.81/291.48	   f(s(x)) -> -(s(x), g(f(x)))
1040.81/291.48	   g(0') -> s(0')
1040.81/291.48	   g(s(x)) -> -(s(x), f(g(x)))
1040.81/291.48	
1040.81/291.48	S is empty.
1040.81/291.48	Rewrite Strategy: INNERMOST
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1040.81/291.48	Infered types.
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(4)
1040.81/291.48	Obligation:
1040.81/291.48	Innermost TRS:
1040.81/291.48	Rules:
1040.81/291.48	-(x, 0') -> x
1040.81/291.48	-(0', s(y)) -> 0'
1040.81/291.48	-(s(x), s(y)) -> -(x, y)
1040.81/291.48	f(0') -> 0'
1040.81/291.48	f(s(x)) -> -(s(x), g(f(x)))
1040.81/291.48	g(0') -> s(0')
1040.81/291.48	g(s(x)) -> -(s(x), f(g(x)))
1040.81/291.48	
1040.81/291.48	Types:
1040.81/291.48	- :: 0':s -> 0':s -> 0':s
1040.81/291.48	0' :: 0':s
1040.81/291.48	s :: 0':s -> 0':s
1040.81/291.48	f :: 0':s -> 0':s
1040.81/291.48	g :: 0':s -> 0':s
1040.81/291.48	hole_0':s1_0 :: 0':s
1040.81/291.48	gen_0':s2_0 :: Nat -> 0':s
1040.81/291.48	
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(5) OrderProof (LOWER BOUND(ID))
1040.81/291.48	Heuristically decided to analyse the following defined symbols:
1040.81/291.48	-, f, g
1040.81/291.48	
1040.81/291.48	They will be analysed ascendingly in the following order:
1040.81/291.48	- < f
1040.81/291.48	- < g
1040.81/291.48	f = g
1040.81/291.48	
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(6)
1040.81/291.48	Obligation:
1040.81/291.48	Innermost TRS:
1040.81/291.48	Rules:
1040.81/291.48	-(x, 0') -> x
1040.81/291.48	-(0', s(y)) -> 0'
1040.81/291.48	-(s(x), s(y)) -> -(x, y)
1040.81/291.48	f(0') -> 0'
1040.81/291.48	f(s(x)) -> -(s(x), g(f(x)))
1040.81/291.48	g(0') -> s(0')
1040.81/291.48	g(s(x)) -> -(s(x), f(g(x)))
1040.81/291.48	
1040.81/291.48	Types:
1040.81/291.48	- :: 0':s -> 0':s -> 0':s
1040.81/291.48	0' :: 0':s
1040.81/291.48	s :: 0':s -> 0':s
1040.81/291.48	f :: 0':s -> 0':s
1040.81/291.48	g :: 0':s -> 0':s
1040.81/291.48	hole_0':s1_0 :: 0':s
1040.81/291.48	gen_0':s2_0 :: Nat -> 0':s
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	Generator Equations:
1040.81/291.48	gen_0':s2_0(0) <=> 0'
1040.81/291.48	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	The following defined symbols remain to be analysed:
1040.81/291.48	-, f, g
1040.81/291.48	
1040.81/291.48	They will be analysed ascendingly in the following order:
1040.81/291.48	- < f
1040.81/291.48	- < g
1040.81/291.48	f = g
1040.81/291.48	
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(7) RewriteLemmaProof (LOWER BOUND(ID))
1040.81/291.48	Proved the following rewrite lemma:
1040.81/291.48	-(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
1040.81/291.48	
1040.81/291.48	Induction Base:
1040.81/291.48	-(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
1040.81/291.48	gen_0':s2_0(0)
1040.81/291.48	
1040.81/291.48	Induction Step:
1040.81/291.48	-(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
1040.81/291.48	-(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) ->_IH
1040.81/291.48	gen_0':s2_0(0)
1040.81/291.48	
1040.81/291.48	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(8)
1040.81/291.48	Complex Obligation (BEST)
1040.81/291.48	
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(9)
1040.81/291.48	Obligation:
1040.81/291.48	Proved the lower bound n^1 for the following obligation:
1040.81/291.48	
1040.81/291.48	Innermost TRS:
1040.81/291.48	Rules:
1040.81/291.48	-(x, 0') -> x
1040.81/291.48	-(0', s(y)) -> 0'
1040.81/291.48	-(s(x), s(y)) -> -(x, y)
1040.81/291.48	f(0') -> 0'
1040.81/291.48	f(s(x)) -> -(s(x), g(f(x)))
1040.81/291.48	g(0') -> s(0')
1040.81/291.48	g(s(x)) -> -(s(x), f(g(x)))
1040.81/291.48	
1040.81/291.48	Types:
1040.81/291.48	- :: 0':s -> 0':s -> 0':s
1040.81/291.48	0' :: 0':s
1040.81/291.48	s :: 0':s -> 0':s
1040.81/291.48	f :: 0':s -> 0':s
1040.81/291.48	g :: 0':s -> 0':s
1040.81/291.48	hole_0':s1_0 :: 0':s
1040.81/291.48	gen_0':s2_0 :: Nat -> 0':s
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	Generator Equations:
1040.81/291.48	gen_0':s2_0(0) <=> 0'
1040.81/291.48	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	The following defined symbols remain to be analysed:
1040.81/291.48	-, f, g
1040.81/291.48	
1040.81/291.48	They will be analysed ascendingly in the following order:
1040.81/291.48	- < f
1040.81/291.48	- < g
1040.81/291.48	f = g
1040.81/291.48	
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(10) LowerBoundPropagationProof (FINISHED)
1040.81/291.48	Propagated lower bound.
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(11)
1040.81/291.48	BOUNDS(n^1, INF)
1040.81/291.48	
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(12)
1040.81/291.48	Obligation:
1040.81/291.48	Innermost TRS:
1040.81/291.48	Rules:
1040.81/291.48	-(x, 0') -> x
1040.81/291.48	-(0', s(y)) -> 0'
1040.81/291.48	-(s(x), s(y)) -> -(x, y)
1040.81/291.48	f(0') -> 0'
1040.81/291.48	f(s(x)) -> -(s(x), g(f(x)))
1040.81/291.48	g(0') -> s(0')
1040.81/291.48	g(s(x)) -> -(s(x), f(g(x)))
1040.81/291.48	
1040.81/291.48	Types:
1040.81/291.48	- :: 0':s -> 0':s -> 0':s
1040.81/291.48	0' :: 0':s
1040.81/291.48	s :: 0':s -> 0':s
1040.81/291.48	f :: 0':s -> 0':s
1040.81/291.48	g :: 0':s -> 0':s
1040.81/291.48	hole_0':s1_0 :: 0':s
1040.81/291.48	gen_0':s2_0 :: Nat -> 0':s
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	Lemmas:
1040.81/291.48	-(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	Generator Equations:
1040.81/291.48	gen_0':s2_0(0) <=> 0'
1040.81/291.48	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	The following defined symbols remain to be analysed:
1040.81/291.48	g, f
1040.81/291.48	
1040.81/291.48	They will be analysed ascendingly in the following order:
1040.81/291.48	f = g
1040.81/291.48	
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(13) RewriteLemmaProof (LOWER BOUND(ID))
1040.81/291.48	Proved the following rewrite lemma:
1040.81/291.48	g(gen_0':s2_0(+(1, n391_0))) -> *3_0, rt in Omega(n391_0)
1040.81/291.48	
1040.81/291.48	Induction Base:
1040.81/291.48	g(gen_0':s2_0(+(1, 0)))
1040.81/291.48	
1040.81/291.48	Induction Step:
1040.81/291.48	g(gen_0':s2_0(+(1, +(n391_0, 1)))) ->_R^Omega(1)
1040.81/291.48	-(s(gen_0':s2_0(+(1, n391_0))), f(g(gen_0':s2_0(+(1, n391_0))))) ->_IH
1040.81/291.48	-(s(gen_0':s2_0(+(1, n391_0))), f(*3_0))
1040.81/291.48	
1040.81/291.48	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(14)
1040.81/291.48	Obligation:
1040.81/291.48	Innermost TRS:
1040.81/291.48	Rules:
1040.81/291.48	-(x, 0') -> x
1040.81/291.48	-(0', s(y)) -> 0'
1040.81/291.48	-(s(x), s(y)) -> -(x, y)
1040.81/291.48	f(0') -> 0'
1040.81/291.48	f(s(x)) -> -(s(x), g(f(x)))
1040.81/291.48	g(0') -> s(0')
1040.81/291.48	g(s(x)) -> -(s(x), f(g(x)))
1040.81/291.48	
1040.81/291.48	Types:
1040.81/291.48	- :: 0':s -> 0':s -> 0':s
1040.81/291.48	0' :: 0':s
1040.81/291.48	s :: 0':s -> 0':s
1040.81/291.48	f :: 0':s -> 0':s
1040.81/291.48	g :: 0':s -> 0':s
1040.81/291.48	hole_0':s1_0 :: 0':s
1040.81/291.48	gen_0':s2_0 :: Nat -> 0':s
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	Lemmas:
1040.81/291.48	-(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
1040.81/291.48	g(gen_0':s2_0(+(1, n391_0))) -> *3_0, rt in Omega(n391_0)
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	Generator Equations:
1040.81/291.48	gen_0':s2_0(0) <=> 0'
1040.81/291.48	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	The following defined symbols remain to be analysed:
1040.81/291.48	f
1040.81/291.48	
1040.81/291.48	They will be analysed ascendingly in the following order:
1040.81/291.48	f = g
1040.81/291.48	
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(15) RewriteLemmaProof (LOWER BOUND(ID))
1040.81/291.48	Proved the following rewrite lemma:
1040.81/291.48	f(gen_0':s2_0(+(1, n4222_0))) -> *3_0, rt in Omega(n4222_0)
1040.81/291.48	
1040.81/291.48	Induction Base:
1040.81/291.48	f(gen_0':s2_0(+(1, 0)))
1040.81/291.48	
1040.81/291.48	Induction Step:
1040.81/291.48	f(gen_0':s2_0(+(1, +(n4222_0, 1)))) ->_R^Omega(1)
1040.81/291.48	-(s(gen_0':s2_0(+(1, n4222_0))), g(f(gen_0':s2_0(+(1, n4222_0))))) ->_IH
1040.81/291.48	-(s(gen_0':s2_0(+(1, n4222_0))), g(*3_0))
1040.81/291.48	
1040.81/291.48	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(16)
1040.81/291.48	Obligation:
1040.81/291.48	Innermost TRS:
1040.81/291.48	Rules:
1040.81/291.48	-(x, 0') -> x
1040.81/291.48	-(0', s(y)) -> 0'
1040.81/291.48	-(s(x), s(y)) -> -(x, y)
1040.81/291.48	f(0') -> 0'
1040.81/291.48	f(s(x)) -> -(s(x), g(f(x)))
1040.81/291.48	g(0') -> s(0')
1040.81/291.48	g(s(x)) -> -(s(x), f(g(x)))
1040.81/291.48	
1040.81/291.48	Types:
1040.81/291.48	- :: 0':s -> 0':s -> 0':s
1040.81/291.48	0' :: 0':s
1040.81/291.48	s :: 0':s -> 0':s
1040.81/291.48	f :: 0':s -> 0':s
1040.81/291.48	g :: 0':s -> 0':s
1040.81/291.48	hole_0':s1_0 :: 0':s
1040.81/291.48	gen_0':s2_0 :: Nat -> 0':s
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	Lemmas:
1040.81/291.48	-(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(0), rt in Omega(1 + n4_0)
1040.81/291.48	g(gen_0':s2_0(+(1, n391_0))) -> *3_0, rt in Omega(n391_0)
1040.81/291.48	f(gen_0':s2_0(+(1, n4222_0))) -> *3_0, rt in Omega(n4222_0)
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	Generator Equations:
1040.81/291.48	gen_0':s2_0(0) <=> 0'
1040.81/291.48	gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))
1040.81/291.48	
1040.81/291.48	
1040.81/291.48	The following defined symbols remain to be analysed:
1040.81/291.48	g
1040.81/291.48	
1040.81/291.48	They will be analysed ascendingly in the following order:
1040.81/291.48	f = g
1040.81/291.48	
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(17) RewriteLemmaProof (LOWER BOUND(ID))
1040.81/291.48	Proved the following rewrite lemma:
1040.81/291.48	g(gen_0':s2_0(+(1, n62853_0))) -> *3_0, rt in Omega(n62853_0)
1040.81/291.48	
1040.81/291.48	Induction Base:
1040.81/291.48	g(gen_0':s2_0(+(1, 0)))
1040.81/291.48	
1040.81/291.48	Induction Step:
1040.81/291.48	g(gen_0':s2_0(+(1, +(n62853_0, 1)))) ->_R^Omega(1)
1040.81/291.48	-(s(gen_0':s2_0(+(1, n62853_0))), f(g(gen_0':s2_0(+(1, n62853_0))))) ->_IH
1040.81/291.48	-(s(gen_0':s2_0(+(1, n62853_0))), f(*3_0))
1040.81/291.48	
1040.81/291.48	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1040.81/291.48	----------------------------------------
1040.81/291.48	
1040.81/291.48	(18)
1040.81/291.48	BOUNDS(1, INF)
1040.96/291.54	EOF